Celebratio Mathematica — Yor — Complete Bibliography

[1] M. Yor: “Sur les intégrales stochastiques à valeurs dans un espace de Banach” [On stochastic integrals with values in a Banach space], C. R. Acad. Sci., Paris, Sér. A 277 (1973), pp. 467–469. A longer article with the same title was published in Ann. Inst. Henri Poincaré 10:1 (1974). MR 41805237 Zbl 0267.60064 article

[2] M. Yor: “Existence et unicité de diffusions à valeurs dans un espace de Hilbert” [The existence and unicity of value diffusions in a Hilbert space], Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 10 : 1 (1974), pp. 55–88. MR 356257 Zbl 0281.60094 article

[3] M. Yor: “Sur les intégrales stochastiques à valeurs dans un espace de Banach” [On stochastic integrals with values in a Banach space], Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 10 : 1 (1974), pp. 31–36. A shorter article with the same title was published in C. R. Acad. Sci., Paris 277 (1973). MR 358986 Zbl 0295.60041 article

[4]M. Yor: “Étude de mesures de probabilité sur \( C(\mathbb{R}^*_+; \mathbb{R}) \) quasi invariantes sous les translations de \( \mathcal{D}(\mathbb{R}^*_+;\mathbb{R}) \)” [Study of probability measures on \( C(\mathbb{R}^*_+;\mathbb{R}) \) quasi-invariant under translations of \( \mathcal{D}(\mathbb{R}^*_+;\mathbb{R}) \)], Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 11 : 2 (1975), pp. 127–171. MR 380965 Zbl 0383.60072 article

@article {key380965m,
	AUTHOR = {Yor, Marc},
	TITLE = {\'{E}tude de mesures de probabilit\'e
		 sur \$C(\mathbb{R}^*_+\$; \$\mathbb{R}\$
		 quasi invariantes sous les translations
		 de \$\mathcal{D}(\mathbb{R}^*_+\$; \$\mathbb{R}\$
		 [Study of probability measures on \$C(\mathbb{R}^*_+\$;
		 \$\mathbb{R}\$ quasi-invariant under translations
		 of \$\mathcal{D}(\mathbb{R}^*_+\$; \$\mathbb{R}\$]},
	JOURNAL = {Ann. Inst. Henri Poincar\'e, Nouv. S\'er.,
		 Sect. B},
	FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
		 Nouvelle S\'erie. Section B. Calcul
		 des Probabilit\'es et Statistique},
	VOLUME = {11},
	NUMBER = {2},
	YEAR = {1975},
	PAGES = {127--171},
	URL = {http://www.numdam.org/item?id=AIHPB_1975__11_2_127_0},
	NOTE = {MR:380965. Zbl:0383.60072.},
	ISSN = {0020-2347},
}

[5] M. Yor: “Représentation des martingales de carré intégrable rélatives aux processus de Wiener et de Poisson à \( n \) paramètres” [Representation of square integrable martingales relativeto Wiener and Poisson processes with \( n \) parameters], C. R. Acad. Sci., Paris, Sér. A 281 : 2–3 (1975), pp. 111–113. A longer article with the same title was published in Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 35:2 (1976). MR 380987 Zbl 0332.60030 article

[6] M. Yor: “Décomposition des tribus \( \mathfrak{F}_{(T+t)^+}^0 \) d’un processus continu à droite” [Family decomposition \( \mathfrak{F}_{(T+t)^+}^0 \) of a right-continuous process], C. R. Acad. Sci., Paris, Sér. A 281 : 12 (1975), pp. 467–470. MR 408003 Zbl 0438.60036 article

[7] P. Priouret and M. Yor: “Processus de diffusion a valeurs dans \( \mathbb{R} \) et mesures quasi-invariantes sur \( C(\mathbb{R} \), \( \mathbb{R}) \)” [Diffusion processes valued in \( \mathbb{R} \) and quasi-invariant measures on \( C(\mathbb{R} \), \( \mathbb{R}) \)], pp. 247–290 in Oscillateur anharmonique, processus de diffusion et mesures quasi-invariantes [Anharmonic oscillator, diffusion processes and quasi-invariant measures]. Edited by P. Courrège. Astérisque 22–23. Société Mathématique de France (Paris), 1975. MR 496179 Zbl 0316.60051 incollection

[8] M. Yor: “Formule de Cauchy rélative à certains lacets browniens” [Cauchy’s formula relative to certain Brownian laces], C. R. Acad. Sci., Paris, Sér. A 281 (1975), pp. 867–870. A longer article with the same title was published in Bull. Soc. Math. France 105:1 (1977). MR 386011 Zbl 0335.60051 article

[9] J. Jacod and M. Yor: “Étude des solutions extremales et représentation intégrale des solutions pour certains problèmes de martingales” [A study of extremal solutions and integral representation of solutions for certain martingale problems], C. R. Acad. Sci., Paris, Sér. A 283 (1976), pp. 523–525. A longer piece with the same title was later published in Z. Wahrscheinlichkeitstheor. Verw. Geb. 38:2 (1977). MR 418224 Zbl 0339.60027 article

[10] M. Yor: “Représentation intégrale des martingales de carré intégrable” [Integral representation of square-integrable martingales], C. R. Acad. Sci., Paris, Sér. A 282 : 16 (1976), pp. 899–901. MR 418229 Zbl 0329.60027 article

[11] M. Yor and P. A. Meyer: “Sur la théorie de la prédiction, et le problème de décomposition des tribus \( \mathfrak{F}^{\circ}_{t^+} \)” [On the theory of prediction, and the problem of family decomposition \( \mathfrak{F}^{\circ}_{t^+} \)], pp. 104–117 in Séminaire de probabilités X [Tenth probability seminar]. Edited by P. A. Meyer. Lecture Notes in Mathematics 511. Springer (Berlin), 1976. MR 438462 Zbl 0332.60025 incollection

[12] M. Yor: “Représentation des martingales de carré intégrable relative aux processus de Wiener et de Poisson à \( n \) paramètres” [Representation of martingales of square integrable relative to Wiener processes and Poisson of \( n \) parameters], Z. Wahrscheinlichkeitstheor. Verw. Geb. 35 : 2 (June 1976), pp. 121–129. A shorter article with the same title was published in C. R. Acad. Sci., Paris 281:2–3 (1975). MR 438470 Zbl 0315.60027 article

[13] M. Yor: “Sur les intégrales stochastiques optionnelles et une suite remarquable de formules exponentielles” [On optional stochastic integrals and a remarkable series of exponential formulas], pp. 481–500 in Séminaire de probabilités X [Tenth probability seminar]. Edited by P. A. Meyer. Lecture Notes in Mathematics 511. Springer (Berlin), 1976. MR 440699 Zbl 0393.60057 incollection

This paper contains several useful results on optional stochastic integrals of local martingales and semimartingales, as well as the first occurence of the well-known formula \[ \mathcal{E}(X)\,\mathcal{E}(Y)=\mathcal{E}(X+Y+[X,Y]) \] where \( \mathcal{E} \) denotes the usual exponential of semimartingales. Also, the s.d.e. \[ Z_t=1+\int_0^t Z_s\,dX_s \] is solved, where \( X \) is a suitable semimartingale, and the integral is an optional one. The Lévy measure of a local martingale is studied, and used to rewrite the Ito formula in a form that involves optional integrals. Finally, a whole family of “exponentials” is introduced, interpolating between the standard one and an exponential involving the Lévy measure, which was used by Kunita–Watanabe in a Markovian set-up.

[14] G. Royer and M. Yor: “Représentation intégrale de certaines mesures quasi-invariantes sur \( \mathcal{C}(\mathbb{R}) \): Mesures extrémales et propriété de Markov” [Integral representation of certain quasi-invariant measures on \( \mathcal{C}(\mathbb{R}) \): Extremal measures and the Markov property], Ann. Inst. Fourier 26 : 2 (1976), pp. 7–24. MR 447517 Zbl 0295.28025 article

[15] M. Yor: “Une remarque sur les formes de Dirichlet et les semi-martingales” [A remark on Dirichlet forms and semi-martingales], pp. 283–292 in Séminaire de théorie du potentiel 2 [Potential theory seminar 2] (Paris, 1975–1976). Edited by F. Hirsch and G. Mokobodzki. Lecture Notes in Mathematics 563. Springer (Berlin), 1976. MR 651572 Zbl 0339.31013 incollection

[16] M. Yor: Calcul stochastique et representations integrales [Stochastic calculus and integral representations]. Ph.D. thesis, Université Pierre-et-Marie-Curie (Paris), June 1976. Advised by P. Priouret. phdthesis

[17] J. Jacod and M. Yor: “Étude des solutions extrémales et représentation intégrale des solutions pour certains problèmes de martingales” [A study of extremal solution and integral representation of solutions for certain martingale problems], Z. Wahrscheinlichkeitstheor. Verw. Geb. 38 : 2 (June 1977), pp. 83–125. A brief piece with the same title was earlier published in C. R. Acad. Sci., Paris, Sér. A 283 (1976). MR 445604 Zbl 0346.60032 article

[18] N. D. Ngoc and M. Yor: “Mesures de Gibbs sur \( \mathbb{R} \)” [Gibbs measures on \( \mathbb{R} \)], C. R. Acad. Sci., Paris, Sér. A 285 : 5 (1977), pp. 395–397. MR 445648 Zbl 0389.60001 article

[19] M. Yor: “Sur quelques approximations d’intégrales stochastiques” [On some approximations of stochastic integrals], pp. 257–297 in Séminaire de probabilités XI [Eleventh probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 581. Springer (Berlin), 1977. MR 448556 Zbl 0367.60058 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[20] M. Yor: “À propos d’un lemme de Ch. Yoeurp” [On a lemma of Ch. Yoeurp], pp. 493–501 in Séminaire de probabilités XI [Eleventh probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 581. Springer (Berlin), 1977. MR 451396 Zbl 0373.60047 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[21] M. Yor: “Étude de certains processus (stochastiquement) différentiables ou holomorphes” [Study of certain (stochastically) differentiable or holomorphic processes], Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 13 : 1 (1977), pp. 1–25. MR 455127 Zbl 0359.60022 article

[22] M. Yor: “Remarques sur la représentation des martingales comme intégrales stochastiques” [Remarks on the the representation of martingales as stochastic integrals], pp. 502–517 in Séminaire de probabilités XI [Eleventh probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 581. Springer (Berlin), 1977. MR 458580 Zbl 0367.60046 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[23] M. Yor: “Sur les théories du filtrage et de la prédiction” [On the theories of filtration and prediction], pp. 257–297 in Séminaire de probabilités XI [Eleventh probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 581. Springer (Berlin), 1977. MR 471060 Zbl 0367.60041 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[24] M. Yor: “Formule de Cauchy rélative à certains lacets browniens” [Cauchy’s formula relative to certain Brownian laces], Bull. Soc. Math. France 105 : 1 (1977), pp. 3–31. A short article with the same title was published in C. R. Acad. Sci., Paris, Sér. A 281 (1975). MR 494477 Zbl 0375.60068 article

[25] M. Yor: “Distributions de Frobénius (d’après S. Lang et H. Trotter)” [Frobenius distributions (after S. Lang and H. Trotter] in Séminaire Delange–Pisot–Poitou, 17e année (1975/76): Théorie des nombres [Delange–Pisot–Poitou seminar, 17th year (1975/76): Number theory] (Paris, 1975–1976), fascicule 2: Exposés 23 à 31 et groupe d’étude. Université Pierre-et-Marie-Curie and Institut Henri Poincaré (Paris), 1977. 10 pp. MR 562501 Zbl 0364.12013 incollection

[26] C. Yoeurp and M. Yor: Espace orthogonal à une semi-martingale et applications. Prépublication, Laboratoire de Probabilités, Université Paris VI, 1977. techreport

[27] M. Yor: “Convergence de martingales dans \( L^1 \) et dans \( H^1 \)” [Convergence of martingales on \( L^1 \) and on \( H^1 \)], C. R. Acad. Sci., Paris, Sér. A 286 : 12 (1978), pp. 571–573. MR 489806 Zbl 0382.60049 article

[28] M. Yor: “Inégalités entre processus minces et applications” [Inequalities between thin processes and applications], C. R. Acad. Sci., Paris, Sér. A 286 : 18 (1978), pp. 799–801. MR 497669 Zbl 0389.60039 article

[29] N. Dang Ngoc and M. Yor: “Champs markoviens et mesures de Gibbs sur \( \mathbb{R} \)” [Markovian fields and Gibbs measures on \( \mathbb{R} \)], Ann. Sci. Éc. Norm. Supér. (4) 11 : 1 (1978), pp. 29–69. MR 504421 Zbl 0391.60013 article

We study the Gibbs probability measures attached to a Markov process, using methods of statistical mechanics. Such a study has already been undertaken, by F. Spitzer and H. Kesten, for Markov chains with discrete state space. Our purpose is to extend their results to general Markov processes.

Some theorems obtained by Ph. Courrège, P. Renouard and G. Royer, on quasi-invariant measures on \( C(\mathbb{R},\mathbb{R}) \) associated to one dimensional quantum fields are also generalized. In particular, we solve some problems raised by Ph. Courrège and P. Renouard on the connection between Markov processes and Markov fields in the sense of E. Nelson.

We show the uniqueness of Gibbs measures in the compact state space case (generalizing a result of R. L. Dobrushin), the uniqueness of invariant Gibbs measures with Markov property for positive recurrent semi-groups (generalizing results of F. Spitzer, H. Kesten and G. Royer) and the absence of Gibbs states for convolution semi-groups (generalizing a result of F. Spitzer).

[30] Temps locaux [Local times] (Paris, 1976–1977). Edited by J. Azéma and M. Yor. Astérisque 52–53. Société Mathématique de France (Paris), 1978. MR 509476 Zbl 0385.60063 book

[31] C. Stricker and M. Yor: “Calcul stochastique dépendant d’un paramètre” [Stochastic calculus dependent on a parameter], Z. Wahrscheinlichkeitstheor. Verw. Geb. 45 : 2 (June 1978), pp. 109–133. MR 510530 Zbl 0388.60056 article

[32] P. Brémaud and M. Yor: “Changes of filtrations and of probability measures,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 45 : 4 (December 1978), pp. 269–295. MR 511775 Zbl 0415.60048 article

[33] M. Yor: “Remarques sur les normes \( H^p \) de (semi-)martingales” [Remarks on the \( H^p \) norms of (semi-)martingales], C. R. Acad. Sci., Paris, Sér. A 287 : 6 (1978), pp. 461–464. MR 517928 Zbl 0387.60050 article

[34] M. Yor: “Grossissement d’une filtration et semi-martingales: Théorèmes généraux” [Enlargement of a filtration and semi-martingales: General theorems], pp. 61–69 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 519996 Zbl 0411.60044 incollection

Given a filtration \( (\mathcal{F}_t) \) and a positive random variable \( L \), the so-called progressively enlarged filtration is the smallest one \( (\mathcal{G}_t) \) containing \( (\mathcal{F}_t) \), and for which \( L \) is a stopping time. The enlargement problem consists in describing the semimartingales \( X \) of \( \mathcal{F} \) which remain semimartingales in \( \mathcal{G} \), and in computing their semimartingale characteristics. In this paper, it is proved that \( X_t\,I_{\{t\lt L\}} \) is a semimartingale in full generality, and that \( X_t\,I_{\{t\geq L\}} \) is a semimartingale whenever \( L \) is honest for \( \mathcal{F} \), i.e., is the end of an \( \mathcal{F} \)-optional set.

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[35] T. Jeulin and M. Yor: “Grossissement d’une filtration et semi-martingales: Formules explicites” [Enlargement of a filtration and semi-martingales: Explicit formulas], pp. 78–97 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 519998 Zbl 0411.60045 incollection

Thierry Jeulin	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[36] C. Dellacherie, P.-A. Meyer, and M. Yor: “Sur certaines propriétés des espaces de Banach \( H^1 \) et BMO” [On certain properties of \( H^1 \) Banach and BMO spaces], pp. 98–113 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 519999 Zbl 0392.60009 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[37] M. Yor: “Sous-espaces denses dans \( L^1 \) ou \( H^1 \) et représentation des martingales” [Dense subspaces in \( L^1 \) or \( H^1 \) and representation of martingales], pp. 265–309 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. With an appendix by the author and J. de Sam Lazaro. MR 520008 Zbl 0391.60046 incollection

This paper was a considerable step in the study of the general martingale problem, i.e., of the set \( \mathcal{P} \) of all laws on a filtered measurable space under which a given set \( \mathcal{N} \) of (adapted, right continuous) processes are local martingales. The starting point is a theorem from measure theory due to R. G. Douglas (Michigan Math. J. 11, 1964), and the main technical difference with preceding papers is the systematic use of stochastic integration in \( H^1 \). The main result can be stated as follows: given a law \( \mathbb{P}\in\mathcal{P} \), the set \( \mathcal{N} \) has the previsible representation property, i.e., \( \mathcal{F}_0 \) is trivial and stochastic integrals with respect to elements of \( \mathcal{N} \) are dense in \( H^1 \), if and only if \( \mathbb{P} \) is an extreme point of \( \mathcal{P} \). Many examples and applications are given.

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[38] M. Yor and P.-A. Meyer: “Sur l’extension d’un théorème de Doob à un noyau \( \sigma \)-fini, d’après G. Mokobodzki” [On the extension of a theorem of Doob to a \( \sigma \)-finite kernel, after G. Mokobodzki], pp. 482–488 in Séminaire de probabilités XII [Twelfth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 649. Springer (Berlin), 1978. MR 520022 Zbl 0391.60047 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[39] T. Jeulin and M. Yor: “Nouveaux résultats sur le grossissement des tribus” [New results on the enlargement of families], Ann. Sci. Éc. Norm. Supér. (4) 11 : 3 (1978), pp. 429–443. MR 521639 Zbl 0414.60054 article

[40] M. Yor: “Quelques résultats sur certaines mesures extrémales: Applications à la représentation des martingales” [Some results on certain extremal measures: Applications to the representation of martingales], pp. 27–36 in Measure theory applications to stochastic analysis (Oberwolfach, Germany, 3–9 July 1977). Edited by G. Kallianpur and D. Kölzow. Lecture Notes in Mathematics 695. Springer (Berlin), 1978. MR 527071 Zbl 0401.60046 incollection

Gopinath Kallianpur	Related
Dietrich Kölzow	Related

[41] M. Yor: “Sur la continuité des temps locaux associés à certaines semi-martingales” [On the continuity of local time associated with certain semi-martingales], pp. 23–26 in Temps locaux [Local times] (Paris, 1976–1977). Edited by J. Azéma and M. Yor. Astérisque 52–53. Société Mathématique de France (Paris), 1978. With an English summary. incollection

[42] M. Yor: “Sur l’étude des martingales continues extrêmales” [On the study of continuous extremal martingales], Stochastics 2 : 3 (1979), pp. 191–196. MR 528910 Zbl 0409.60043 article

[43] M. Yor: “Quelques intéractions entre mesures vectorielles et intégrales stochastiques” [Some interactions between vectorial measures and stochastic integrals], pp. 264–281 in Séminaire de théorie du potentiel, no. 4 [Potential theory seminar, no. 4] (Paris, 1977 — 1978). Edited by F. Hirsch and G. Mokobodzki. Lecture Notes in Mathematics 713. Springer (Berlin), 1979. MR 543655 Zbl 0444.60038 incollection

[44] J. Azéma and M. Yor: “Une solution simple au problème de Skorokhod” [A simple solution to a problem of Skorokhod], pp. 90–115 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544782 Zbl 0414.60055 incollection

Jacques Azéma	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[45] T. Jeulin and M. Yor: “Inégalité de Hardy, semimartingales, et faux-amis” [Hardy’s inequality, semimartingales and false friends], pp. 332–359 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544805 Zbl 0419.60049 incollection

The main purpose of this paper is to warn against “obvious” statements which are in fact false. Let \( (\mathcal{G}_t) \) be an enlargement of \( (\mathcal{F}_t) \). Assume that \( \mathcal{F} \) has the previsible representation property with respect to a martingale \( X \) which is a \( \mathcal{G} \)-semimartingale. Then it does not follow that every \( \mathcal{F} \)-martingale \( Y \) is a \( \mathcal{G} \)-semimartingale. Also, even if \( Y \) is a \( \mathcal{G} \)-semimartingale, its \( \mathcal{G} \)-compensator may have bad absolute continuity properties. The counterexample to the first statement involves a detailed study of the initial enlargement of the filtration of Brownian motion \( (B_t)_{t\leq 1} \) by the random variable \( B_1 \), which transforms it into the Brownian bridge, a semimartingale. Then the stochastic integrals with respect to \( B \) which are \( \mathcal{G} \)-semimartingales are completely described, and this is the place where the classical Hardy inequality appears.

Thierry Jeulin	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[46] T. Jeulin and M. Yor: “Sur l’expression de la dualité entre \( H^1 \) et \( BMO \)” [On the expression of the duality between \( H^1 \) and \( BMO \)], pp. 360–370 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544806 Zbl 0426.60046 incollection

Thierry Jeulin	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[47] M. Yor: “Quelques épilogues” [Some conclusions], pp. 400–406 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544810 Zbl 0427.60040 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[48] M. Yor: “In search of a natural definition of optional stochastic integrals,” pp. 407–426 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544811 Zbl 0439.60041 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[49] M. Yor: “Les filtrations de certaines martingales du mouvement brownien dans \( \mathbb{R}^n \)” [The filtrations of certain martingales of Brownian motion on \( \mathbb{R}^n \)], pp. 427–440 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544812 Zbl 0418.60057 incollection

The problem is to study the filtration generated by real valued stochastic integrals \[ Y=\int_0^t(AX_s, dX_s) ,\] where \( X \) is a \( n \)-dimensional Brownian motion, \( A \) is a \( n{\times}n \)-matrix, and \( (\,\cdot,\cdot\,) \) is the scalar product. If \( A \) is the identity matrix we thus get (squares of) Bessel processes. If \( A \) is symmetric, we can reduce it to diagonal form, and the filtration is generated by a Brownian motion, the dimension of which is the number of different non-zero eigenvalues of \( A \). In particular, this dimension is 1 if and only if the matrix is equivalent to \( cI_r \), a diagonal with \( r \) ones and \( n{-}r \) zeros. This is also (even if the symmetry assumption is omitted) the only case where \( Y \) has the previsible representation property.

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[50] M. Yor: “Sur le balayage des semi-martingales continues” [On the balayage of continuous semi-martingales], pp. 453–471 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544815 Zbl 0409.60042 incollection

This paper is independent from the preceding one [El Karoui 1979], and some overlap occurs. The balayage formula is extended to processes \( Z \) which are not locally bounded, and the local time of the semimartingale \( Y \) is computed. The class of continuous semimartingales \( X \) with canonical decomposition \( X=M+V \) such that \( dV \) is carried by \[ H=\{X=0\} \] is introduced and studied. It turns out to be an important class, closely related to “relative martingales” [Azéma et al. 1992]. A number of results are given, too technical to be stated here. Stopping previsible, optional and progressive processes at the last exit time \( L \) from \( H \) leads to three \( \sigma \)-fields, \( \mathcal{F}_L^p \), \( \mathcal{F}_L^o \), \( \mathcal{F}_L^{\pi} \), and it was considered surprising that the last two could be different (see [Dellacherie 1978]). Here it is shown that if \( X \) is a continuous uniformly integrable martingale with \( X_0=0 \), \[ \mathbb{E}[X_{\infty}|\mathcal{F}_L^o]=0\neq \mathbb{E}[X_{\infty}|\mathcal{F}_L^{\pi}] .\]

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[51] P.-A. Meyer, C. Stricker, and M. Yor: “Sur une formule de la théorie du balayage” [On a formula from balayage theory], pp. 478–487 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544817 Zbl 0415.60031 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[52] M. Yor: “Sur le supremum du mouvement brownien: Les théorèmes de P. Lévy et J. Pitman” [On the supremum of Brownian motion: The theorems of P. Lévy and J. Pitman], pp. 528–532 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. This is an appendix of T. Jeunlin’s article “Un théorème de J.W. Pitman”. MR 544821 Zbl 0422.60028 incollection

Thierry Jeulin	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

@incollection {key544821m,
	AUTHOR = {Yor, Marc},
	TITLE = {Sur le supremum du mouvement brownien:
		 {L}es th\'eor\`emes de {P}. {L}\'evy
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		 {B}rownian motion: {T}he theorems of
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	ISSN = {0075-8434},
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[53] M. Yor: “Un exemple de J. Pitman” [An example of J. Pitman], pp. 624 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. MR 544831 Zbl 0422.60029 incollection

Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

[54] J. Azéma and M. Yor: “Le problème de Skorokhod: Compléments à l’exposé précédent” [The problem of Skorokhod: Supplement to the previous talk], pp. 625–633 in Séminaire de probabilités XIII [Thirteenth probability seminar]. Edited by C. Dellacherie, P. A. Meyer, and M. Weil. Lecture Notes in Mathematics 721. Springer (Berlin), 1979. This is a supplement to the preceding article in Séminaire de probabilités XIII 721 (1979). MR 544832 Zbl 0414.60056 incollection

Jacques Azéma	Related
Paul-André Meyer	Related
M. Weil	Related
Claude Dellacherie	Related

@incollection {key544832m,
	AUTHOR = {Az\'ema, Jacques and Yor, Marc},
	TITLE = {Le probl\`eme de {S}korokhod: {C}ompl\'ements
		 \`a l'expos\'e pr\'ec\'edent [The problem
		 of {S}korokhod: {S}upplement to the
		 previous talk]},
	BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
		 [Thirteenth probability seminar]},
	EDITOR = {Dellacherie, C. and Meyer, P. A. and
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	SERIES = {Lecture Notes in Mathematics},
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	NOTE = {This is a supplement to the preceding
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		 XIII} \textbf{721} (1979). MR:544832.
		 Zbl:0414.60056.},
	ISSN = {0075-8434},
	ISBN = {9783540095057},
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[55] M. Yor: “Les inégalités de sous-martingales, comme conséquences de la relation de domination” [Inequalities of sub-martingales, as consequences of the domination relation], Stochastics 3 : 1 (1979), pp. 1–15. MR 546696 Zbl 0437.60038 article

[56] M. Yor: “Fonctions et processus de Bessel” [Bessel functions and processes], C. R. Acad. Sci., Paris, Sér. A 289 : 16 (1979), pp. 817–819. MR 558806 Zbl 0419.60031 article

[57] M. Yor: “Loi de l’indice du lacet brownien, et distribution de Hartman–Watson” [Law of indices of Brownian laces, and the Hartman–Watson distribution], Z. Wahrscheinlichkeitstheor. Verw. Geb. 53 : 1 (January 1980), pp. 71–95. MR 576898 Zbl 0436.60057 article

[58] Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580103 Zbl 0416.00014 book

[59] J. Azéma, R. F. Gundy, and M. Yor: “Sur l’intégrabilité uniforme des martingales continues” [On the uniform integrability of continuous martingales], pp. 53–61 in Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580108 Zbl 0442.60046 incollection

Richard Floyd Gundy	Related
Jacques Azéma	Related

[60] M. T. Barlow and M. Yor: “Sur la construction d’une martingale continue, de valeur absolue donnée” [On the construction of a continuous martingale of given absolute value], pp. 62–75 in Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580109 Zbl 0426.60047 incollection

This paper consists of two notes on Gilat’s theorem (Ann. Prob. 5, 1977, see also [Maisonneuve 1979]). The problem consists in constructing, given a continuous positive submartingale \( Y \), a continuous martingale \( X \) (possibly on a different space) such that \( |X| \) has the same law as \( Y \). Let \( A \) be the increasing process associated with \( Y \); it is necessary for the existence of \( X \) that \( dA \) should be carried by \( \{Y=0\} \). This is shown by the first note (Yor’s) to be also sufficient — more precisely, in this case the solutions of Gilat’s problem are all continuous. The second note (Barlow’s) shows how to construct a Gilat martingale by “putting a random \( \pm \) sign in front of each excursion of \( Y \)”, a simple intuitive idea and a delicate proof.

Martin Thomas Barlow	Related
Jacques Azéma	Related

[61] M. Yor: “Application d’un lemme de T. Jeulin au grossissement de la filtration brownienne” [Application of a lemma of T. Jeulin to the enlargement of the Brownian filtration], pp. 189–199 in Séminaire de probabilités XIV [Fourteenth probability seminar], vol. 784. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580123 Zbl 0427.60041 incollection

[62] J. Auerhan, D. Lépingle, and M. Yor: “Construction d’une martingale réelle continue, de filtration naturelle donnée” [Construction of a real continuous martingale, of given natural filtration], pp. 200–204 in Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580124 Zbl 0432.60056 incollection

[63] M. Yor: “Remarques sur une formule de Paul Lévy” [Remarks on a formula of Paul Lévy], pp. 343–346 in Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580140 Zbl 0429.60045 incollection

[64] C. Cocozza and M. Yor: “Démonstration d’un théorème de F. Knight à l’aide de martingales exponentielles” [Demonstration of a theorem of F. Knight using exponential martingales], pp. 496–499 in Séminaire de probabilités XIV [Fourteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 784. Springer (Berlin), 1980. MR 580150 Zbl 0432.60057 incollection

[65]D. W. Stroock and M. Yor: “On extremal solutions of martingale problems,” Ann. Sci. École Norm. Sup. (4) 13 : 1 (1980), pp. 95–164. MR 584083 Zbl 0447.60034

[66] J. W. Pitman and M. Yor: “Processus de Bessel, et mouvement brownien, avec ‘drift’” [Bessel processes, and Brownian motion, with ‘drift’], C. R. Acad. Sci., Paris, Sér. A 291 : 2 (1980), pp. 151–153. MR 605004 Zbl 0438.60063 article

[67] M. T. Barlow and M. Yor: “(Semi-) martingale inequalities and local times,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 55 : 3 (1981), pp. 237–254. MR 608019 Zbl 0451.60050 article

[68]N. Bouleau and M. Yor: “Sur la variation quadratique des temps locaux de certaines semimartingales” [On the quadratic variation of local times of some semimartingales], C. R. Acad. Sci., Paris, Sér. I 292 : 9 (1981), pp. 491–494. MR 612544 Zbl 0476.60046 article

[69] J. Pitman and M. Yor: “Bessel processes and infinitely divisible laws,” pp. 285–370 in Stochastic integrals (Durham, UK, 7–17 July 1980). Edited by D. Williams. Lecture Notes in Mathematics 851. Springer (Berlin), 1981. MR 620995 Zbl 0469.60076 incollection

James W. Pitman	Related
David Williams	Related

[70] Séminaire de probabilités XV [Fifteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 850. Springer (Berlin), 1981. With a general index to the papers from the Seminars 1966/67–1978/79. MR 622550 Zbl 0447.00009 book

[71] T. Jeulin and M. Yor: “Sur les distributions de certaines fonctionnelles du mouvement brownien” [On the distribution of certain functionals of Brownian motion], pp. 210–226 in Séminaire de probabilités XV [Fifteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 850. Springer (Berlin), 1981. MR 622565 Zbl 0462.60077 incollection

Thierry Jeulin	Related
Jacques Azéma	Related

[72] M. Yor: “Sur certains commutateurs d’une filtration” [On certain commutators of a filtration], pp. 526–528 in Séminaire de Probabilités XV [Fifteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 850. Springer (Berlin), 1981. MR 622585 Zbl 0456.60057 incollection

[73]D. W. Stroock and M. Yor: “Some remarkable martingales,” pp. 590–603 in Seminar on Probability (Univ. Strasbourg, 1979/1980), vol. XV. Lecture Notes in Math. 850. Springer (Berlin), 1981. MR 622590 Zbl 0456.60048

[74] D. Lépingle, P.-A. Meyer, and M. Yor: “Extrémalité et remplissage de tribus pour certaines martingales purement discontinues” [Extremality and the filling of families for certain purely discontinuous martingales], pp. 604–617 in Séminaire de probabilités XV [Fifteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 850. Springer (Berlin), 1981. MR 622591 Zbl 0461.60064 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[75] P. J. Bickel, N. El Karoui, and M. Yor: Ecole d’eté de probabilités de Saint-Flour IX–1979 [Ninth Saint-Flour probability summer school–1979] (Saint-Flour, France, 1979). Edited by P. L. Hennequin. Lecture Notes in Mathematics 876. Springer (Berlin), 1981. Yor’s contribution to this book was reprinted in Stochastic filtering at Saint-Flour (2012). MR 637469 Zbl 0455.00015 book

Paul-Louis Hennequin	Related
Peter J. Bickel	Related

[76] M. Yor: “Sur la théorie du filtrage” [On the theory of filtration], pp. 239–280 in P. J. Bickel, N. El Karoui, and M. Yor: Ecole d’eté de probabilités de Saint-Flour IX–1979 [Ninth Saint-Flour probability summer school–1979] (Saint-Flour, France, 1979). Edited by P. L. Hennequin. Lecture Notes in Mathematics 876. Springer (Berlin), 1981. Reprinted in Stochastic filtering at Saint-Flour (2012). MR 637472 Zbl 0474.60032 incollection

Paul-Louis Hennequin	Related
Peter J. Bickel	Related

[77] J. Pitman and M. Yor: “A decomposition of Bessel bridges,” Z. Wahrsch. Verw. Gebiete 59 : 4 (December 1982), pp. 425–457. English translation of French original from Functional analysis in Markov processes (1982). MR 656509 Zbl 0484.60062 article

[78] Séminaire de probabilités XVI [Sixteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 920. Springer (Berlin), 1982. MR 658668 Zbl 0471.00023 book

[79] N. Falkner, C. Stricker, and M. Yor: “Temps d’arrêt riches et applications” [Rich stopping times and applications], pp. 213–218 in Séminaire de probabilités XVI [Sixteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 920. Springer (Berlin), 1982. MR 658683 Zbl 0485.60034 incollection

This paper starts from the existence of increasing left-continuous processes \( (A_t) \) which generate the previsible \( \sigma \)-field, i.e., every previsible process can be represented as \( f(X_t) \) for some Borel function \( f \) (see [Dellacherie and Stricker 1982]), to prove the existence (discovered by the first named author) of “rich” stopping times \( T \), i.e., previsible stopping times which encode the whole past up to time \[ T: \sigma(T)=\mathcal{F}_{T-} \] (a few details are omitted here). This result leads to counterexamples: a non-reversible semimartingale (see the preceding paper [Walsh 1982]) and a stopping time \( T \) for Brownian motion such that \( L^a_T \) is not a semimartingale in its space variable \( a \).

[80] M. Yor: “Application de la rélation de domination à certains renforcements des inégalités de martingales” [Applications of the domination relation for certain strengthenings of martingale inequalities], pp. 221–233 in Séminaire de probabilités XVI [Sixteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 920. Springer (Berlin), 1982. MR 658685 Zbl 0479.60050 incollection

The domination relation [Lenglart 1977] between a positive, right-continuous process \( X \) and a previsible increasing process \( A \) holds whenever \[ \mathbb{E}[X_T]\leq \mathbb{E}[A_T] \] at stopping times. It plays an important role in the paper [1980] of Lenglart–Lepingle–Pratelli on martingale inequalities. Here it is shown to imply a general inequality involving \( X^*_{\infty} \) and \( 1/A_{\infty} \), from which follow a number of inequalities for a continuous local martingale \( M \). Among them, estimates on the ratios of the three quantities \( M^*_{\infty} \), \( \langle M\rangle_{\infty} \), \( \sup_{a,t} L^a_t \). One can recover also the stronger version of Doob’s inequality, proved by Pitman [1981].

[81] M. Yor: “Sur la transformée de Hilbert des temps locaux browniens, et une extension de la formule d’Itô” [On the Hilbert transformation of Brownian local times, and an extension of Itô’s formula], pp. 238–247 in Séminaire de probabilités XVI [Sixteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 920. Springer (Berlin), 1982. MR 658687 Zbl 0495.60080 incollection

[82] Séminaire de probabilités XVI: Supplément: Géométrie différentielle stochastique [Sixteenth probability seminar: Supplement: Stochastic differential geometry]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 921. Springer (Berlin), 1982. MR 658721 Zbl 0471.00024 book

[83] J. Pitman and M. Yor: “Sur une décomposition des ponts de Bessel” [On a decomposition of Bessel bridges], pp. 276–285 in Functional analysis in Markov processes (Katata and Kyoto, Japan, 21–29 August 1981). Edited by M. Fukushima. Lecture Notes in Mathematics 923. Springer (Berlin), 1982. An English translation was published in Z. Wahrsch. Verw. Gebiete 59:4 (1982). MR 661630 Zbl 0499.60082 incollection

James W. Pitman	Related
Masatoshi Fukushima	Related

[84] M. T. Barlow and M. Yor: “Application du lemme de Garsia–Rodemich–Rumsey à certaines inégalités de (semi) martingales continues” [Application of the lemma of Garsia–Rodemich–Rumsey to certain inequalities for continuous (semi) martingales], C. R. Acad. Sci., Paris, Sér. I 294 : 19 (1982), pp. 665–668. MR 664645 Zbl 0488.60060 article

[85] P. Messulam and M. Yor: “D. Williams’ ‘pinching method’ and some applications,” J. Lond. Math. Soc., II. Ser. 26 : 2 (1982), pp. 348–364. MR 675178 Zbl 0518.60088 article

[86] M. T. Barlow and M. Yor: “Semimartingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local times,” J. Funct. Anal. 49 : 2 (November 1982), pp. 198–229. MR 680660 Zbl 0505.60054 article

[87] M. Yor: “Introduction au calcul stochastique” [Introduction to stochastic calculus], pp. 275–292 in Séminaire Bourbaki: 1981/82, exposés 579–596. Astérisque 92–93. Société Mathématique de France (Paris), 1982. Exposé no. 590. MR 689534 Zbl 0497.60055 incollection

[88] M. Yor: “Sur un processus associé aux temps locaux browniens” [On a process associated with Brownian local times], Ann. Sci. Univ. Clermont-Ferrand II, Math. 71 : 20 (1982), pp. 140–148. MR 707543 Zbl 0523.60069 article

[89] J.-M. Bismut and M. Yor: “An inequality for processes which satisfy Kolmogorov’s continuity criterion: Application to continuous martingales,” J. Funct. Anal. 51 : 2 (1983), pp. 166–173. MR 701054 Zbl 0524.60020 article

Theorem 1 below is a by-product of the continuous martingale inequalities obtained in [Yor 1985], using the technique of enlargment of filtrations. The main purpose of this paper is to give an entirely new proof and some extensions of Theorem 1; some global stability properties of classes of processes which satisfy Kolmogorov’s criterion (\( K^{\phi} \)) play a crucial role in our proof.

Let \( k\in (0,\infty) \), and \( p\in (1,\infty) \). There exists a constant \( C_{k,p} \) such that for any positive, finite variable \( L \), and any continuous local martingale \( (X_t)_{t\geq 0} \), with \( X(0) = 0 \), we have

\( E[(X^*(L))^k]\leq C_{k,p} \bigl\| \langle X\rangle_L^{k/2} \bigr\|_p \),
\( E\bigl[ \langle X\rangle_L^{k/2} \bigr] \leq C_{k,p} \bigl\| (X^*(L))^k \bigr\|_p \), where \( X*(t)\equiv \sup_{s\leq t} |X(s)| \),
and \( (\langle X\rangle_t) \) is the increasing process of \( X \).

[90] M. Yor: “Une inégalité optimale pour le mouvement brownien arrêté à un temps quelconque” [An optimal inequality for Brownian motion stopped at a given time], C. R. Acad. Sci., Paris, Sér. I 296 : 9 (1983), pp. 407–409. MR 703908 Zbl 0532.60071 article

[91] Séminaire de probabilités XVII [Seventeenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 986. Springer (Berlin), 1983. MR 770391 Zbl 0498.00003 book

[92] J.-F. Le Gall and M. Yor: “Sur l’équation stochastique de Tsirelson” [On Tsirelson’s stochastic equation], pp. 81–88 in Séminaire de probabilités XVII [Seventeenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 986. Springer (Berlin), 1983. MR 770399 Zbl 0535.60052 incollection

Jean-François Le Gall	Related
Jacques Azéma	Related

[93] M. Yor: “Le drap brownien comme limite en loi de temps locaux linéaires” [The Brownian sheet as a limit in law of local linear time], pp. 89–105 in Séminaire de probabilités XVII [Seventeenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 986. Springer (Berlin), 1983. MR 770400 Zbl 0514.60075 incollection

[94] J. Rosen: “A local time approach to the self-intersections of Brownian paths in space,” Comm. Math. Phys. 88 : 3 (1983), pp. 327–338. MR 701921 Zbl 0534.60070 article

[95] J. W. Pitman and M. Yor: “The asymptotic joint distribution of windings of planar Brownian motion,” Bull. Am. Math. Soc., New Ser. 10 : 1 (January 1984), pp. 109–111. MR 722863 Zbl 0535.60073 article

[96] M. T. Barlow, S. D. Jacka, and M. Yor: “Inégalités pour un couple de processus arrêtés à un temps quelconque” [Inequalities for a pair of processes stopped at a random time], C. R. Acad. Sci., Paris, Sér. I 299 : 8 (1984), pp. 351–354. MR 761264 Zbl 0574.60022 article

[97] Séminaire de probabilités XVIII [Eighteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1059. Springer (Berlin), 1984. MR 770944 Zbl 0527.00020 book

[98] M. Yor: “Square-root boundaries for Bessel processes, and pole-seeking Brownian motion,” pp. 100–107 in Stochastic analysis and application (Swansea, UK, 11–15 April 1983). Edited by A. Truman and D. Williams. Lecture Notes in Mathematics 1095. Springer (Berlin), 1984. MR 777516 Zbl 0598.60086 incollection

[99] M. Yor: “À propos de l’inverse du mouvement brownien dans \( \mathbb{R}^n\;(n\geq 3) \)” [The inverse of Brownian motion on \( \mathbb{R}^n\;(n\geq 3) \)], Ann. Inst. Henri Poincaré, Probab. Stat. 21 : 1 (1985), pp. 27–38. MR 791267 Zbl 0556.60032 article

[100] P. Biane and M. Yor: “Valeurs principales associées aux temps locaux browniens et processus stables symétriques” [Principal values associated with Brownian local times and stable symmetric processes], C. R. Acad. Sci., Paris, Sér. I 300 : 20 (1985), pp. 695–698. MR 799465 Zbl 0583.60080 article

[101] M. Yor: “Une décomposition asymptotique du nombre de tours du mouvement brownien complexe” [An asymptotic decomposition of the winding number of complex Brownian motion], pp. 103–126 in Colloque en l’honneur de Laurent Schwartz [Colloquium in honor of Laurent Schwartz] (Paris, 30 May–3 June 1983), vol. 2. Astérisque 132. Société Mathematique de France (Paris), 1985. MR 816763 Zbl 0583.60077 incollection

[102] Grossissements de filtrations: Exemples et applications [Enlargements of filtrations: Examples and applications] (Paris, 1982–1983). Edited by T. Jeulin and M. Yor. Lecture Notes in Mathematics 1118. Springer (Berlin), 1985. Proceedings of a seminar on stochastic calculus. MR 884713 Zbl 0547.00034 book

[103] Séminaire de probabilités XIX [Nineteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1123. Springer (Berlin), 1985. MR 889463 Zbl 0549.00021 book

[104] M. Yor: “Compléments aux formules de Tanaka–Rosen” [Complements to the Tanaka–Rosen formulas], pp. 332–349 in Séminaire de probabilités XIX [Nineteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1123. Springer (Berlin), 1985. MR 889493 Zbl 0563.60073 incollection

[105] M. Yor: “Renormalisation et convergence en loi pour les temps locaux d’intersection du mouvement brownien dans \( \mathbb{R}^3 \)” [Renormalization and convergence in law for local times of intersection of Brownian motion on \( \mathbb{R}^3 \)], pp. 350–365 in Séminaire de probabilités XIX [Nineteenth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1123. Springer (Berlin), 1985. MR 889494 Zbl 0569.60075 incollection

[106] M. Yor: “Inegalités de martingales continues arretées à un temps quelconque” [Inequalities of continuous martingales stopping at a random time], pp. 110–146 in Grossissements de filtrations: Exemples et applications [Enlargements of filtrations: Examples and applications] (Paris, 1982–1983). Edited by T. Jeulin and M. Yor. Lecture Notes in Mathematics 1118. Springer (Berlin), 1985. A follow-up paper was also published in Grossissements de filtrations (1985). Zbl 0563.60045 incollection

[107] M. Yor: “Entropie d’une partition, et grossissement initial d’une filtration” [Entropy of a partition, and initial enlargement of a filtration], pp. 45–58 in Grossissements de filtrations: Exemples et applications [Enlargements of filtrations: Examples and applications] (Paris, 1982–1983). Edited by T. Jeulin and M. Yor. Lecture Notes in Mathematics 1118. Springer (Berlin), 1985. Zbl 0568.60050 incollection

[108] M. Yor: “Inegalités de martingales continues arretées à un temps quelconque: Le rôle de certains espaces BMO” [Inequalities of continuous martingales stopping at a random time: The role of certain BMO spaces], pp. 147–171 in Grossissements de filtrations: Exemples et applications [Enlargements of filtrations: Examples and applications]. Edited by T. Jeulin and M. Yor. Lecture Notes in Mathematics 1118. Springer, 1985. Follow-up to a paper also published in Grossissements de filtrations (1985). Zbl 0568.60051 incollection

[109] M. Yor: “Grossissement de filtrations et absolue continuite de noyaux” [Enlargement of filtrations and absolute continuity of kernels], pp. 6–14 in Grossissements de filtrations: Exemples et applications [Enlargements of filtrations: Examples and applications] (Paris, 1982–1983). Edited by T. Jeulin and M. Yor. Lecture Notes in Mathematics 1118. Springer (Berlin), 1985. Zbl 0576.60038 incollection

[110] M. T. Barlow, S. D. Jacka, and M. Yor: “Inequalities for a pair of processes stopped at a random time,” Proc. Lond. Math. Soc. (3) 52 : 1 (1986), pp. 142–172. MR 812449 Zbl 0585.60055 article

[111] J.-F. Le Gall and M. Yor: “Étude asymptotique de certains mouvements browniens complexes avec drift” [Asymptotic study of certain complex Brownian motions with drift], Probab. Theory Relat. Fields 71 : 2 (January 1986), pp. 183–229. MR 816704 Zbl 0579.60077 article

[112] J. Pitman and M. Yor: “Asymptotic laws of planar Brownian motion,” Ann. Probab. 14 : 3 (1986), pp. 733–779. A follow-up to this was published in Ann. Probab. 17:3 (1989). MR 841582 Zbl 0607.60070 article

[113] J. Pitman and M. Yor: “Level crossings of a Cauchy process,” Ann. Probab. 14 : 3 (1986), pp. 780–792. MR 841583 Zbl 0602.60059 article

[114] J.-F. Le Gall and M. Yor: “Excursions browniennes et carrés de processus de Bessel” [Brownian excursions and squares of Bessel processes], C. R. Acad. Sci., Paris, Sér. I 303 : 3 (1986), pp. 73–76. MR 851079 Zbl 0589.60070 article

[115] J. W. Pitman and M. Yor: “Some divergent integrals of Brownian motion,” pp. 109–116 in Analytic and geometric stochastics: Papers in honour of G. E. H. Reuter, published as Adv. Appl. Probab. 18. Issue edited by D. G. Kendall, J. F. C. Williams, and D. Williams. Applied Probability Trust (Sheffield, UK), December 1986. Supplementary issue. MR 868512 Zbl 0618.60074 incollection

Gerd Edzard Harry Reuter	Related
David George Kendall	Related
James W. Pitman	Related
David Williams	Related

[116] Séminaire de probabilités XX [Twentieth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1204. Springer (Berlin), 1986. MR 942008 Zbl 0593.00014 book

[117] M. Yor: “Sur la représentation comme intégrales stochastiques des temps d’occupation du mouvement Brownien dans \( \mathbb{R}^d \)” [On the stochastic integral representation of occupation times of Brownian motion in \( \mathbb{R}^d \)], pp. 543–552 in Séminaire de probabilités XX [Twentieth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1204. Springer (Berlin), 1986. MR 942043 Zbl 0611.60067 incollection

[118] M. Yor: “Précisions sur l’existence et la continuité des temps locaux d’intersection du mouvement brownien dans \( \mathbb{R}^2 \)” [Details on the existence and continuity of intersection local times of Brownian motion in \( \mathbb{R}^2 \)], pp. 532–542 in Séminaire de probabilités XX [Twentieth probability seminar]. Edited by J. Azéma and M. Yor. Lecture Notes in Mathematics 1204. Springer (Berlin), 1986. MR 942042 Zbl 0611.60066 incollection

[119] K. Burdzy, J. W. Pitman, and M. Yor: Some asymptotic laws for crossings and excursions. Technical report 112, Department of Statistics, UC-Berkeley, September 1987. Also published in Colloque Paul Lévy sur les processus stochastiques (1988). techreport

James W. Pitman	Related
Krzysztof Burdzy	Related

[120] J.-F. Le Gall and M. Yor: “Étude asymptotique des enlacements du mouvement brownien autour des droites de l’espace” [Asymptotic study of windings of Brownian motion around straight lines], Probab. Theory Relat. Fields 74 : 4 (April 1987), pp. 617–635. MR 876259 Zbl 0594.60083 article

[121] P. Biane and M. Yor: “Valeurs principales associées aux temps locaux browniens” [Principal values associated to Brownian local times], Bull. Sci. Math., II. Sér. 111 : 1 (1987), pp. 23–101. MR 886959 Zbl 0619.60072 article

[122] P. Biane and M. Yor: “Variations sur une formule de Paul Lévy” [Variations on a formula of Paul Lévy], Ann. Inst. Henri Poincaré, Probab. Stat. 23 : S2 (1987), pp. 359–377. MR 898500 Zbl 0623.60099 article

[123] M. Yor: “Some recent studies of Brownian paths intersections,” pp. 439–446 in VIIIth international congress on mathematical physics (Marseille, 16–25 July 1986). Edited by M. Mebkhout and R. Sénéor. World Scientific (Singapore), 1987. MR 915590 incollection

Roland Sénéor	Related
Mohammed Mebkhout	Related

[124] J. Pitman and M. Yor: “Compléments à l’étude asymptotique des nombres de tours du mouvement brownien complexe autour d’un nombre fini de points” [Complements to the study of the winding numbers of complex Brownian motion around a finite set of points], C. R. Acad. Sci., Paris, Sér. I 305 : 17 (1987), pp. 757–760. MR 921145 Zbl 0624.60088 article

[125] Séminaire de probabilités XXI [Twenty-first probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1247. Springer (Berlin), 1987. MR 941971 Zbl 0606.00022 book

Paul-André Meyer	Related
Jacques Azéma	Related

[126] S. Song and M. Yor: “Inégalités pour les processus self-similaires arrêtés à un temps quelconque” [Inequalities for self-similar processes stopping at an arbitrary time], pp. 230–245 in Séminaire de probabilités XXI [Twenty-first probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1247. Springer (Berlin), 1987. MR 941987 Zbl 0624.60065 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[127] J. Azéma and M. Yor: “Interprétation d’un calcul de H. Tanaka en théorie générale des processus” [Interpretation of a calculus of H. Tanaka in the general theory of processes], pp. 262–269 in Séminaire de probabilités XXI [Twenty-first probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1247. Springer (Berlin), 1987. MR 941989 Zbl 0621.60085 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[128] P. Biane, J.-F. Le Gall, and M. Yor: “Un processus qui ressemble au pont brownien” [A process that resembles the Brownian bridge], pp. 270–275 in Séminaire de probabilités XXI [Twenty-first probability seminar]. Edited by J. Az’e\ma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1247. Springer (Berlin), 1987. MR 941990 Zbl 0621.60086 incollection

Philippe Biane	Related
Jacques Azéma	Related
Paul-André Meyer	Related
Jean-François Le Gall	Related

[129] J. Y. Calais and M. Yor: “Renormalisation et convergence en loi pour certaines intégrales multiples associées au mouvement brownien dans \( \mathbb{R}^d \)” [Renormalisation and convergence in law for certain multiple integrals associated with Brownian motion in \( \mathbb{R}^d \)], pp. 375–403 in Séminaire de probabilités XXI [Twenty-first probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1247. Springer (Berlin), 1987. MR 941995 Zbl 0624.60067 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

@incollection {key941995m,
	AUTHOR = {Calais, J. Y. and Yor, M.},
	TITLE = {Renormalisation et convergence en loi
		 pour certaines int\'egrales multiples
		 associ\'ees au mouvement brownien dans
		 \$\mathbb{R}^d\$ [Renormalisation and
		 convergence in law for certain multiple
		 integrals associated with {B}rownian
		 motion in \$\mathbb{R}^d\$]},
	BOOKTITLE = {S\'eminaire de probabilit\'es {XXI}
		 [Twenty-first probability seminar]},
	EDITOR = {Az\'ema, Jacques and Meyer, Paul Andr\'e
		 and Yor, Marc},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1247},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {1987},
	PAGES = {375--403},
	DOI = {10.1007/BFb0077646},
	URL = {http://www.numdam.org/item?id=SPS_1987__21__375_0},
	NOTE = {MR:941995. Zbl:0624.60067.},
	ISSN = {0075-8434},
	ISBN = {9783540177685},
}

[130] P. Biane and M. Yor: “Quelques précisions sur le méandre brownien” [Some details on the Brownian meander], Bull. Sci. Math., II. Sér. 112 : 1 (1988), pp. 101–109. MR 942801 Zbl 0665.60081 article

[131] Séminaire de probabilités XXII [Twenty-second probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1321. Springer (Berlin), 1988. MR 960506 Zbl 0635.00013 book

Paul-André Meyer	Related
Jacques Azéma	Related

[132] M. Yor: “Remarques sur certaines constructions des mouvements browniens fractionnaires” [Remarks on certain constructions of fractional Brownian motions], pp. 217–224 in Séminaire de probabilités XXII [Twenty-second probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1321. Springer (Berlin), 1988. MR 960530 Zbl 0656.60087 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[133] S. Weinryb and M. Yor: “Le mouvement brownien de Lévy indexé par \( \mathbb{R}^3 \) comme limite centrale de temps locaux d’intersection” [Lévy Brownian motion for \( \mathbb{R}^3 \) as central limit of intersection local times], pp. 225–248 in Séminaire de probabilités XXII [Twenty-second probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1321. Springer (Berlin), 1988. MR 960531 Zbl 0653.60074 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[134] P. Biane and M. Yor: “Sur la loi des temps locaux browniens pris en un temps exponentiel” [On the law of Brownian local times at an exponential time], pp. 454–466 in Séminaire de probabilités XXII [Twenty-second probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1321. Springer (Berlin), 1988. MR 960541 Zbl 0652.60081 incollection

Paul-André Meyer	Related
Philippe Biane	Related
Jacques Azéma	Related

[135] K. Burdzy, J. W. Pitman, and M. Yor: “Some asymptotic laws for crossings and excursions,” pp. 59–74 in Colloque Paul Lévy sur les processus stochastiques [Paul Lévy colloquium on stochastic processes] (Palaiseau, France, 22–26 June 1987). Edited by M. Métivier. Astérisque 157–158. Société Mathématique de France (Paris), 1988. Also published as a 1988 UC-Berkeley technical report. MR 976213 Zbl 0666.60070 incollection

Krzysztof Burdzy	Related
Paul Lévy	Related
Michel E. Métivier	Related
James W. Pitman	Related

[136] M. Yor: “Une extension markovienne de l’algèbre des lois béta-gamma” [A Markovian extension of the algebra of the beta-gamma law], C. R. Acad. Sci., Paris, Sér. I Math. 308 : 8 (1989), pp. 257–260. MR 1006074 Zbl 0664.60073 article

[137] M. Yor: “On stochastic areas and averages of planar Brownian motion,” J. Phys. A 22 : 15 (1989), pp. 3049–3057. MR 1007232 Zbl 0717.60095 article

Duplantier [1989] has computed the characteristic function of the stochastic area of planar Brownian motion taken with respect to its centre of gravity. In this paper it is shown that this result can be deduced from Lévy’s stochastic area formula. Moreover, Duplantier’s stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval \( [0,1] \) of the Brownian ring defined on the time interval \( [0,4/3] \).

The same method also applies to the stochastic area formulae, obtained by Biane and Yor [1986], which are relative to certain planar processes associated with the orthogonal decomposition of Brownian motion along the \( L^2[0,1] \) basis of Legendre polynomials.

[138] J. Pitman and M. Yor: “Further asymptotic laws of planar Brownian motion,” Ann. Probab. 17 : 3 (1989), pp. 965–1011. This was a follow-up to an article published in Ann. Probab. 14:3 (1986). MR 1009441 Zbl 0686.60085 article

[139] Séminaire de probabilités XXIII [Twenty-third probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1372. Springer (Berlin), 1989. MR 1022893 Zbl 0722.00030 book

Paul-André Meyer	Related
Jacques Azéma	Related

[140] J. Azéma and M. Yor: “Étude d’une martingale remarquable” [Study of a remarkable martingale], pp. 88–130 in Séminaire de probabilités XXIII [Twenty-third probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1372. Springer (Berlin), 1989. MR 1022900 Zbl 0743.60045 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[141] M. Barlow, J. Pitman, and M. Yor: “On Walsh’s Brownian motions,” pp. 275–293 in Séminaire de probabilités XXIII [Twenty-third probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1372. Springer (Berlin), 1989. MR 1022917 Zbl 0747.60072 incollection

Jacques Azéma	Related
Paul-André Meyer	Related
James W. Pitman	Related
Martin Thomas Barlow	Related

[142] M. Barlow, J. Pitman, and M. Yor: “Une extension multidimensionnelle de la loi de l’arc sinus” [A multidimensional extension of the arcsine law], pp. 294–314 in Séminaire de probabilités XXIII [Twenty-third probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1372. Springer (Berlin), 1989. MR 1022918 Zbl 0738.60072 incollection

Jacques Azéma	Related
Paul-André Meyer	Related
James W. Pitman	Related
Martin Thomas Barlow	Related

[143] C. Donati-Martin and M. Yor: “Mouvement brownien et inégalité de Hardy dans \( L^2 \)” [Brownian motion and the Hardy inequality on \( L^2 \)], pp. 315–323 in Séminaire de probabilités XXIII [Twenty-third probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1372. Springer (Berlin), 1989. MR 1022919 Zbl 0739.60073 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[144] M. Yor: “De nouveaux résultats sur l’équation de Tsirel’son” [Some new results on Tsirel’son’s equation], C. R. Acad. Sci., Paris, Sér. I 309 : 7 (1989), pp. 511–514. MR 1055470 Zbl 0697.60062 article

[145] J.-F. Le Gall and M. Yor: “Enlacements du mouvement brownien autour des courbes de l’espace” [Winding of Brownian motion around space curves], Trans. Am. Math. Soc. 317 : 2 (February 1990), pp. 687–722. MR 946219 Zbl 0696.60072 article

[146] K. Burdzy, J. Pitman, and M. Yor: “Brownian crossings between spheres,” J. Math. Anal. Appl. 148 : 1 (May 1990), pp. 101–120. MR 1052048 Zbl 0713.60082 article

James W. Pitman	Related
Krzysztof Burdzy	Related

[147] Séminaire de probabilités XXIV [Twenty-fourth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1426. Springer (Berlin), 1990. MR 1071527 Zbl 0695.00024 book

Paul-André Meyer	Related
Jacques Azéma	Related

[148] J. Azéma and M. Yor: “Dérivation par rapport au processus de Bessel” [Derivation with respect to a Bessel process], pp. 210–226 in Séminaire de probabilités XXIV [Twenty-fourth probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1426. Springer (Berlin), 1990. MR 1071542 Zbl 0723.60094 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[149] T. Jeulin and M. Yor: “Filtration des ponts browniens et équations différentielles stochastiques linéaires” [Filtering of Brownian bridges and linear stochastic differential equations], pp. 227–265 in Séminaire de probabilités XXIV [Twenty-fourth probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1426. Springer (Berlin), 1990. MR 1071543 Zbl 0699.60075 incollection

Paul-André Meyer	Related
Thierry Jeulin	Related
Jacques Azéma	Related

[150] D. Revuz and M. Yor: Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften 293. Springer (Berlin), 1991. A 2nd edition was published in 1994. A 3rd edition was published in 1999. MR 1083357 Zbl 0731.60002 book

[151] J. Rosen and M. Yor: “Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane,” Ann. Probab. 19 : 1 (1991), pp. 142–159. MR 1085330 Zbl 0719.60084 article

[152] C. Donati-Martin and M. Yor: “Fubini’s theorem for double Wiener integrals and the variance of the Brownian path,” Ann. Inst. Henri Poincaré, Probab. Stat. 27 : 2 (1991), pp. 181–200. MR 1118933 Zbl 0738.60074 article

[153] M. Yor: “Une explication du théorème de Ciesielski–Taylor” [An explanation of the Ciesielski–Taylor theorem], Ann. Inst. Henri Poincaré, Probab. Stat. 27 : 2 (1991), pp. 201–213. MR 1118934 Zbl 0743.60080 article

[154] M. Yor: “Étude asymptotique des nombres de tours de plusieurs mouvements browniens complexes corrélés” [Asymptotic study of the winding numbers of several correlated complex Brownian motions], pp. 441–455 in Random walks, Brownian motion, and interacting particle systems: Festschrift in honor of Frank Spitzer. Edited by R. Durrett and H. Kesten. Progress in Probability 28. Birkhäuser (Boston, MA), 1991. MR 1146463 Zbl 0747.60076 incollection

Frank Ludvig Spitzer	Related
Harry Kesten	Related
Richard Timothy Durrett	Related

[155] M. Yor: “The laws of some Brownian functionals,” pp. 1105–1112 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 2. Edited by I. Satake. Springer (Tokyo), 1991. MR 1159295 Zbl 0751.60077 incollection

Thanks mainly to the relationship between the heat equation, newtonian potential theory and Brownian motion, the laws of a large number of Brownian functionals have been obtained during the last fifty years, at least via explicit expressions of their Laplace and Fourier transforms. Much pioneering work in this area was done by Paul Lévy.

Gradually, with the development of Itô’s stochastic calculus, excursion theory, path decompositions and the technique of enlargement of nitrations, these studies of individual distributions on \( \mathbb{R} \), sometimes exhibiting identities between two laws, which looked a priori to be mere “coincidences”, have been understood in a deeper way, in fact often by showing that two processes are identical in law; see [Biane 1990] for a recent survey in that spirit.

The most elementary examples of Brownian functionals are linear functionals: if \[ f\in L^2(\mathbb{R}_+,dt) ,\] and \( (B_t \), \( t\geq 0) \) is a real-valued BM, the Wiener integral \[ \int_0^{\infty} f(t)\,dB_t \] is a centered Gaussian variable, with variance \[ \int_0^{\infty}f^2(t)\,dt .\] Quadratic functionals of BM represent the next level of complexity; those functionals are of great interest as, somewhat surprisingly, they occur in a number of very different studies of Brownian motion, such as the Ray–Knight theorems for Brownian local times, the Ciesielski–Taylor identities, some limiting laws of planar BM, and principal values of Brownian local times.

We shall take here, as a prototype of a quadratic Brownian functional, the stochastic area of planar BM, and it will be shown how Paul Lévy’s formula for this stochastic area appears again and again in most of the above mentioned studies of Brownian motion.

[156] Séminaire de probabilités XXV [Twenty-fifth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1485. Springer (Berlin), 1991. MR 1187763 Zbl 0733.00018 book

Paul-André Meyer	Related
Jacques Azéma	Related

[157] L. E. Dubins, M. Émery, and M. Yor: “A continuous martingale in the plane that may spiral away to infinity,” pp. 284–290 in Séminaire de probabilités XXV [Twenty-fifth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1485. Springer (Berlin), 1991. MR 1187786 Zbl 0737.60036 incollection

Lester Eli Dubins	Related
Jacques Azéma	Related
Michel Emery	Related
Paul-André Meyer	Related

[158] M. Yor: “Tsirel’son’s equation in discrete time,” Probab. Theory Relat. Fields 91 : 2 (June 1992), pp. 135–152. MR 1147613 Zbl 0744.60033 article

Motivated by Tsirel’son’s equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solutions. In general, for any time \( n \), the \( \sigma \)-field generated by the past of a solution up to time \( n \) is shown to be equal, up to negligible sets, to the \( \sigma \)-field generated by the 3 following components: the infinitely remote past of the solution, the past of the noise up to time \( n \), together with an adequate independent complement.

[159] M. Yor: “Sur certaines fonctionnelles exponentielles du mouvement brownien réel” [On certain exponential functionals of real-valued Brownian motion], J. Appl. Probab. 29 : 1 (1992), pp. 202–208. Translated into English in Exponential functionals of Brownian motion and related processes (2001). MR 1147781 Zbl 0758.60085 article

[160] H. Geman and M. Yor: “Quelques relations entre processus de Bessel, options asiatiques et fonctions confluentes hypergéométriques” [Some relations between Bessel processes, Asian options and confluent hypergeometric functions], C. R. Acad. Sci., Paris, Sér. I 314 : 6 (1992), pp. 471–474. Translated into English in Exponential functionals of Brownian motion and related processes (2001). MR 1154389 Zbl 0759.60084 article

[161] M. Perman, J. Pitman, and M. Yor: “Size-biased sampling of Poisson point processes and excursions,” Probab. Theory Relat. Fields 92 : 1 (March 1992), pp. 21–39. MR 1156448 Zbl 0741.60037 article

Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

James W. Pitman	Related
Mihael Perman	Related

[162] J. Pitman and M. Yor: “Arcsine laws and interval partitions derived from a stable subordinator,” Proc. London Math. Soc. (3) 65 : 2 (September 1992), pp. 326–356. MR 1168191 Zbl 0769.60014 article

Lévy discovered that the fraction of time a standard one-dimensional Brownian motion \( B \) spends positive before time \( t \) has arcsine distribution, both for \( t \) a fixed time when \( B_t \neq 0 \) almost surely, and for \( t \) an inverse local time, when \( B_t = 0 \) almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of \( B \) away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension \( d \) for \( 0 \lt d \lt 2 \).

[163] M. Yor: “Sur les lois des fonctionnelles exponentielles du mouvement brownien, considérées en certains instants aléatoires” [The laws of exponential functionals of Brownian motion, taken at various random times], C. R. Acad. Sci., Paris, Sér. I 314 : 12 (1992), pp. 951–956. Abridged English version reprinted in Exponential functionals of Brownian motion and related processes (2001). MR 1168332 Zbl 0751.60076 article

[164] S. M. Kozlov, J. W. Pitman, and M. Yor: “Brownian interpretations of an elliptic integral,” pp. 83–95 in Seminar on stochastic processes, 1991 (Los Angeles, 23–25 March 1991). Edited by E. Çinlar, K. L. Chung, and M. J. Sharpe. Progress in Probability 29. Birkhäuser (Boston, MA), 1992. Proceedings dedicated to the memory of Steven Orey. MR 1172145 Zbl 0765.60082 incollection

Erhan Çinlar	Related
Kai Lai Chung	Related	Volume
James W. Pitman	Related
M. J. Sharpe	Related
Steven Orey	Related

[165] M. Yor: “On some exponential functionals of Brownian motion,” Adv. Appl. Probab. 24 : 3 (1992), pp. 509–531. Reprinted in Exponential functionals of Brownian motion and related processes (2001). MR 1174378 Zbl 0765.60084 article

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval \( [0,T] \) of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time \( T \) is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.

[166] C. Donati-Martin and M. Yor: “Extension d’une formule de Paul Lévy pour la variation quadratique du mouvement brownien plan” [Extension of a formula of Paul Lévy for the quadratic variation of planar Brownian motion], Bull. Sci. Math., II Sér. 116 : 3 (1992), pp. 353–382. MR 1177286 Zbl 0763.60036 article

[167] M. Yor: Some aspects of Brownian motion, part 1: Some special functionals. Lectures in Mathematics ETH Zürich. Birkhäuser (Basel), 1992. MR 1193919 Zbl 0779.60070 book

[168] S. M. Kozlov, J. Pitman, and M. Yor: “Wiener football,” Teor. Veroyatnost. i Primenen. 37 : 3 (1992), pp. 562–564. An English translation was published in Theory Probab. Appl. 37:3 (1992). A letter concerning this article appeared in Teor. Veroyatnost. i Primenen. 38:1 (1993). MR 1214362 article

[169] Séminaire de probabilités XXVI [Twenty-sixth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1526. Springer (Berlin), 1992. MR 1231978 Zbl 0754.00008 book

Paul-André Meyer	Related
Jacques Azéma	Related

[170] J. Azéma and M. Yor: “Sur les zéros des martingales continues” [On the zeros of continuous martingales], pp. 248–306 in Séminaire de probabilités XXVI [Twenty-sixth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1526. Springer (Berlin), 1992. MR 1231999 Zbl 0765.60038 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[171] J. Azéma, P.-A. Meyer, and M. Yor: “Martingales relatives” [Relative martingales], pp. 307–321 in Séminaire de probabilités XXVI [Twenty-sixth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1526. Springer (Berlin), 1992. MR 1232000 Zbl 0765.60037 incollection

Paul-André Meyer	Related
Jacques Azéma	Related

[172] T. Jeulin and M. Yor: “Une décomposition non-canonique du drap brownien” [A non-canonical decomposition of the Brownian sheet], pp. 322–347 in Séminaire de probabilités XXVI [Twenty-sixth seminar on probability]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1526. Springer (Berlin), 1992. MR 1232001 Zbl 0767.60082 incollection

Paul-André Meyer	Related
Thierry Jeulin	Related
Jacques Azéma	Related

[173] S. M. Kozlov, J. B. Pitman, and M. Yor: “Wiener football,” Theory Probab. Appl. 37 : 3 (1992), pp. 550–553. Russian original was published in Teor. Veroyatnost. i Primenen. 37:3 (1992). Zbl 0773.60079 article

The paper suggests an abstract model of the game of soccer (football) on an infinite field, where the path of the ball is the planar Wiener trajectory. The asymptotical distribution of the soccer score is obtained for large time \( T \). It shows that the variance of the score is proportional to \( (K^{\prime}/K)\log T \), where \( K \) and \( K^{\prime} \) are the periods of an elliptic integral with singularities at the locations of the goal posts. In particular this relation shows how the score in this game depends on the width of the goals. It is interesting to note that today FIFA is considering a possible increase of the goal’s width in connection with the fall of attendance at soccer matches even during the world championships. Of course, the suggested model is one among many possible models taking more and more soccer rules into account. For instance, the slow logarithmic growth of the variance in the given model is a result of the unboundedness of the soccer field. A more realistic model of the same game would be on a rectangular field where the Wiener motion is reflected from the boundary along the normal; one may obtain the central limit theorem for the score in this model by analogy with [Kozlov 1985] since this is actually Wiener soccer on a torus due to the symmetry. In such a model the variance of the score increases as fast as \( T \) and the limiting distribution turns out to be Gaussian. However there is no such explicit formula for the limiting variance as we have in the suggested model of planar Wiener soccer.

[174] J. W. Pitman and M. Yor: “Dilatations d’espace-temps, réarrangements des trajectoires browniennes, et quelques extensions d’une identité de Knight” [Spacetime dilations, rearrangements of Brownian trajectories, and some extensions of an identity of Knight], C. R. Acad. Sci., Paris, Sér. I 316 : 7 (1993), pp. 723–726. MR 1214423 Zbl 0789.60059 article

[175] M. Yor: “On an identity in law obtained by A. Földes and P. Révész,” Ann. Inst. Henri Poincaré, Probab. Stat. 29 : 2 (1993), pp. 321–324. MR 1227422 Zbl 0776.60102 article

[176] C. Donati-Martin and M. Yor: “On some examples of quadratic functionals of Brownian motion,” Adv. Appl. Probab. 25 : 3 (1993), pp. 570–584. MR 1234297 Zbl 0781.60066 article

During the last few years, several variants of P. Lévy’s formula for the stochastic area of complex Brownian motion have been obtained. These are of interest in various domains of applied probability, particularly in relation to polymer studies. The method used by most authors is the diagonalization procedure of Paul Lévy.

Here we derive one such variant of Lévy’s formula, due to Chan, Dean, Jansons and Rogers, via a change of probability method, which reduces the computation of Laplace transforms of Brownian quadratic functionals to the computations of the means and variances of some adequate Gaussian variables.

We then show that with the help of linear algebra and invariance properties of the distribution of Brownian motion, we are able to derive simply three other variants of Lévy’s formula.

[177] S. D. Jacka and M. Yor: “Inequalities for non-moderate functions of a pair of stochastic processes,” Proc. London Math. Soc. (3) 67 : 3 (November 1993), pp. 649–672. MR 1238048 Zbl 0789.60016 article

[178] M. Yor: “From planar Brownian windings to Asian options,” Insurance Math. Econom. 13 : 1 (September 1993), pp. 23–34. MR 1242683 Zbl 0792.60074 article

[179] S. Weinryb and M. Yor: “Théorème central limite pour l’intersection de deux saucisses de Wiener indépendantes” [Central limit theorem for the intersection of two independent Wiener sausages], Probab. Theory Relat. Fields 97 : 3 (September 1993), pp. 383–401. MR 1245251 Zbl 0792.60019 article

[180] P. Fitzsimmons, J. Pitman, and M. Yor: “Markovian bridges: Construction, Palm interpretation, and splicing,” pp. 101–134 in Seminar on stochastic processes, 1992 (Seattle, WA, 26–28 March 1992). Edited by E. Çinlar, K. L. Chung, and M. J. Sharpe. Progress in Probability 33. Birkhäuser (Boston, MA), 1993. MR 1278079 Zbl 0844.60054 incollection

Erhan Çinlar	Related
Kai Lai Chung	Related	Volume
P. J. Fitzsimmons	Related
James W. Pitman	Related
M. J. Sharpe	Related

[181] Séminaire de probabilités XXVII [Twenty-seventh probability seminar]. Edited by J. Azéma, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1557. Springer (Berlin), 1993. MR 1308545 Zbl 0780.00013 book

Paul-André Meyer	Related
Jacques Azéma	Related

[182] T. Jeulin and M. Yor: “Moyennes mobiles et semimartingales” [Moving averages and semimartingales], pp. 53–77 in Séminaire de probabilités XXVII [Twenty-seventh probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1557. Springer (Berlin), 1993. MR 1308553 Zbl 0788.60059 incollection

Paul-André Meyer	Related
Thierry Jeulin	Related
Jacques Azéma	Related

[183] L. E. Dubins, M. Émery, and M. Yor: “On the Lévy transformation of Brownian motions and continuous martingales,” pp. 122–132 in Séminaire de probabilités XXVII [Twenty-seventh probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1557. Springer (Berlin), 1993. A correction was published in Séminaire de probabilités XLIV (2012). MR 1308559 Zbl 0844.60055 incollection

Lester Eli Dubins	Related
Jacques Azéma	Related
Michel Emery	Related
Paul-André Meyer	Related

[184] J. Azéma, T. Jeulin, F. Knight, and M. Yor: “Le théorème d’arrêt en une fin d’ensemble prévisible” [The stopping time theorem of a predictable set], pp. 133–158 in Séminaire de probabilités XXVII [Twenty-seventh probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1557. Springer (Berlin), 1993. MR 1308560 Zbl 0798.60048 incollection

Paul-André Meyer	Related
Thierry Jeulin	Related
Jacques Azéma	Related

[185] K. D. Elworthy and M. Yor: “Conditional expectations for derivatives of certain stochastic flows,” pp. 159–172 in Séminaire de probabilités XXVII [Twenty-seventh probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1557. Springer (Berlin), 1993. MR 1308561 Zbl 0795.60046 incollection

Paul-André Meyer	Related
K. David Elworthy	Related
Jacques Azéma	Related

[186] S. M. Kozlov: “Letter to the editors: ‘Wiener soccer’,” Teor. Veroyatnost. i Primenen. 38 : 1 (1993), pp. 204. This concerns an article published in Teor. Veroyatnost. i Primenen. 37:3 (1992). MR 1317797 article

[187] H. Geman and M. Yor: “Bessel processes, Asian options, and perpetuities,” Math. Finance 3 : 4 (October 1993), pp. 349–375. Zbl 0884.90029 article

Using Bessel processes, one can solve several open problems involving the integral of an exponential of Brownian motion. This point will be illustrated with three examples. The first one is a formula for the Laplace transform of an Asian option which is “out of the money.” The second example concerns volatility misspecification in portfolio insurance strategies, when the stochastic volatility is represented by the Hull and White model. The third one is the valuation of perpetuities or annuities under stochastic interest rates within the Cox–Ingersoll–Ross framework. Moreover, without using time changes or Bessel processes, but only simple probabilistic methods, we obtain further results about Asian options: the computation of the moments of all orders of an arithmetic average of geometric Brownian motion; the property that, in contrast with most of what has been written so far, the Asian option may be more expensive than the standard option (e.g., options on currencies or oil spreads); and a simple, closed-form expression of the Asian option price when the option is “in the money,” thereby illuminating the impact on the Asian option price of the revealed underlying asset price as time goes by. This formula has an interesting resemblance with the Black–Scholes formula, even though the comparison cannot be carried too far.

[188] M. Yor: “On some exponential-integral functionals of Bessel processes,” pp. 231–240, published as Math. Finance 3 : 2 (April 1993). Also published in Exponential functionals of Brownian motion and related processes (2001). Zbl 0884.90056 incollection

[189] F. Knight: “Some aspects of Brownian motion, part 1 (Marc Yor),” SIAM Rev. 36 : 3 (September 1994), pp. 511–512. article

[190] C. Donati-Martin, S. Song, and M. Yor: “On symmetric stable random variables and matrix transposition,” Ann. Inst. Henri Poincaré, Probab. Stat. 30 : 3 (1994), pp. 397–413. MR 1288357 Zbl 0856.60020 article

[191] P. Carmona, F. Petit, and M. Yor: “Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion,” Probab. Theory Related Fields 100 : 1 (March 1994), pp. 1–29. MR 1292188 Zbl 0808.60066 article

[192] D. Revuz and M. Yor: Continuous martingales and Brownian motion, 2nd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer (Berlin), 1994. Republication of 1991 original. A 3rd edition was published in 1999. MR 1303781 Zbl 0804.60001 book

[193] Séminaire de probabilités XXVIII [Twenty-eighth probability seminar]. Edited by J. Azéma, P.-A. Meyer, and M. Yor. Lecture Notes in Mathematics 1583. Springer (Berlin), 1994. MR 1329096 Zbl 0797.00020 book

Paul-André Meyer	Related
Jacques Azéma	Related

[194] C. Donati-Martin, S. Song, and M. Yor: “Symmetric stable processes, Fubini’s theorem, and some extensions of the Ciesielski–Taylor identities in law,” Stochastics Stochastics Rep. 50 : 1–2 (1994), pp. 1–33. MR 1784742 Zbl 0831.60049 article

[195] P. Carmona, F. Petit, and M. Yor: “Sur les fonctionnelles exponentielles de certains processus de Lévy” [On the exponential functionals of certain Lévy processes], Stochastics Stochastics Rep. 47 : 1–2 (1994), pp. 71–101. MR 1787143 Zbl 0830.60072 article

[196] M. Yor: “The distribution of Brownian quantiles,” J. Appl. Probab. 32 : 2 (1995), pp. 405–416. MR 1334895 Zbl 0829.60065 article

[197] B. Rajeev and M. Yor: “Local times and almost sure convergence of semi-martingales,” Ann. Inst. Henri Poincaré, Probab. Stat. 31 : 4 (1995), pp. 653–667. MR 1355611 Zbl 0834.60047 article

[198] P. Embrechts, L. C. G. Rogers, and M. Yor: “A proof of Dassios’ representation of the \( \alpha \)-quantile of Brownian motion with drift,” Ann. Appl. Probab. 5 : 3 (1995), pp. 757–767. MR 1359828 Zbl 0844.60044 article

[199] M. Yor: “Random Brownian scaling and some absolute continuity relationships,” pp. 243–252 in Seminar on stochastic analysis, random fields and applications (Ascona, Switzerland, 7–12 June 1993). Edited by E. Bolthausen, M. Dozzi, and F. Russo. Progress in Probability 36. Birkhäuser (Basel), 1995. MR 1360280 Zbl 0827.60010 incollection

Consider \( (X_t \), \( t\geq 0) \) a one-dimensional, or \( d \)-dimensional Brownian motion, or \( d \)-dimensional Bessel process, starting from 0, and let \( a \) and \( b \) be two random times with \( a \lt b \), almost surely. One finds, in the literature, a number of examples of such random couples \( (a,b) \) such that the law of the process \[ X_t^{[a,b]}\equiv \frac{1}{\sqrt{b-a}}X_{a+t(b-a)},\qquad t\leq 1, \] which is the transform of \( X \) by random Brownian scaling on the interval \( [a,b] \), is absolutely continuous with respect to the law of \( (X_t \), \( t\leq 1) \), or the law of another interesting process \( (Y_t \), \( t\leq 1) \), with a particularly simple density. In the following notes, I shall

present a list of such results; this is done in Section 1;
attempt to unify their proofs; this is partly done in Section 2;
show how these absolute continuity results may be applied to give a deep insight into P. Lévy’s arc sine law for Brownian motion; this is done in Section 3;
develop some examples of applications to Bessel processes, in Section 4.

[200] Séminaire de probabilités XXIX [Twenty-ninth probability seminar]. Edited by J. Azéma, M. Emery, P. A. Meyer, and M. Yor. Lecture Notes in Mathematics 1613. Springer (Berlin), 1995. MR 1459442 Zbl 0826.00027 book

Paul-André Meyer	Related
Michel Emery	Related
Jacques Azéma	Related

[201] M. Yor: Local times and excursions for Brownian motion: A concise introduction. Lecciones en Matemáticas. Universidad Central de Venezuela (Caracas), 1995. book

[202] M. Yor: Local times and excursions for Brownian motion: A concise introduction. Lecciones en Matemáticas. Universidad Central de Venezuela (Caracas), 1995. book

[203] J. Bertoin and M. Yor: “Some independence results related to the arc-sine law,” J. Theoret. Probab. 9 : 2 (April 1996), pp. 447–458. MR 1385407 Zbl 0876.60060 article

[204] J. Pitman and M. Yor: “Random discrete distributions derived from self-similar random sets,” Electron. J. Probab. 1 : 4 (1996). Article no. 4, 28 pp. MR 1386296 Zbl 0891.60042 article

A model is proposed for a decreasing sequence of random variables \( (V_1 \), \( V_2 \), \( \dots) \) with \( \sum_n V_n = 1 \), which generalizes the Poisson–Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let \( V_n \) be the length of the \( n \)th longest component interval of \( [0,1]\backslash Z \), where \( Z \) is an a.s. non-empty random closed of \( (0,\infty) \) of Lebesgue measure 0, and \( Z \) is self-similar, i.e., \( cZ \) has the same distribution as \( Z \) for every \( c \geq 0 \). Then for \( 0 \leq a \lt b \leq 1 \) the expected number of \( n \)’s such that \( V_n \in (a,b) \) equals \[ \int_a^b v^{-1} F(dv) \] where the structural distribution \( F \) is identical to the distribution of \[ 1-\sup(Z\cap [0,1]) .\] Then \( F(dv) = f(v)\,dv \) where \( (1-v)f(v) \) is a decreasing function of \( v \), and every such probability distribution \( F \) on \( [0,1] \) can arise from this construction.

[205] J. W. Pitman and M. Yor: “Quelques identités en loi pour les processus de Bessel” [Some identities in law for Bessel processes], pp. 249–276 in Hommage à P. A. Meyer et J. Neveu [In homage to P. A. Meyer and J. Neveu]. Astérisque 236. Société Mathématique de France (Paris), 1996. MR 1417987 Zbl 0863.60035 incollection

James W. Pitman	Related
Paul-André Meyer	Related
Jacques E. Neveu	Related

[206] J. Pitman and M. Yor: “Decomposition at the maximum for excursions and bridges of one-dimensional diffusions,” pp. 293–310 in Itô’s stochastic calculus and probability theory. Edited by N. Ikeda. Springer (Tokyo), 1996. Book dedicated to dedicated to Kiyosi Itô on the occasion of his 80th birthday. MR 1439532 Zbl 0877.60053 incollection

In his fundamental paper [1971], Itô showed how to construct a Poisson point process of excursions of a strong Markov process \( X \) over time intervals when \( X \) is away from a recurrent point a of its statespace. The point process is parameterized by the local time process of \( X \) at \( a \). Each point of the excursion process is a path in a suitable space of possible excursions of \( X \), starting at \( a \) at time 0, and returning to \( a \) for the first time at some strictly positive time \( \zeta \), called the lifetime of the excursion. The intensity measure of the Poisson process of excursions is a \( \sigma \)-finite measure on the space of excursions, known as Itô’s excursion law. Accounts of Itô’s theory of excursions can now be found in several textbooks [Rogers and Williams 1987; Revuz and Yor 1994; Blumenthal 1992]. His theory has also been generalized to excursions of Markov processes away from a set of states [Maisonneuve 1975; Getoor and Sharpe 1982; Blumenthal 1992] and to excursions of stationary, not necessarily Markovian processes [Pitman 1986].

James W. Pitman	Related
Nobuyuki Ikeda	Related
Kiyoshi Itō	Related

[207] Séminaire de probabilités XXX [Thirtieth probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1626. Springer (Berlin), 1996. MR 1459471 Zbl 0840.00041 book

Michel Emery	Related
Jacques Azéma	Related

[208] J. Azéma, C. Rainer, and M. Yor: “Une propriété des martingales pures” [A property of pure martingales], pp. 243–254 in Séminaire de probabilités XXX [Thirtieth probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1626. Springer (Berlin), 1996. MR 1459487 Zbl 0857.60076 incollection

Michel Emery	Related
Jacques Azéma	Related

[209] J. Azéma, T. Jeulin, F. Knight, G. Mokobodzki, and M. Yor: “Sur les processus croissants de type injectif” [On growing processes which are injective], pp. 312–343 in Séminaire de probabilités XXX [Thirtieth probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1626. Springer (Berlin), 1996. MR 1459491 Zbl 0859.60035 incollection

Michel Emery	Related
Thierry Jeulin	Related
Jacques Azéma	Related

[210] Séminaire de probabilités XXX [Thirtieth probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1626. Springer (Berlin), 1996. MR 1478710 Zbl 0864.00069 book

Michel Emery	Related
Jacques Azéma	Related

[211] H. Geman and M. Yor: “Pricing and hedging double-barrier options: A probabilistic approach,” Math. Finance 6 : 4 (October 1996), pp. 365–378. Zbl 0915.90016 article

Barrier options have become increasingly popular over the last few years. Less expensive than standard options, they may provide the appropriate hedge in a number of risk management strategies. In the case of a single-barrier option, the valuation problem is not very difficult (see [Merton 1973] and [Goldman et al. 1979]). the situation where the option gets knocked out when the underlying instrument hits either of two well-defined boundaries is less straightforward. Kunitomo and Ikeda [1992] provide a pricing formula expressed as the sum of an infinite series whose convergence is studied through numerical procedures and suggested to be rapid. We follow a methodology which proved quite successful in the case of Asian options (see Geman and Yor [1992, 1993]) and which has its roots in some fundamental properties of Brownian motion. This methodology permits the derivation of a simple expression of the Laplace transform of the double-barrir price with respect to its maturity date. the inversion of the Laplace transform using techniques developed by Geman and Eydeland [1995], is then fairly easy to perform.

[212] Z. Shi and M. Yor: “On an identity in law for the variance of the Brownian bridge,” Bull. London Math. Soc. 29 : 1 (1997), pp. 103–108. MR 1416415 Zbl 0956.60085 article

[213] M. Chesney, M. Jeanblanc-Picqué, and M. Yor: “Brownian excursions and Parisian barrier options,” Adv. Appl. Probab. 29 : 1 (March 1997), pp. 165–184. MR 1432935 Zbl 0882.60042 article

In this paper we study a new kind of option, called hereinafter a Parisian barrier option. This option is the following variant of the so-called barrier option: a down-and-out barrier option becomes worthless as soon as a barrier is reached, whereas a down-and-out Parisian barrier option is lost by the owner if the underlying asset reaches a prespecified level and remains constantly below this level for a time interval longer than a fixed number, called the window. Properties of durations of Brownian excursions play an essential role. We also study another kind of option, called here a cumulative Parisian option, which becomes worthless if the total time spent below a certain level is too long.

[214] J. Pitman and M. Yor: “The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator,” Ann. Probab. 25 : 2 (1997), pp. 855–900. MR 1434129 Zbl 0880.60076 article

The two-parameter Poisson-Dirichlet distribution, denoted \( \textrm{PD}(\alpha,\theta) \) is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson–Dirichlet distribution with a single parameter \( \theta \), introduced by Kingman, is \( \textrm{PD}(0,\theta) \). Known properties of \( \textrm{PD}(0,\theta) \), including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of \( \textrm{PD}(\alpha,\theta) \) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For \( 0\lt \alpha\lt 1 \), \( \textrm{PD}(\alpha,0) \) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index \( \alpha \). Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is \( \textrm{PD}(1/2,0) \), and the corresponding distribution for a Brownian bredge is \( \textrm{PD}(1/2,1/2) \). The \( \textrm{PD}(\alpha,0) \) and \( \textrm{PD}(\alpha,\alpha) \) distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index \( \alpha \).

[215] M. Yor: Some aspects of Brownian motion, part 2: Some recent martingale problems. Lectures in Mathematics ETH Zürich. Birkhäuser (Basel), 1997. MR 1442263 Zbl 0880.60082 book

[216] Z. Shi and M. Yor: “Integrability and lower limits of the local time of iterated Brownian motion,” Studia Sci. Math. Hung. 33 : 1–3 (1997), pp. 279–298. Dedicated to Endre Csáki on his sixtieth birthday. MR 1454115 Zbl 0909.60068 article

We study the extraordinarily large and small values of the local time of iterated Brownian motion. It is known that the local time has exponential moments for deterministic times. We prove that, taken at appropriate random times, the local time has finite \( \alpha \)-th moment if and only if \( \alpha \lt 2/3 \). We also investigate the almost sure lower asymptotics of both the local time at a fixed level and the maximum local time. The critical rate functions for these two processes are obtained, which improves previous results of Csáki et al. [1996]. Our approach essentially relies on Ray–Knight theorems and the general theory of stochastic calculus, notably some refined martingale inequalities.

[217] M. Yor: “Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer’s asymptotic result on Brownian windings,” Studia Sci. Math. Hung. 33 : 1–3 (1997), pp. 339–343. Dedicated to Professor E. Csáki on his sixtieth birthday. MR 1454119 Zbl 0909.60070 article

[218] C. Donati-Martin and M. Yor: “Some Brownian functionals and their laws,” Ann. Probab. 25 : 3 (1997), pp. 1011–1058. MR 1457611 Zbl 0885.60072 article

[219] M. Jeanblanc, J. Pitman, and M. Yor: “The Feynman–Kac formula and decomposition of Brownian paths,” Comput. Appl. Math. 16 : 1 (1997), pp. 27–52. MR 1458521 Zbl 0877.60027 article

[220] K. D. Elworthy, X. M. Li, and M. Yor: “On the tails of the supremum and the quadratic variation of strictly local martingales,” pp. 113–125 in Séminaire de probabilités XXXI [Thirty-first probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1655. Springer (Berlin), 1997. MR 1478722 Zbl 0886.60035 incollection

Michel Emery	Related
K. David Elworthy	Related
Jacques Azéma	Related

[221] J. Pitman and M. Yor: “On the lengths of excursions of some Markov processes,” pp. 272–286 in Séminaire de probabilités XXXI [Thirty-first probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1655. Springer (Berlin), 1997. MR 1478737 Zbl 0884.60071 incollection

Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Lévy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azéma and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transoformation. In particular, results for the zero set of an Ornstein–Uhlenbeck process or a Cox–Ingersoll–Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.

James W. Pitman	Related
Michel Emery	Related
Jacques Azéma	Related

[222] J. Pitman and M. Yor: “On the relative lengths of excursions derived from a stable subordinator,” pp. 287–305 in Séminaire de probabilités XXXI [Thirty-first probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1655. Springer (Berlin), 1997. MR 1478738 Zbl 0884.60072 incollection

James W. Pitman	Related
Michel Emery	Related
Jacques Azéma	Related

@incollection {key1478738m,
	AUTHOR = {Pitman, Jim and Yor, Marc},
	TITLE = {On the relative lengths of excursions
		 derived from a stable subordinator},
	BOOKTITLE = {S\'eminaire de probabilit\'es {XXXI}
		 [Thirty-first probability seminar]},
	EDITOR = {Az\'ema, J. and Emery, M. and Yor, M.},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1655},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {1997},
	PAGES = {287--305},
	DOI = {Results are obtained concerning the
		 distribution of ranked relative lengths
		 of excursions of a recurrent Markov
		 process from a point in its state space
		 whose inverse local time process is
		 a stable subordinator. It is shown that
		 for a large class of random times \$T\$
		 the distribution of relative excursion
		 lengths prior to \$T\$ is the same as
		 if \$T\$ were a fixed time. It follows
		 that the generalized arc-sine laws of
		 Lamperti extend to such random times
		 \$T\$. For some other random times \$T\$,
		 absolute continuity relations are obtained
		 which relate the law of the relative
		 lengths at time \$T\$ to the law at a
		 fixed time.},
	URL = {http://www.numdam.org/item?id=SPS_1997__31__287_0},
	NOTE = {MR:1478738. Zbl:0884.60072.},
	ISSN = {0075-8434},
	ISBN = {9783540626343},
}

[223] M. Yor: “Some remarks about the joint law of Brownian motion and its supremum,” pp. 306–314 in Séminaire de probabilités XXXI [Thirty-first probability seminar]. Edited by J. Azéma, M. Emery, and M. Yor. Lecture Notes in Mathematics 1655. Springer (Berlin), 1997. MR 1478739 Zbl 0885.60071 incollection

Michel Emery	Related
Jacques Azéma	Related

[224] M. Yor: “On certain discounted arc-sine laws,” Stochastic Process. Appl. 71 : 1 (October 1997), pp. 111–122. MR 1480642 Zbl 0943.60069 article

[225] J. Bertoin, L. Chaumont, and M. Yor: “Two chain-transformations and their applications to quantiles,” J. Appl. Probab. 34 : 4 (1997), pp. 882–897. MR 1484022 Zbl 0904.60059 article

[226] M. Yor, M. Chesney, H. Geman, and M. Jeanblanc-Picqué: “Some combinations of Asian, Parisian and barrier options,” pp. 61–87 in Mathematics of derivative securities (Cambridge, UK, January–June 1995). Edited by M. Dempster and S. R. Pleska. Publications of the Newton Institute 15. Cambridge University Press, 1997. MR 1491368 Zbl 0911.90036 incollection

[227] H. Geman and M. Yor: “Stochastic time changes in catastrophe option pricing,” Insurance Math. Econom. 21 : 3 (December 1997), pp. 185–193. MR 1614517 Zbl 0894.90046 article

Catastrophe insurance derivatives (Futures and options) were introduced in December 1992 by the Chicago Board of Trade in order to offer insurers new ways of hedging their underwriting risk. Only CAT options and combinations of options such as call spreads are traded today, and the ISO index has been replaced by the PCS index. Otherwise, the economic goal of these instruments continues to be for insurers an alternative to reinsurance and for portfolio managers a new class of assets to invest in.

The pricing methodology of these derivatives relies on some crucial elements:

the choice of the stochastic modelling of the aggregate reported claim index dynamics (since the terminal value of this index defines the pay-off of the CAT options);
the decision of a financial versus actuarial approach to the valuation;
the number of sources of randomness in the model and the determination of a “martingale measure” for insurance and reinsurance instruments.

We represent in this paper the dynamics of the aggregate claim index by the sum of a geometric Brownian motion which accounts for the randomness in the reporting of the claims and a Poisson process which accounts for the occurrence of catastrophes (only catastrophic claims are incorporated in the index). Geman [1994] and Cummins and Geman [1995] took this modelling for the instantaneous claim process. Our choice here is closer to the classical actuarial representation while preserving the quasi-completeness of insurance derivative markets obtained by applying the Delbaen and Haezendonck [1989] methodology to the class of layers of reinsurance replicating the call spreads. Moreover, we obtain semi-analytical solutions for the CAT options and call spreads by extending to the jump-diffusion case the method of the Laplace transform and stochastic time changes introduced in Geman and Yor [1993, 1996] in order to price financial path-dependent options through the properties of excursion theory.

[228] Exponential functionals and principal values related to Brownian motion. Edited by M. Yor. Biblioteca de la Revista Matemática Iberoamericana. Universidad Autónoma de Madrid, 1997. MR 1648653 Zbl 0889.00015 book

[229] L. Alili, D. Dufresne, and M. Yor: “Sur l’identité de Bougerol pour les fonctionnelles exponentielles du mouvement brownien avec drift” [On the Bougerol identity for the exponential functionals of Brownian motion with drift], pp. 3–14 in Exponential functionals and principal values related to Brownian motion. Edited by M. Yor. Biblioteca de la Revista Matemática Iberoamericana. Universidad Autónoma de Madrid, 1997. MR 1648654 Zbl 0905.60059 incollection

[230] P. Carmona, F. Petit, and M. Yor: “On the distribution and asymptotic results for exponential functionals of Lévy processes,” pp. 73–130 in Exponential functionals and principal values related to Brownian motion. Edited by M. Yor. Biblioteca de la Revista Matemática Iberoamericana. Universidad Autónoma de Madrid, 1997. MR 1648657 Zbl 0905.60056 incollection

The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional \[ A_t := \int_0^t e^{\xi_s}\,ds ,\] where \( (\xi_s \), \( s\geq 0) \) denotes a Lévy process. When \( A_{\infty} \lt \infty \), we show that in most cases, the law of \( A_{\infty} \) is a solution of an integro-differential equation; moreover, this law is characterized by its integral moments. When the process \( \xi \) is asymptotically \( \alpha \)-stable, we prove that \( t^{-1/\alpha}\log A_t \) converges in law, as \( t\to\infty \), to the supremum of an \( \alpha \)-stable Lévy process; in particular, if \( \mathbb{E}[\xi_1] \gt 0 \), then \( \alpha = 1 \) and \( (1/t)\log A_t \) converges almost surely to \( \mathbb{E}[\xi_1] \). Eventually, we use Girsanov’s transform to give the explicit behavior of \[ \mathbb{E}[(a + A_t(\xi))^{-1}] \] as \( t\to\infty \), where \( a \) is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random Lévy environment.

[231] L. Alili, C. Donati-Martin, and M. Yor: “Une identité en loi remarquable pour l’excursion brownienne normalisée” [A remarkable identity in law for the normalised Brownian excursion], pp. 155–180 in Exponential functionals and principal values related to Brownian motion. Edited by M. Yor. Biblioteca de la Revista Matemática Iberoamericana. Universidad Autónoma de Madrid, 1997. MR 1648659 Zbl 0916.60072 incollection

[232] Y. Hu, Z. Shi, and M. Yor: “Some applications of Lévy’s area formula to pseudo-Brownian and pseudo-Bessel bridges,” pp. 181–209 in Exponential functionals and principal values related to Brownian motion. Biblioteca de la Revista Matemática Iberoamericana. Universidad Autónoma de Madrid, 1997. MR 1648660 Zbl 0911.60072 incollection

[233] J. Warren and M. Yor: Skew-products involving Bessel and Jacobi processes. Technical report, Statistics Group, University of Bath, 1997. techreport

[234] P. Chassaing, J. Marckert, and M. Yor: Stochastic properties of two algorithms searching the zeros of simple random walks, and applications, June 1998. unpublished

[235] A. Comtet, C. Monthus, and M. Yor: “Exponential functionals of Brownian motion and disordered systems,” J. Appl. Probab. 35 : 2 (June 1998), pp. 255–271. MR 1641852 Zbl 0929.60063 ArXiv cond-mat/9601014 article

[236] R. A. Doney and M. Yor: “On a formula of Takács for Brownian motion with drift,” J. Appl. Probab. 35 : 2 (June 1998), pp. 272–280. MR 1641856 Zbl 0914.60044 article

[237] J. Pitman and M. Yor: “Ranked functionals of Brownian excursions,” C. R. Acad. Sci., Paris, Sér. I 326 : 1 (1998), pp. 93–97. With French summary. MR 1649517 Zbl 0924.60073 article

[238] Séminaire de probabilités XXXII [Thirty-second probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1651223 Zbl 0893.00035 book

Michel Emery	Related
Jacques Azéma	Related

[239] P. Carmona, F. Petit, and M. Yor: “Beta-gamma random variables and intertwining relations between certain Markov processes,” Rev. Mat. Iberoamericana 14 : 2 (1998), pp. 311–367. MR 1654531 Zbl 0919.60074 article

[240] R. A. Doney, J. Warren, and M. Yor: “Perturbed Bessel processes,” pp. 237–249 in Séminaire de probabilités XXXII [Thirty-second probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1655297 Zbl 0924.60039 incollection

There has been some interest in the literature in Brownian motion perturbed at its maximum; that is a process \( (X_t \); \( t\geq 0) \) satisfying \[ X_t = B_t + \alpha M_t^X, \] where \[ M_t^X = \sup_{0\leq s\leq t} X_s ,\] and \( (B_t \); \( t\geq 0) \) is Brownian motion issuing from zero. The parameter \( \alpha \) must satisfy \( \alpha \lt 1 \). For example arc-sine laws and Ray–Knight theorems have been obtained for this process; see [Carmona et al. 1994; Werner 1995; Doney 1998]. Our initial aim was to identify a process which could be considered as the process \( X \) conditioned to stay positive. This new process behaves like the Bessel process of dimension three except when at its maximum and we call it a perturbed three-dimensional Bessel process. We establish Ray–Knight theorems for the local times of this process, up to a first passage time and up to infinity, and observe that these descriptions coincide with those of the local times of two processes that have been considered in [Yor 1992]. We give an explanation for this coincidence by showing that these processes are linked to the perturbed three dimensional Bessel process by space-time transformations and time-reversal.

Michel Emery	Related
Jacques Azéma	Related

[241] M. T. Barlow, M. Émery, F. B. Knight, S. Song, and M. Yor: “Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes” [On a theorem of Tsirelson concerning Brownian and non-Brownian filtrations], pp. 264–305 in Séminaire de probabilités XXXII [Thirty-second probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1655299 Zbl 0914.60064 incollection

Michel Emery	Related
Martin Thomas Barlow	Related
Jacques Azéma	Related

@incollection {key1655299m,
	AUTHOR = {Barlow, M. T. and \'Emery, M. and Knight,
		 F. B. and Song, S. and Yor, M.},
	TITLE = {Autour d'un th\'eor\`eme de {T}sirelson
		 sur des filtrations browniennes et non
		 browniennes [On a theorem of {T}sirelson
		 concerning {B}rownian and non-{B}rownian
		 filtrations]},
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		 [Thirty-second probability seminar]},
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	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1686},
	PUBLISHER = {Springer},
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	NOTE = {MR:1655299. Zbl:0914.60064.},
	ISSN = {0075-8434},
	ISBN = {9783540643760},
}

[242] M. Émery and M. Yor: “Sur un théorème de Tsirelson relatif à des mouvements browniens corrélés et à la nullité de certains temps locaux” [On a theorem of Tsirelson with respect to correlated Brownian motion and the nullity of certain local times], pp. 306–312 in Séminaire de probabilités XXXII [Thirty-second probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1655300 Zbl 0949.60088 incollection

Michel Emery	Related
Jacques Azéma	Related

@incollection {key1655300m,
	AUTHOR = {\'Emery, M. and Yor, M.},
	TITLE = {Sur un th\'eor\`eme de {T}sirelson relatif
		 \`a des mouvements browniens corr\'el\'es
		 et \`a la nullit\'e de certains temps
		 locaux [On a theorem of {T}sirelson
		 with respect to correlated {B}rownian
		 motion and the nullity of certain local
		 times]},
	BOOKTITLE = {S\'eminaire de probabilit\'es {XXXII}
		 [Thirty-second probability seminar]},
	EDITOR = {Az\'ema, J. and \'Emery, M. and Ledoux,
		 M. and Yor, M.},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1686},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
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	PAGES = {306--312},
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	NOTE = {MR:1655300. Zbl:0949.60088.},
	ISSN = {0075-8434},
	ISBN = {9783540643760},
}

[243] J. Azéma, T. Jeulin, F. Knight, and M. Yor: “Quelques calculs de compensateurs impliquant l’injectivité de certains processus croissants” [Some computations including the injectivity of certain increasing processes], pp. 316–327 in Séminaire de probabilités XXXII [Thirty-second probability seminar], vol. 1686. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1655302 Zbl 0915.60058 incollection

Michel Emery	Related
Thierry Jeulin	Related
Jacques Azéma	Related

@incollection {key1655302m,
	AUTHOR = {Az\'ema, J. and Jeulin, T. and Knight,
		 F. and Yor, M.},
	TITLE = {Quelques calculs de compensateurs impliquant
		 l'injectivit\'e de certains processus
		 croissants [Some computations including
		 the injectivity of certain increasing
		 processes]},
	BOOKTITLE = {S\'eminaire de probabilit\'es {XXXII}
		 [Thirty-second probability seminar]},
	EDITOR = {Az\'ema, J. and \'Emery, M. and Ledoux,
		 M. and Yor, M.},
	VOLUME = {1686},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1686},
	PUBLISHER = {Springer},
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	NOTE = {MR:1655302. Zbl:0915.60058.},
	ISSN = {0075-8434},
	ISBN = {9783540643760},
}

[244] J. Warren and M. Yor: “The Brownian burglar: Conditioning Brownian motion by its local time process,” pp. 328–342 in Séminaire de probabilités XXXII [Thirty-second probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1686. Springer (Berlin), 1998. MR 1655303 Zbl 0924.60072 incollection

Michel Emery	Related
Jacques Azéma	Related

[245] S. Sato and M. Yor: “Computations of moments for discounted Brownian additive functionals,” J. Math. Kyoto Univ. 38 : 3 (1998), pp. 475–486. MR 1661228 Zbl 0943.60068 article

[246] Y. Hu and M. Yor: “Convergence in law and convergence of moments: An example related to Bessel processes,” pp. 387–397 in Asymptotic methods in probability and statistics (Ottawa, ON, 8–13 July 1997). Edited by B. Szyszkowicz. North-Holland/Elsevier (Amsterdam), 1998. A volume in honour of Miklós Csörgő. MR 1661495 Zbl 0933.60021 incollection

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

[247] P. Carmona, F. Petit, and M. Yor: “Beta variables as times spent in \( [0,\infty[ \) by certain perturbed Brownian motions,” J. Lond. Math. Soc., II. Ser. 58 : 1 (August 1998), pp. 239–256. MR 1670130 Zbl 0924.60067 article

[248] J. Pitman and M. Yor: “Random Brownian scaling identities and splicing of Bessel processes,” Ann. Probab. 26 : 4 (1998), pp. 1683–1702. MR 1675059 Zbl 0937.60079 article

An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

[249] H. Matsumoto and M. Yor: “On Bougerol and Dufresne’s identities for exponential Brownian functionals,” Proc. Japan Acad. Ser. A Math. Sci. 74 : 10 (1998), pp. 152–155. MR 1675456 Zbl 0942.60063 article

[250] B. Leblanc and M. Yor: “Lévy processes in finance: A remedy to the non-stationarity of continuous martingales,” Finance Stoch. 2 : 4 (August 1998), pp. 399–408. MR 1809527 Zbl 0909.90025 article

[251] J. Pitman and M. Yor: Laws of homogeneous functionals of Brownian motion, 1998. unpublished

[252] Y. Hu, Z. Shi, and M. Yor: “Rates of convergence of diffusions with drifted Brownian potentials,” Trans. Am. Math. Soc. 351 : 10 (1999), pp. 3915–3934. MR 1637078 Zbl 0932.60083 article

[253] P. Carmona, F. Petit, and M. Yor: “An identity in law involving reflecting Brownian motion, derived from generalized arc-sine laws for perturbed Brownian motions,” Stochastic Processes Appl. 79 : 2 (February 1999), pp. 323–333. MR 1671824 Zbl 0965.60074 article

[254] J. Pitman and M. Yor: “Laplace transforms related to excursions of a one-dimensional diffusion,” Bernoulli 5 : 2 (1999), pp. 249–255. MR 1681697 Zbl 0921.60015 article

[255] M. Balazard, E. Saias, and M. Yor: “Notes sur la fonction \( \zeta \) de Riemann, 2” [Notes on the Riemann \( \zeta \)-function, 2], Adv. Math. 143 : 2 (May 1999), pp. 284–287. MR 1686420 Zbl 0937.11032 article

[256] H. Matsumoto and M. Yor: “A version of Pitman’s \( 2M{-}X \) theorem for geometric Brownian motions,” C. R. Acad. Sci., Paris, Sér. I 328 : 11 (June 1999), pp. 1067–1074. With French summary. MR 1696208 Zbl 0936.60076 article

[257] C. Donati-Martin and M. Yor: “Intégrales stochastiques de processus anticipants et projections duales prévisibles” [Stochastic integrals of anticipating processes and predictable dual projections], Publ. Mat. 43 : 1 (1999), pp. 281–301. MR 1697526 Zbl 0936.60051 article

[258] J. Pitman and M. Yor: “The law of the maximum of a Bessel bridge,” Electron. J. Probab. 4 (1999). Article no. 15, 35 pp. MR 1701890 Zbl 0943.60084 article

[259] M. Gradinaru, B. Roynette, P. Vallois, and M. Yor: “Abel transform and integrals of Bessel local times,” Ann. Inst. Henri Poincaré, Probab. Stat. 35 : 4 (July 1999), pp. 531–572. MR 1702241 Zbl 0937.60080 article

[260] H. Matsumoto and M. Yor: “Some changes of probabilities related to a geometric Brownian motion version of Pitman’s \( 2M{-}X \) theorem,” Electron. Commun. Probab. 4 (1999), pp. 15–23. MR 1703607 Zbl 0929.60031 article

[261] H. Föllmer, C.-T. Wu, and M. Yor: “Canonical decomposition of linear transformations of two independent Brownian motions motivated by models of insider trading,” Stochastic Processes Appl. 84 : 1 (November 1999), pp. 137–164. MR 1720102 Zbl 0996.60079 article

[262] D. Revuz and M. Yor: Continuous martingales and Brownian motion, 3rd edition. Grundlehren der Mathematischen Wissenschaften 293. Springer (Berlin), 1999. Republication of 1991 original. A 2nd edition was published in 1994. MR 1725357 Zbl 0917.60006 book

[263] K. D. Elworthy, X.-M. Li, and M. Yor: “The importance of strictly local martingales: Applications to radial Ornstein–Uhlenbeck processes,” Probab. Theory Relat. Fields 115 : 3 (1999), pp. 325–355. MR 1725406 Zbl 0960.60046 article

[264] M. Csörgő, Z. Shi, and M. Yor: “Some asymptotic properties of the local time of the uniform empirical process,” Bernoulli 5 : 6 (1999), pp. 1035–1058. MR 1735784 Zbl 0960.60023 article

[265] R. Douady, M. Yor, and A. N. Shiryaev: “On the probability characteristics of ‘drop’ variables in standard Brownian motion,” Teor. Veroyatnost. i Primenen. 44 : 1 (1999), pp. 3–13. An English translation was published in Theory Probab. Appl. 44:1 (1999). MR 1751185 article

[266] Y. Hu and M. Yor: “Asymptotic studies of Brownian functionals,” pp. 187–217 in Random walks (Budapest, 13–24 July 1998). Edited by P. Révész and T. Bálint. Bolyai Society Mathematical Studies 9. János Bolyai Mathematical Society (Budapest), 1999. MR 1752895 Zbl 0973.60084 incollection

[267] J. Pitman and M. Yor: “Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude,” Studia Sci. Math. Hung. 35 : 3–4 (1999), pp. 457–474. MR 1761927 Zbl 0973.60082 article

[268] P. Carmona, F. Petit, J. Pitman, and M. Yor: “On the laws of homogeneous functionals of the Brownian bridge,” Studia Sci. Math. Hung. 35 : 3–4 (1999), pp. 445–455. MR 1762255 Zbl 0980.60099 article

[269] Séminaire de probabilités XXXIII [Thirty-third probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1709. Springer (Berlin), 1999. MR 1768019 Zbl 0924.00016 book

Michel Emery	Related
Jacques Azéma	Related

[270] M. Gradinaru, B. Roynette, P. Vallois, and M. Yor: “The laws of Brownian local time integrals,” Comput. Appl. Math. 18 : 3 (1999), pp. 259–331. MR 1993869 Zbl 1123.60065 article

[271] N. Tsilevich, A. Vershik, and M. Yor: Quasi-invariance of the gamma process and multiplicative properties of the Poisson–Dirichlet measures. Preprint, 1999. techreport

In this paper we describe new fundamental properties of the law \( P_{\Gamma} \) of the classical gamma process and related properties of the Poisson–Dirichlet measures \( PD(\theta) \). We prove the quasi-invariance of the measure \( P_{\Gamma} \) with respect to an infinite-dimensional multiplicative group and the Markov–Krein identity as corollaries of the formula for the Laplace transform of \( P_{\Gamma} \). The quasi-invariance of the measure \( P_{\Gamma} \) allows us to obtain new quasi-invariance properties of the measure \( PD(\theta) \). The corresponding invariance properties hold for \( \sigma \)-finite analogues of \( P_{\Gamma} \) and \( PD(\theta) \). We also show that the measure \( P_{\Gamma} \) can be considered as a limit of measures corresponding to the \( \alpha \)-stable Lévy processes when parameter \( \alpha \) tends to zero. Our approach is based on simultaneous considering the gamma process (especially its Laplace transform) and its simplicial part–the Poisson–Dirichlet measures.

[272] R. Douady, A. N. Shiryaev, and M. Yor: “On probability characteristics of ‘downfalls’ in a standard Brownian motion,” Theory Probab. Appl. 44 : 1 (1999), pp. 29–38. Original Russian version was published in Teor. Veroyatnost. i Primenen. 44:1 (1999). Zbl 0959.60073 article

For a Brownian motion \( B=(B_t)_{t\leq 1} \) with \( B_0=0 \), \( \textbf{\textrm{E}}B_t=0 \), \( \textbf{\textrm{E}}B_t^2=t \) problems of probability distributions and their characteristics are considered for the variables \begin{align*} & \mathbb{D} =\sup_{0\leq t\leq t^{\prime}\leq 1}(B_t-B_{t^{\prime}}),\\ & \mathbb{D}_1 =B_\sigma-\inf_{\sigma\leq t^{\prime}\leq 1}B_{t^{\prime}}, \\ & \mathbb{D}_2 =\sup_{0\leq t\leq\sigma^{\prime}}B_{t}-B_{\sigma^{\prime}}, \end{align*} where \( \sigma \) and \( \sigma^{\prime} \) are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on \( [0,1] \), i.e., \[ B_{\sigma}=\sup_{0\leq t\leq 1}B_t, \quad B_{\sigma^{\prime}}=\inf_{0\leq t^{\prime}\leq 1}B_{t^{\prime}} .\]

[273] C. Donati-Martin, Z. Shi, and M. Yor: “The joint law of the last zeros of Brownian motion and of its Lévy transform,” Ergodic Theory Dyn. Syst. 20 : 3 (2000), pp. 709–725. MR 1764924 Zbl 0965.60080 article

[274] L. Vostrikova and M. Yor: “Some invariance properties (of the laws) of Ocone’s martingales,” pp. 417–431 in Séminaire de probabilités XXXIV [Thirty-fourth probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1729. Springer (Berlin), 2000. MR 1768078 Zbl 0965.60047 incollection

Michel Emery	Related
Jacques Azéma	Related

[275] Séminaire de probabilités XXXIV [Thirty-fourth probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1729. Springer (Berlin), 2000. MR 1768089 Zbl 0940.00007 book

Michel Emery	Related
Jacques Azéma	Related

[276] C. Donati-Martin and M. Yor: “Some measure-valued Markov processes attached to occupation times of Brownian motion,” Bernoulli 6 : 1 (February 2000), pp. 63–72. MR 1781182 Zbl 0956.60086 article

[277] H. Matsumoto and M. Yor: “An analogue of Pitman’s \( 2M{-}X \) theorem for exponential Wiener functionals, I: A time-inversion approach,” Nagoya Math. J. 159 (2000), pp. 125–166. MR 1783567 Zbl 0963.60076 article

[278] H. Föllmer, C.-T. Wu, and M. Yor: “On weak Brownian motions of arbitrary order,” Ann. Inst. Henri Poincaré, Probab. Stat. 36 : 4 (2000), pp. 447–487. MR 1785391 Zbl 0968.60069 article

[279] C. Donati-Martin, H. Matsumoto, and M. Yor: “On positive and negative moments of the integral of geometric Brownian motions,” Statist. Probab. Lett. 49 : 1 (August 2000), pp. 45–52. MR 1789663 Zbl 0974.60069 article

[280] M. Yor: “Le mouvement brownien: Quelques développements de 1950 à 1995” [Brownian motion: Some developments from 1950 to 1995], pp. 1187–1202 in Development of mathematics 1950–2000. Edited by J.-P. Pier. Birkhäuser (Basel), 2000. MR 1796872 Zbl 0968.60007 incollection

[281] P. Chassaing, J. F. Marckert, and M. Yor: “The height and width of simple trees,” pp. 17–30 in Mathematics and computer science: Algorithms, trees, combinatorics and probabilities (Versailles, France, 18–20 September 2000), vol. 1. Edited by D. Gardy and A. Mokkadem. Trends in Mathematics. Birkhäuser (Basel), 2000. MR 1798284 Zbl 0965.68067 incollection

[282] C. Donati-Martin, R. Ghomrasni, and M. Yor: “Affine random equations and the stable \( (1/2) \) distribution,” Studia Sci. Math. Hung. 36 : 3–4 (2000), pp. 387–405. MR 1798746 Zbl 0980.60023 article

[283] R. J. Elliott, M. Jeanblanc, and M. Yor: “On models of default risk,” pp. 179–195 in Tenth INFORMS applied probability conference (Ulm, Germany, 26–28 July 1999), published as Math. Finance 10 : 2. Issue edited by W. J. Runggaldier. Wiley-Blackwell (Hoboken, NJ), 2000. MR 1802597 Zbl 1042.91038 incollection

[284] C. Donati-Martin, H. Matsumoto, and M. Yor: “On striking identities about the exponential functionals of the Brownian bridge and Brownian motion,” Period. Math. Hung. 41 : 1–2 (November 2000), pp. 103–119. Dedicated to Professor Endre Csáki on the occasion of his 65th birthday. MR 1812799 Zbl 1062.60080 article

[285] G. Pap and M. Yor: “The accuracy of Cauchy approximation for the windings of planar Brownian motion,” Period. Math. Hung. 41 : 1–2 (November 2000), pp. 213–226. Dedicated to Professor Endre Csáki on the occasion of his 65th birthday. MR 1812807 Zbl 1074.60507 article

[286] M. Yor: “Présentation du pli cacheté” [Presentation of the sealed envelope], C. R. Acad. Sci., Paris, Sér. I 331 : 12, part 2 (special issue) (2000), pp. 1033–1035. Part of the multi-author piece “Sur l’équation de Kolmogoroff, par W. Doeblin”. This forms the first part of a three-author item. MR 1835814 Zbl 1035.01517 article

[287] B. Bru, W. Doeblin, and M. Yor: “Sur l’équation de Kolmogoroff, par W. Doeblin” [On Kolmogoroff’s equation, for W. Doeblin], C. R. Acad. Sci., Paris, Sér. I 331 : 12, part 2 (special issue) (2000), pp. i–ii, 1033–1187. The first part of this is a piece authored by Yor alone. MR 1835815 Zbl 0973.00016 article

Wolfgang Doeblin	Related
Bernard Bru	Related

[288] N. Tsilevich, A. Vershik, and M. Yor: Distinguished properties of the gamma process and related topics. Preprint 575, March 2000. ArXiv math.PR/0005287 techreport

We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasi-invariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson–Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the Poisson–Dirichlet measures.

[289] H. Geman and M. Yor: “Some relations between Bessel processes, Asian options and confluent hypergeometric functions,” pp. 49–54 in M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. English translation of an article published in C. R. Acad. Sci., Paris, Sér. I 314:6 (1992). incollection

[290] H. Geman, D. B. Madan, and M. Yor: “Time changes for Lévy processes,” Math. Finance 11 : 1 (2001), pp. 79–96. MR 1807849 Zbl 0983.60082 article

Hélyette Geman	Related
Dilip B. Madan	Related

[291] J. Pitman and M. Yor: “On the distribution of ranked heights of excursions of a Brownian bridge,” Ann. Probab. 29 : 1 (2001), pp. 361–384. MR 1825154 Zbl 1033.60050 article

The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge \( (B^{\mathrm{br}}_t \), \( 0\leq t\leq 1) \) is described. The height \( M^{\mathrm{br}+}_j \) of the \( j \)-th highest maximum over a positive excursion of the bridge has the same distribution as \( M^{\mathrm{br}+1}_1/j \), where the distribution of \[ M^{\mathrm{br}+1}_1=\sup_{0\lt t\lt 1}B^{\mathrm{br}}_t \] is given by Lévy’s formula \[ P(M^{\mathrm{br}+}_1 \gt x) = e^{-2x^2} .\] The probability density of the height \( M^{\mathrm{br}}_j \) of the \( j \)th highest maximum of excursions of the reflecting Brownian bridge \( (|B^{\mathrm{br}}_t| \), \( 0\leq t\leq 1) \) is given by a modification of the known \( \theta \)-function series for the density of \[ M^{\mathrm{br}}_1 = \sup_{0\leq t\leq 1}|B^{\mathrm{br}}_t| .\] These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process.

[292] M. Yor and M. Zani: “Large deviations for the Bessel clock,” Bernoulli 7 : 2 (April 2001), pp. 351–362. MR 1828510 Zbl 0993.60082 article

[293] R. Pemantle, Y. Peres, J. Pitman, and M. Yor: “Where did the Brownian particle go?,” Electron. J. Probab. 6 (2001). Article no. 10, 22 pp. MR 1831805 Zbl 0977.60071 ArXiv math/0404097 article

Consider the radial projection onto the unit sphere of the path a \( d \)-dimensional Brownian motion \( W \), started at the center of the sphere and run for unit time. Given the occupation measure \( \mu \) of this projected path, what can be said about the terminal point \( W(1) \), or about the range of the original path? In any dimension, for each Borel set \( A \) in \( S^{d-1} \), the conditional probability that the projection of \( W(1) \) is in \( A \) given \( \mu(A) \) is just \( \mu(A) \). Nevertheless, in dimension \( d\geq 3 \), both the range and the terminal point of \( W \) can be recovered with probability 1 from \( \mu \). In particular, for \( d\geq 3 \) the conditional law of the projection of \( W(1) \) given \( \mu \) is not \( \mu \). In dimension 2 we conjecture that the projection of \( W(1) \) cannot be recovered almost surely from \( \mu \), and show that the conditional law of the projection of \( W(1) \) given \( \mu \) is not \( \mu \).

James W. Pitman	Related
Yuval Peres	Related
Robin Alexander Pemantle	Related

[294] H. Matsumoto and M. Yor: “A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration,” Osaka J. Math. 38 : 2 (2001), pp. 383–398. MR 1833628 Zbl 0981.60078 article

[295] P. Carmona, F. Petit, and M. Yor: “Exponential functionals of Lévy processes,” pp. 41–55 in Lévy processes: Theory and applications. Edited by O. E. Barndorff-Nielsen, S. I. Resnick, and T. Mikosch. Birkhäuser (Boston, MA), 2001. MR 1833691 Zbl 0979.60038 incollection

[296] M. Yor: “Interpretations in terms of Brownian and Bessel meanders of the distribution of a subordinated perpetuity,” pp. 361–375 in Lévy processes: Theory and applications. Edited by O. E. Barndorff-Nielsen, S. I. Resnick, and T. Mikosch. Birkhäuser (Boston), 2001. MR 1833705 Zbl 0982.60080 incollection

[297] H. Matsumoto and M. Yor: “An analogue of Pitman’s \( 2M{-}X \) theorem for exponential Wiener functionals, II: The role of the generalized inverse Gaussian laws,” Nagoya Math. J. 162 (June 2001), pp. 65–86. MR 1836133 Zbl 0983.60075 article

[298] Séminaire de probabilités XXXV [Thirty-fifth probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1755. Springer (Berlin), 2001. MR 1837273 Zbl 0960.00020 book

Michel Emery	Related
Jacques Azéma	Related

[299] L. Chaumont, D. G. Hobson, and M. Yor: “Some consequences of the cyclic exchangeability property for exponential functionals of Lévy processes,” pp. 334–347 in Séminaire de probabilités XXXV [Thirty-fifth probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1755. Springer (Berlin), 2001. MR 1837296 Zbl 0982.60020 incollection

Michel Emery	Related
Jacques Azéma	Related

[300] C. Donati-Martin, R. Ghomrasni, and M. Yor: “On certain Markov processes attached to exponential functionals of Brownian motion: Application to Asian options,” Rev. Mat. Iberoamericana 17 : 1 (2001), pp. 179–193. MR 1846094 Zbl 0979.60073 article

[301] P. Biane, J. Pitman, and M. Yor: “Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions,” Bull. Am. Math. Soc. (N.S.) 38 : 4 (2001), pp. 435–465. MR 1848256 Zbl 1040.11061 ArXiv math.PR/9912170 article

James W. Pitman	Related
Philippe Biane	Related

[302] N. Tsilevich, A. Vershik, and M. Yor: “An infinite-dimensional analogue of the Lebesgue measure and distinguished properties of the gamma process,” J. Funct. Anal. 185 : 1 (September 2001), pp. 274–296. MR 1853759 Zbl 0990.60053 article

We define a one-parameter family \( \mathscr{L}_{\theta} \) of sigma-finite (finite on compact sets) measures in the space of distributions. These measures are equivalent to the laws of the classical gamma processes and invariant under an infinite-dimensional abelian group of certain positive multiplicators. This family of measures was first discovered by Gelfand–Graev–Vershik in the context of the representation theory of current groups; here we describe it in direct terms using some remarkable properties of the gamma processes. We show that the class of multiplicative measures coincides with the class of zero-stable measures which is introduced in the paper. We give also a new construction of the canonical representation of the current group \( \operatorname{SL}(2,\mathbb{R})^X \).

[303] M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. With an introductory chapter by Hélyette Geman. MR 1854494 Zbl 0999.60004 book

[304] C. Donati-Martin, H. Matsumoto, and M. Yor: “Some absolute continuity relationships for certain anticipative transformations of geometric Brownian motions,” Publ. Res. Inst. Math. Sci. 37 : 3 (2001), pp. 295–326. MR 1855425 Zbl 1033.60085 article

We present some absolute continuity relationships between the probability laws of a geometric Brownian motion \( e^{(\mu)} = (e_t^{(\mu)} \), \( t\geq 0 ) \) and its images by certain transforms \( T_a \) involving \( e^{(\mu)} \) and its quadratic variation. These results are derived from, and shown to be closely related to, our previous results about the generalized Dufresne’s identity and the exponential type extensions of Pitman’s \( M {-} X \) theorem for \( X \), a Brownian motion with constant drift \( \mu \), and its one-sided supremum \( M \). These absolute continuity results are then shown to be particular cases of those by Ramer–Kusuoka for non-linear transformations of the Wiener space and by Buckdahn–Föllmer for solutions of certain stochastic differential equations with anticipative drifts.

[305] N. O’Connell and M. Yor: “Brownian analogues of Burke’s theorem,” Stochastic Process. Appl. 96 : 2 (December 2001), pp. 285–304. MR 1865759 Zbl 1058.60078 article

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke’s theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.

[306] J. Bertoin and M. Yor: “On subordinators, self-similar Markov processes and some factorizations of the exponential variable,” Electron. Commun. Probab. 6 (2001), pp. 95–106. Article no. 10. MR 1871698 Zbl 1024.60030 article

Let \( \xi \) be a subordinator with Laplace exponent \( \Phi \), \[ I=\int_0^{\infty}\exp(-\xi_s)\,ds \] the so-called exponential functional, and \( X \) (respectively, \( \hat{X} \)) the self-similar Markov process obtained from \( \xi \) (respectively, from \( \hat{\xi} = -\xi \)) by Lamperti’s transformation. We establish the existence of a unique probability measure \( \rho \) on \( ]0,\infty[ \) with \( k \)-th moment given for every \( k\in\mathbb{N} \) by the product \( \Phi(1)\cdots\Phi(k) \), and which bears some remarkable connections with the preceding variables. In particular we show that if \( R \) is an independent random variable with law \( \rho \) then \( IR \) is a standard exponential variable, that the function \( t\to\mathbb{E}(1/X_t) \) coincides with the Laplace transform of \( \rho \), and that \( \rho \) is the 1-invariant distribution of the sub-markovian process \( \hat{X} \). A number of known factorizations of an exponential variable are shown to be of the preceding form \( IR \) for various subordinators \( \xi \).

[307] B. De Meyer, B. Roynette, P. Vallois, and M. Yor: “Sur l’indépendance d’un temps d’arrêt \( T \) et de la position \( B_T \) d’un mouvement brownien \( (B_u \), \( u\geqq 0) \)” [On the independence of a stopping time \( T \) and the position \( B_T \) of a Brownian motion \( (B_u \), \( u\geqq 0) \)], C. R. Acad. Sci., Paris, Sér. I 333 : 11 (December 2001), pp. 1017–1022. MR 1872465 Zbl 0995.60077 article

[308] H. Geman, D. B. Madan, and M. Yor: “Asset prices are Brownian motion: Only in business time,” pp. 103–146 in Quantitative analysis in financial markets: Collected papers of the New York University mathematical finance seminar (New York, 1995–1998), vol. 2. Edited by M. Avellaneda. World Scientific (River Edge, NJ), 2001. MR 1886692 Zbl 1134.91019 incollection

Hélyette Geman	Related
Dilip B. Madan	Related

[309] P. Baldi, E. Casadio Tarabusi, A. Figà-Talamanca, and M. Yor: “Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities,” Rev. Mat. Iberoamericana 17 : 3 (2001), pp. 587–605. MR 1900896 Zbl 1001.60018 article

[310] A. Rouault, M. Zani, and M. Yor: A LDP related to the Jacobi processes, January 2001. unpublished

[311] M. Yor: “On certain exponential functionals of real-valued Brownian motion,” pp. 14–22 in M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. English translation of French original published in J. Appl. Probab. 29:1 (1992). incollection

[312] M. Yor: “On some exponential functionals of Brownian motion,” pp. 23–48 in M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. This article was originally published in Adv. Appl. Probab. 24:3 (1992). incollection

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval \( [0,T] \) of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time \( T \) is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit time distributions and the fixed time case is recovered by inverting Laplace transforms.

[313] M. Yor: “On some exponential-integral functionals of Bessel processes,” pp. 172–181 in M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. Originally published in Math. Finance 3:2 (1993). incollection

[314] M. Yor: “The laws of exponential functionals of Brownian motion, taken at various random times,” pp. 55–62 in M. Yor: Exponential functionals of Brownian motion and related processes. Springer Finance. Springer (Berlin), 2001. Abridged English translation of an article published in C. R. Acad. Sci., Paris, Sér. I 314:12 (1992). incollection

[315] P. Carr, H. Geman, D. B. Madan, and M. Yor: “The fine structure of asset returns: An empirical investigation,” J. Business 75 : 2 (April 2002), pp. 305–332. article

Hélyette Geman	Related
Dilip B. Madan	Related

[316] B. Bru and M. Yor: “Comments on the life and mathematical legacy of Wolfgang Doeblin,” Finance Stoch. 6 : 1 (January 2002), pp. 3–47. MR 1885582 Zbl 1046.01009 article

Wolfgang Doeblin	Related
Bernard Bru	Related

[317] H. Geman, D. B. Madan, and M. Yor: “Stochastic volatility, jumps and hidden time changes,” Finance Stoch. 6 : 1 (January 2002), pp. 63–90. MR 1885584 Zbl 1006.60026 article

Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e., the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.

Hélyette Geman	Related
Dilip B. Madan	Related

[318] N. O’Connell and M. Yor: “A representation for non-colliding random walks,” Electron. Commun. Probab. 7 (2002), pp. 1–12. Article no. 1. MR 1887169 Zbl 1037.15019 article

We define a sequence of mappings \[ \Gamma_k:D_0(\mathbb{R}_+)^k\to D_0(\mathbb{R}_+)^k \] and prove the following result: Let \( N_1,\dots \), \( N_n \) be the counting functions of independent Poisson processes on \( \mathbb{R}_+ \) with respective intensities \( \mu_1 \lt{} \) \( \mu_2 \lt \cdots \lt \mu_n \). The conditional law of \( N_1,\dots \), \( N_n \), given that \[ N_1(t)\leq \cdots \leq N_n(t)\quad\textrm{for all } t\geq 0, \] is the same as the unconditional law of \( \Gamma_n(N) \). From this, we deduce the corresponding results for independent Poisson processes of equal rates and for independent Brownian motions (in both of these cases the conditioning is in the sense of Doob). This extends a recent observation, independently due to Baryshnikov [2001] and Gravner, Tracy and Widom [2001], which relates the law of a certain functional of Brownian motion to that of the largest eigenvalue of a GUE random matrix. Our main result can also be regarded as a generalisation of Pitman’s representation for the 3-dimensional Bessel process.

[319] C.-T. Wu and M. Yor: “Linear transformations of two independent Brownian motions and orthogonal decompositions of Brownian filtrations,” Publ. Mat., Barc. 46 : 1 (2002), pp. 237–256. MR 1904865 Zbl 1007.60024 article

[320] B. Roynette, P. Vallois, and M. Yor: “A solution to Skorokhod’s embedding for linear Brownian motion and its local time,” Studia Sci. Math. Hung. 39 : 1–2 (May 2002), pp. 97–127. MR 1909150 Zbl 1026.60047 article

[321] D. B. Madan and M. Yor: “Making Markov martingales meet marginals: With explicit constructions,” Bernoulli 8 : 4 (2002), pp. 509–536. MR 1914701 Zbl 1009.60037 article

[322] J. Bertoin and M. Yor: “The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes,” Potential Anal. 17 : 4 (December 2002), pp. 389–400. MR 1918243 Zbl 1004.60046 article

[323] M. Jeanblanc, J. Pitman, and M. Yor: “Self-similar processes with independent increments associated with Lévy and Bessel processes,” Stochastic Process. Appl. 100 : 1–2 (2002), pp. 223–231. MR 1919614 Zbl 1059.60052 article

[324] A. Rouault, M. Yor, and M. Zani: “A large deviations principle related to the strong arc-sine law,” J. Theoret. Probab. 15 : 3 (July 2002), pp. 793–815. MR 1922447 Zbl 1011.60006 article

[325] Séminaire de probabilités, 1967–1980: A selection in martingale theory [Probability seminars, 1967–1980: A selection in martingale theory]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics 1771. Springer (Berlin), 2002. MR 1925827 Zbl 0977.00031 book

[326] F. Delbaen and M. Yor: “Passport options,” Math. Finance 12 : 4 (2002), pp. 299–328. MR 1926234 Zbl 1048.91063 article

[327] B. De Meyer, B. Roynette, P. Vallois, and M. Yor: “On independent times and positions for Brownian motions,” Rev. Mat. Iberoamericana 18 : 3 (2002), pp. 541–586. MR 1954864 Zbl 1055.60078 article

Let \( (B_t \); \( t\geq 0) \), resp. \( ((X_t,Y_t) \); \( t\geq 0) \), be a one, resp. two, dimensional Brownian motion started at 0. Let \( T \) be a stopping time such that \( (B_{t\wedge T} \); \( t\geq 0) \), resp. \( (X_{t \wedge T} \); \( t\geq 0) \), \( (Y_{t\wedge T} \); \( t \geq 0)) \), is uniformly integrable. The main results obtained in the paper are:

if \( T \) and \( B_T \) are independent and \( T \) has all exponential moments, then \( T \) is constant.
If \( X_T \) and \( Y_T \) are independent and have all exponential moments, then \( X_T \) and \( Y_T \) are Gaussian.

We also give a number of examples of stopping times \( T \), with only some exponential moments, such that \( T \) and \( B_T \) are independent, and similarly for \( X_T \) and \( Y_T \). We also exhibit bounded non-constant stopping times \( T \) such that \( X_T \) and \( Y_T \) are independent and Gaussian.

[328] C. Donati-Martin, H. Matsumoto, and M. Yor: “The law of geometric Brownian motion and its integral, revisited: Application to conditional moments,” pp. 221–243 in Mathematical finance — Bachelier Congress, 2000 (Paris, 29 June–1 July 2000). Edited by H. Geman, D. Madan, S. R. Pliska, and T. Vorst. Springer Finance. Springer (Berlin), 2002. MR 1960566 Zbl 1030.91029 incollection

Stanley R. Pliska	Related
Hélyette Geman	Related
Dilip B. Madan	Related
Ton Vorst	Related

[329] A. S. Cherny, A. N. Shiryaev, and M. Yor: “Limit behaviour of the ‘horizontal-vertical’ random walk and some extensions of the Donsker–Prokhorov invariance principle,” Teor. Veroyatnost. i Primenen. 47 : 3 (2002), pp. 498–517. An English translation was published in Theory Probab. Appl. 47:3 (2002). MR 1975425 article

We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane \( \{y\gt x\} \) and in the vertical direction on the half-plane \( \{y\leq x\} \). The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.

[330] E. Csáki, Z. Shi, and M. Yor: “Fractional Brownian motions as ‘higher-order’ fractional derivatives of Brownian local times,” pp. 365–387 in Limit theorems in probability and statistics: Fourth Hungarian colloquium on limit theorems in probability and statistics (Balatonlelle, Hungary, 28 June–2 July 1999), vol. 1. Edited by I. Berkes, E. Csáki, and M. Csörgő. János Bolyai Mathematical Society (Budapest), 2002. Dedicated to Pál Révész on the occasion of his 65th birthday. MR 1979974 Zbl 1030.60073 incollection

Pál Révész	Related
István Berkes	Related
Endre Csáki	Related
Miklós Csörgő	Related

[331] J. Bertoin and M. Yor: “On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes,” Ann. Fac. Sci. Toulouse Math. (6) 11 : 1 (2002), pp. 33–45. MR 1986381 Zbl 1031.60038 article

[332] Stochastic processes and related topics (Siegmundsburg, Germany, 27 February–4 March 2000). Edited by R. Buckdahn, H.-J. Engelbert, and M. Yor. Stochastics Monographs 12. Taylor and Francis (London), 2002. MR 1987308 Zbl 1103.60007 book

[333] H. Matsumoto, L. Nguyen, and M. Yor: “Subordinators related to the exponential functionals of Brownian bridges and explicit formulae for the semigroups of hyperbolic Brownian motions,” pp. 213–235 in Stochastic processes and related topics (Siegmundsburg, Germany, 27 February–4 March 2000). Edited by R. Buckdahn, H.-J. Engelbert, and M. Yor. Stochastics Monographs 12. Taylor & Francis (London), 2002. MR 1987318 incollection

[334] M. Yor: “Three intertwined Brownian topics: Exponential functionals, winding numbers, and Ray–Knight theorems on local time,” pp. 269–272 in Stochastic processes and related topics (Siegmundsburg, Germany, 27 February–4 March 2000). Edited by R. Buckdahn, H.-J. Engelbert, and M. Yor. Stochastics Monographs 12. Taylor & Francis (London), 2002. MR 1987321 incollection

[335] B. Bru and M. Yor: “Wolfgang Doeblin et le pli cacheté 11668” [Wolfgang Doeblin and sealed envelope 11668], Matapli 68 (2002), pp. 75–92. MR 2061581 article

Wolfgang Doeblin	Related
Bernard Bru	Related

[336] M. Yor: Une promenade probabiliste [A probabilistic walk], January 2002. unpublished

[337] A. S. Cherny, A. N. Shiryaev, and M. Yor: “Limit behaviour of the ‘horizontal-vertical’ random walk and some extensions of the Donsker–Prokhorov invariance principle,” Theory Probab. Appl. 47 : 3 (2002), pp. 377–394. English translation of Russian original published in Teor. Veroyatnost. i Primenen. 47:3 (2002). Zbl 1034.60076 article

We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane \( \{y\gt x\} \) and in the vertical direction on the half-plane \( \{y\leq x\} \). The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.

[338] H. Matsumoto and M. Yor: “Interpretation via Brownian motion of some independence properties between GIG and gamma variables,” Statist. Probab. Lett. 61 : 3 (February 2003), pp. 253–259. MR 1959132 Zbl 1039.60074 article

[339] J. Pitman and M. Yor: “Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches,” Bernoulli 9 : 1 (2003), pp. 1–24. MR 1963670 Zbl 1024.60032 article

[340] J. Pitman and M. Yor: “Infinitely divisible laws associated with hyperbolic functions,” Canad. J. Math. 55 : 2 (2003), pp. 292–330. MR 1969794 Zbl 1039.11054 article

The infinitely divisible distributions on \( \mathbb{R}^+ \) of random variables \( C_t \), \( S_t \) and \( T_t \), with Laplace transforms \[ \Bigl(\frac{1}{\cosh\sqrt{2\lambda}}\Bigr)^t, \quad \Bigl(\frac{\sqrt{2\lambda}}{\sinh\sqrt{2\lambda}}\Bigr)^t, \quad \textrm{and} \quad \Bigl(\frac{\tanh\sqrt{2\lambda}}{\sqrt{2\lambda}}\Bigr)^t \] respectively, are characterized for various \( t \gt 0 \) in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for \( t=1 \) or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of \( C_1 \) and \( S_2 \) are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet \( L \)-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from \( S_t \) and \( C_t \) by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

[341] Séminaire de probabilités XXXVI [Thirty-sixth seminar on probability]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1801. Springer (Berlin), 2003. MR 1971581 Zbl 1003.00010 book

Michel Emery	Related
Jacques Azéma	Related

[342] H. Matsumoto and M. Yor: “On Dufresne’s relation between the probability laws of exponential functionals of Brownian motions with different drifts,” Adv. Appl. Probab. 35 : 1 (2003), pp. 184–206. In honor of Joseph Mecke. MR 1975510 Zbl 1030.60077 article

Denote by \( \alpha_t^{(\mu)} \) the probability law of \[ A_t^{(\mu)} = \int_0^t\exp\bigl(2(B_s + \mu s)\bigr)ds \] for a Brownian motion \( \{B_s \), \( s\geq 0\} \). It is well known that \( \alpha_t^{(\mu)} \) is of interest in a number of domains, e.g., mathematical finance, diffusion processes in random environments, stochastic analysis on hyperbolic spaces and so on, but that it has complicated expressions. Recently, Dufresne obtained some remarkably simple expressions for \( \alpha_t^{(0)} \) and \( \alpha_t^{(1)} \), as well as an equally remarkable relationship between \( \alpha_t^{(\mu)} \) and \( \alpha_t^{(\nu)} \) for two different drifts \( \mu \) and \( \nu \). In this paper, hinging on previous results about \( \alpha_t^{(\mu)} \), we give different proofs of Dufresne’s results and present extensions of them for the processes \( \{A_t^{(\mu)} \), \( t\geq 0\} \).

[343] M. Jacobsen and M. Yor: “Multi-self-similar Markov processes on \( \mathbb{R}_+^n \) and their Lamperti representations,” Probab. Theory Relat. Fields 126 : 1 (May 2003), pp. 1–28. MR 1981630 Zbl 1031.60029 article

A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti’s result to processes in \( n \) dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper.

The special case where the Lévy process \( \xi \) is standard Brownian motion in \( n \) dimensions is studied in detail. There are also specific comments on the case where \( \xi \) is an \( n \)-dimensional compound Poisson process with drift.

Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to \( n \)-dimensional subordinators.

[344] V. Bentkus, G. Pap, and M. Yor: “Optimal bounds for Cauchy approximations for the winding distribution of planar Brownian motion,” J. Theor. Probab. 16 : 2 (April 2003), pp. 345–361. MR 1982031 Zbl 1027.60089 article

[345] P. Carr, H. Geman, D. B. Madan, and M. Yor: “Stochastic volatility for Lévy processes,” Math. Finance 13 : 3 (2003), pp. 345–382. EFA 2002 Berlin meetings, presented paper. MR 1995283 Zbl 1092.91022 article

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein–Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.

Hélyette Geman	Related
Dilip B. Madan	Related

[346] A. Göing-Jaeschke and M. Yor: “A survey and some generalizations of Bessel processes,” Bernoulli 9 : 2 (2003), pp. 313–349. MR 1997032 Zbl 1038.60079 article

Bessel processes play an important role in financial mathematics because of their strong relation to financial models such as geometric Brownian motion or Cox–Ingersoll–Ross processes. We are interested in the first time Bessel processes and, more generally, radial Ornstein–Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein–Uhlenbeck processes, that is, Cox–Ingersoll–Ross processes. As a natural extension we study squared Bessel processes and squared Ornstein–Uhlenbeck processes with negative dimensions or negative starting points and derive their properties.

[347] F. Trèves, G. Pisier, and M. Yor: “Laurent Schwartz (1915–2002),” Notices Am. Math. Soc. 50 : 9 (2003), pp. 1072–1084. Yor’s contribution is an English translation of a piece published in Gaz. Math. 98 (2003). MR 2002753 Zbl 1159.01350 article

Laurent Schwartz	Related
François Trèves	Related
Gilles Pisier	Related

[348] A. N. Shiryaev and M. Yor: “On stochastic integral representations of functionals of Brownian motion, I,” Teor. Veroyatnost. i Primenen. 48 : 2 (2003), pp. 375–385. An English translation was published in Theory Probab. Appl. 48:2 (2003). MR 2015458 article

For functionals \( S=S(\omega) \) of the Brownian motion \( B \), we propose a method for finding stochastic integral representations based on the Itô formula for the stochastic integral associated with \( B \). As an illustration of the method, we consider functionals of the “maximal” type: \( S_T \), \( S_{T_{-a}} \), \( S_{g_T} \), and \( S_{\theta_T} \), where \[ S_T=\max_{t\leq T}B_t, \qquad S_{T_{-a}}=\max_{t\leq T_{-a}}B_t \] with \( T_{-a}=\inf\{t \gt 0\mid B_t=-a\} \), \( a\gt 0 \), and \[ S_{g_T}=\max_{t\leq g_T} B_t, \qquad S_{\theta_T}=\max_{t\leq \theta_T}B_t ,\] \( g_T \) and \( \theta_T \) are non-Markov times: \( g_T \) is the time of the last zero of Brownian motion on \( [0,T] \) and \( \theta_T \) is a time when the Brownian motion achieves its maximal value on \( [0,T] \).

[349] L. Chaumont and M. Yor: Exercises in probability: A guided tour from measure theory to random processes, via conditioning. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge University Press, 2003. Reprinted in 2009. 2nd edition published in 2012. MR 2016344 Zbl 1065.60001 book

[350] P. Chassaing, J. F. Marckert, and M. Yor: “A stochastically quasi-optimal search algorithm for the maximum of the simple random walk,” Ann. Appl. Probab. 13 : 4 (2003), pp. 1264–1295. MR 2023877 Zbl 1084.68145 article

[351] B. Roynette, P. Vallois, and M. Yor: “Limiting laws associated with Brownian motion perturbated by normalized exponential weights,” C. R., Math., Acad. Sci. Paris 337 : 10 (November 2003), pp. 667–673. With French summary. MR 2030109 Zbl 1031.60021 ArXiv math/0510550 article

[352] Séminaire de probabilités XXXVII [Thirty-seventh probability seminar]. Edited by J. Azéma, M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1832. Springer (Berlin), 2003. MR 2053038 Zbl 1027.00025 book

Michel Emery	Related
Jacques Azéma	Related

[353] M. Yor: “Deux maîtres ès probabilités: Laurent Schwartz et Paul-André Meyer” [Two masters of probability: Laurent Schwartz and Paul-André Meyer], Gaz. Math., Soc. Math. Fr. 98 (2003), pp. 119–122. An English translation appeared as part of a multi-author tribute to Schwartz in Notices Am. Math. Soc. 50:9 (2003). MR 2067357 article

Laurent Schwartz	Related
Paul-André Meyer	Related

[354] H. Matsumoto and M. Yor: The Gauss transform and Bougerol’s identity for the arithmetic-geometric mean of Brownian motion, March 2003. unpublished

[355] J. Pitman and M. Yor: Polynomials associated with Lévy processes, 2003. unpublished

[356] A. N. Shiryaev and M. Yor: “On the problem of stochastic integral representations of functionals of the Brownian motion, I,” Theory Probab. Appl. 48 : 2 (2003), pp. 304–313. English translation of Russian original published in Teor. Veroyatnost. i Primenen. 48:2 (2003). Zbl 1057.60057 article

[357] A. Göing-Jaeschke and M. Yor: “A clarification note about hitting times densities for Ornstein–Uhlenbeck processes,” Finance Stoch. 7 : 3 (July 2003), pp. 413–415. Zbl 1064.60026 article

[358] J. Bertoin, B. Roynette, and M. Yor: Some connections between (sub)critical branching mechanisms and Bernstein functions. Preprint, December 2004. ArXiv math/0412322 techreport

Jean Bertoin	Related
Bernard Roynette	Related

[359] L. Chaumont, L. Mazliak, and M. Yor: “Quelques aspects de l’oeuvre probabiliste” [Some aspects of the probabilistic work], Chapter 3 in L’Héritage de Kolmogorov en mathématiques [Kolmogorov’s heritage in mathematics]. Edited by É. Charpentier, A. Lesne, and N. Nikolski. Echelles. Belin (Paris), 2004. An English translation was published in Kolmogorov’s heritage in mathematics (2007). incollection

Andreï N. Kolmogorov (1903–1987) fut l’un des plus brillants mathématiciens du XXe siècle. Esprit exceptionnellement profond et original, il savait poser un regard neuf sur chacun des sujets qu’il abordait, pour bien souvent en changer radicalement le paysage. À l’occasion du centenaire de sa naissance, une vingtaine de mathématiciens de renom ont voulu lui rendre hommage en revisitant son oeuvre des plus foisonnantes. Qu’on en juge: séries de Fourier, logique, probabilités, statistique, topologie, systèmes dynamiques, complexité, etc., sans oublier les résultats obtenus en écologie théorique ou en théorie algorithmique! Destiné aux étudiants comme aux chercheurs, cet ouvrage présente en outre diverses applications marquantes et très récentes des méthodes de Kolmogorov - parfois dans des domaines inattendus comme les réseaux de neurones ou le théorème de Godel–qui passionneront tous les amateurs d’idées mathématiques. Il fait suite à L’héritage de Kolmogorov en physique publié dans la même collection en 2003.

[360] C. Donati-Martin, Y. Doumerc, H. Matsumoto, and M. Yor: Some asymptotic laws for Wishart processes, January 2004. unpublished

[361] A. M. Vershik, M. Yor, and N. V. Tsilevich: “On the Markov–Krein identity and the quasi-invariance of the gamma process,” J. Math. Sci., New York 121 : 3 (2004), pp. 2303–2310. English translation of an article published in Representation theory, dynamical systems, combinatorial and algorithmic methods 6 (2001). MR 1879060 article

[362] T. Fujita, F. Petit, and M. Yor: “Pricing path-dependent options in a Black–Scholes market from the distribution of homogeneous Brownian functionals,” J. Appl. Probab. 41 : 1 (2004), pp. 1–18. MR 2036268 Zbl 1049.60070 article

[363] J. Obłój and M. Yor: “An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale,” Stochastic Process. Appl. 110 : 1 (March 2004), pp. 83–110. MR 2052138 Zbl 1075.60038 article

A general methodology allowing to solve the Skorokhod stopping problem for positive functionals of Brownian excursions, with the help of Brownian local time, is developed. The stopping times we consider have the following form: \[ T_{\mu} = \inf\{t\gt 0 \mid F_t\geq\phi_{\mu}^F(L_t)\} .\] As an application, the Skorokhod embedding problem for a number of functionals \( (F_t \), \( t\geq 0) \), including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times \( T_{\mu} \), such that \( F_{T_{\mu}}\sim\mu \), are given. It is shown that the function \( \phi_{\mu}^F \) is the same for the maximum and for the age, \( \phi_{\mu} = \psi_{\mu}^{-1} \), where \[ \psi_{\mu}(x) = \int_{[0,x]}(y/\overline{\mu}(y)) \,d\mu(y) .\] The joint law of \( (g_{T_{\mu}} \), \( T_{\mu} \), \( L_{T_{\mu}}) \), in the case of the age functional, is characterized. Examples for specific measures \( \mu \) are discussed. Finally, a randomized solution to the embedding problem for Azéma martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel.

[364] Y. Hariya and M. Yor: “On an extension of Dufresne’s relation between exponential Brownian functionals from opposite drifts to two different drifts: A short proof,” Statist. Probab. Lett. 67 : 4 (May 2004), pp. 331–341. MR 2060133 Zbl 1042.60051 article

[365] C. Donati-Martin, A. Rouault, M. Yor, and M. Zani: “Large deviations for squares of Bessel and Ornstein–Uhlenbeck processes,” Probab. Theory Relat. Fields 129 : 2 (June 2004), pp. 261–289. MR 2063378 Zbl 1055.60017 article

[366] M. Émery and M. Yor: “A parallel between Brownian bridges and gamma bridges,” Publ. Res. Inst. Math. Sci. 40 : 3 (2004), pp. 669–688. MR 2074696 Zbl 1074.60054 article

[367] Y. Hariya and M. Yor: “Limiting distributions associated with moments of exponential Brownian functionals,” Studia Sci. Math. Hung. 41 : 2 (2004), pp. 193–242. MR 2082657 Zbl 1115.60005 article

[368] M. Yor and L. Zambotti: “A remark about the norm of a Brownian bridge,” Statist. Probab. Lett. 68 : 3 (July 2004), pp. 297–304. MR 2083898 Zbl 1080.60037 article

[369] P. Carmona, F. Petit, and M. Yor: “A trivariate law for certain processes related to perturbed Brownian motions,” Ann. Inst. Henri Poincaré, Probab. Stat. 40 : 6 (November–December 2004), pp. 737–758. MR 2096216 Zbl 1060.60082 article

D. Williams’ path decomposition and Pitman’s representation theorem for \( BES(3) \) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process.

In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101].

Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions.

The results provide some new understanding of the generalizations of Lévy’s arc sine law for perturbed Brownian motions previously obtained by the second author.

[370] J. Bertoin, P. Biane, and M. Yor: “Poissonian exponential functionals, \( q \)-series, \( q \)-integrals, and the moment problem for log-normal distributions,” pp. 45–56 in Seminar on stochastic analysis, random fields and applications IV (Ascona, Switzerland, 20–24 May 2002). Edited by R. C. Dalang, M. Dozzi, and F. Russo. Progress in Probability 58. Birkhäuser (Basel), 2004. MR 2096279 Zbl 1056.60046 incollection

Philippe Biane	Related
Jean Bertoin	Related

[371] C. Donati-Martin, Y. Doumerc, H. Matsumoto, and M. Yor: “Some properties of the Wishart processes and a matrix extension of the Hartman–Watson laws,” Publ. Res. Inst. Math. Sci. 40 : 4 (2004), pp. 1385–1412. MR 2105711 Zbl 1076.60067 article

[372] G. Peccati and M. Yor: “Hardy’s inequality in \( L^2([0,1]) \) and principal values of Brownian local times,” pp. 49–74 in Asymptotic methods in stochastics: Festschrift for Miklós Csörgő (Ottawa, 23–25 May 2002). Edited by L. Horvárth and B. Szyszkowicz. Fields Institute Communications 44. American Mathematical Society (Providence, RI), 2004. MR 2106848 Zbl 1074.60029 incollection

We present in a unified framework two examples of a random function \( \phi(\omega,s) \) on \( \mathfrak{R}_+ \) such that

the integral \[ \int_0^{\infty}\phi(\omega,s)g(s)\,ds \] is well defined and finite (at least, as a limit in probability) for every deterministic and square integrable function \( g \), and
\( \phi \) does not belong to \( L^2([0,\infty) \), \( ds) \) with probability one.

In particular, the second example is related to the existence of principal values of Brownian local times. Our key tools are Hardy’s inequality, some semimartingale representation results for Brownian local times due to Ray, Knight and Jeulin, and the reformulation of certain theorems of Jeulin–Yor [1979] and Cherny [2001] in terms of bounded \( L^2 \) operators. We also establish, in the last paragraph, several weak convergence results.

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

[373] G. Peccati and M. Yor: “Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge,” pp. 75–87 in Asymptotic methods in stochastics: Festschrift for Miklós Csörgő (Ottawa, 23–25 May 2002). Edited by L. Horvárth and B. Szyszkowicz. Fields Institute Communications 44. American Mathematical Society (Providence, RI), 2004. MR 2106849 Zbl 1071.60017 incollection

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

[374] Z. J. Jurek and M. Yor: “Selfdecomposable laws associated with hyperbolic functions,” Probab. Math. Statist. 24 : 1 (2004), pp. 181–190. MR 2108161 Zbl 1066.60019 ArXiv 1009.3542 article

[375] P. Barrieu, A. Rouault, and M. Yor: “A study of the Hartman–Watson distribution motivated by numerical problems related to the pricing of Asian options,” J. Appl. Probab. 41 : 4 (December 2004), pp. 1049–1058. MR 2122799 Zbl 1064.60021 article

[376] J. Pitman and M. Yor: “Some properties of the arc-sine law related to its invariance under a family of rational maps,” pp. 126–137 in A festschrift for Herman Rubin. Edited by A. Dasgupta. IMS Lecture Notes Monograph Series 45. Institute of Mathematical Statistics (Beachwood, OH), 2004. MR 2126891 Zbl 1268.37071 incollection

James W. Pitman	Related
Herman Rubin	Related

[377] P. Salminen and M. Yor: “On Dufresne’s perpetuity, translated and reflected,” pp. 337–354 in Stochastic processes and applications to mathematical finance (Kusatsu, Japan, 5–9 March 2003). Edited by J. Akahori, S. Ogawa, and S. Watanabe. World Scientific (River Edge, NJ), 2004. MR 2202705 Zbl 1323.60113 incollection

Let \( B(\mu) \) denote a Brownian motion with drift \( \mu \). In this paper we study two perpetual integral functionals of \( B(\mu) \). The first one, introduced and investigated by Dufresne in [1990], is \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,ds,\quad \mu\lt 0. \] It is known that this functional is identical in law with the first hitting time of 0 for a Bessel process with index \( \mu \). In particular, we analyze the following reflected (or one-sided) variants of Dufresne’s functional \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,\mathbf{1}_{\{B_s^{(\mu)}\gt 0\}}\,ds, \] and \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,\mathbf{1}_{\{B_s^{(\mu)}\lt 0\}}\,ds, \] We shall show in this paper how these functionals can also be connected to hitting times. Our second functional, which we call Dufresne’s translated functional, is \[ \hat{D}_c^{(\nu)} := \int_0^{\infty}\bigl(c+\exp(B_s^{(\nu)}) \bigr)^{-2}ds, \] where \( c \) and \( \mu \) are positive. This functional has all its moments finite, in contrast to Dufresne’s functional which has only some finite moments. We compute explicitly the Laplace transform of \( \hat{D}_c^{(\nu)} \) in the case \( \nu = 1/2 \) (other parameter values do not seem to allow explicit solutions) and connect this variable, as well as its reflected variants, to hitting times.

[378] P. Carr, H. Geman, D. B. Madan, and M. Yor: “From local volatility to local Lévy models,” Quant. Finance 4 : 5 (October 2004), pp. 581–588. MR 2241297 article

Hélyette Geman	Related
Dilip B. Madan	Related

[379] D. Madan and M. Yor: “On the Itô–Tanaka formula for strictly local martingales: Does it need a correction term?,” Obwerwolfach Rep. 1 : 2 (2004), pp. 1365–1366. Report no. 26/2004. Brief summary of lecture given by Yor at the mini-workshop “Local time-space calculus with applications ,” Oberwolfach, Germany, 16–22 May 2004. article

[380] D. Madan and M. Yor: Mimicking the one-dimensional marginals of the Heston model, January 2004. unpublished

[381] B. Roynette, P. Vallois, and M. Yor: Feynman–Kac asymptotics and the corresponding renormalizations of the Wiener measure, February 2004. Submitted to Studia Sci. Math. Hungarica. unpublished

[382] B. Roynette, P. Vallois, and M. Yor: Examples of \( BM-BES(3) \) concatenations related to renormalization of Brownian motion induced by its maximum and local time processes, May 2004. unpublished

[383] W. Schachermayer and M. Yor: Hiding a drift, March 2004. unpublished

[384] M. Atlan, H. Geman, and M. Yor: Options on hedge funds under the high water mark rule. Preprint, October 2005. ArXiv math/0510497 techreport

The rapidly growing hedge fund industry has provided individual and institutional investors with new investment vehicles and styles of management. It has also brought forward a new form of performance contract: hedge fund managers receive incentive fees which are typically a fraction of the fund net asset value (NAV) above its starting level–a rule known as high water mark. Options on hedge funds are becoming increasingly popular, in particular because they allow investors with limited capital to get exposure to this new asset class. The goal of the paper is to propose a valuation of plain-vanilla options on hedge funds which accounts for the high water market rule. Mathematically, this valuation leads to an interesting use of local times of Brownian motion. Option prices are numerically computed by inversion of their Laplace transforms.

Hélyette Geman	Related
Marc Atlan	Related

[385] R. Mansuy and M. Yor: “Harnesses, Lévy bridges and Monsieur Jourdain,” Stochastic Processes Appl. 115 : 2 (February 2005), pp. 329–338. MR 2111197 Zbl 1070.60041 ArXiv math/0406563 article

[386] Séminaire de probabilités XXXVIII [Thirty-eighth probability seminar]. Edited by M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1857. Springer (Berlin), 2005. Dedicated to Jacques Azéma on the occasion on his 65th birthday. MR 2126961 Zbl 1055.60001 book

Michel Emery	Related
Jacques Azéma	Related

[387] L. Nguyen-Ngoc and M. Yor: “Some martingales associated to reflected Lévy processes,” pp. 42–69 in Séminaire de probabilités XXXVIII [Thirty-eighth probability seminar]. Edited by M. Émery, M. Ledoux, and M. Yor. Lecture Notes in Mathematics 1857. Springer (Berlin), 2005. MR 2126966 Zbl 1079.60048 incollection

[388] L. Gallardo and M. Yor: “Some new examples of Markov processes which enjoy the time-inversion property,” Probab. Theory Relat. Fields 132 : 1 (2005), pp. 150–162. MR 2136870 Zbl 1087.60058 article

[389] P. Salminen and M. Yor: “Properties of perpetual integral functionals of Brownian motion with drift,” Ann. Inst. Henri Poincaré, Probab. Stat. 41 : 3 (May–June 2005), pp. 335–347. MR 2139023 Zbl 1082.60073 article

[390] P. Salminen and M. Yor: “Perpetual integral functionals as hitting and occupation times,” Electron. J. Probab. 10 (2005), pp. 371–419. Article no. 11. MR 2147313 Zbl 1110.60078 ArXiv math/0403069 article

Let \( X \) be a linear diffusion and \( f \) a non-negative, Borel measurable function. We are interested in finding conditions on \( X \) and \( f \) which imply that the perpetual integral functional \[ I^X_{\infty}(f) := \int_0^{\infty} f(X_t) \,dt \] is identical in law with the first hitting time of a point for some other diffusion. This phenomenon may often be explained using random time change. Because of some potential applications in mathematical finance, we are considering mainly the case when \( X \) is a Brownian motion with drift \( \mu \gt 0 \) denoted \( (B^{(\mu)}_t \); \( t\geq 0) \), but it is obvious that the method presented is more general. We also review the known examples and give new ones. In particular, results concerning one-sided functionals \[ \int_0^{\infty} f(B^{(\mu)}_t)\,\mathbf{1}_{\{B^{(\mu)}_t \lt 0\}} dt \qquad \mathrm{ and } \qquad \int_0^{\infty} f(B^{(\mu)}_t)\,\mathbf{1}_{\{B^{(\mu)}_t \gt 0\}} dt \] are presented. This approach generalizes the proof, based on the random time change techniques, of the fact that the Dufresne functional (this corresponds to \( f(x) = \exp(-2x) \)), playing quite an important role in the study of geometric Brownian motion, is identical in law with the first hitting time for a Bessel process. Another functional arising naturally in this context is \[ \int_0^{\infty} (a + \exp(B^{(\mu)}_t))^{-2} dt, \] which is seen, in the case \( \mu = 1/2 \), to be identical in law with the first hitting time for a Brownian motion with drift \( \mu = a/2 \). The paper is concluded by discussing how the Feynman–Kac formula can be used to find the distribution of a perpetual integral functional.

[391] P. Cheridito, D. Filipović, and M. Yor: “Equivalent and absolutely continuous measure changes for jump-diffusion processes,” Ann. Appl. Probab. 15 : 3 (2005), pp. 1713–1732. MR 2152242 Zbl 1082.60034 ArXiv math/0508450 article

[392] B. Roynette and M. Yor: “Couples de Wald indéfiniment divisibles: Exemples liés à la fonction gamma d’Euler et à la fonction zeta de Riemann” [Infinitely divisible Wald couples: Examples linked with the Euler gamma function and the Riemann zeta function], Ann. Inst. Fourier (Grenoble) 55 : 4 (2005), pp. 1219–1283. MR 2157168 Zbl 1083.60012 article

To any positive measure \( c \) on \( \mathbb{R}_+ \), such that: \[ \int^{\infty}_0 (x \wedge x^2) \,c(dx) \lt \infty \] we associate an infinitely divisible Wald couple, i.e., a couple of random variables \( (X,H) \) such that \( X \) and \( H \) are infinitely divisible, \( H \geq 0 \), and for any \( \lambda \geq 0 \), \[ E\bigl(e^{\lambda X}\bigr) \cdot E\bigl(e^{-\frac{\lambda^2}{2} H}\bigr)=1 .\] More generally, to a positive measure \( c \) on \( \mathbb{R}_+ \) which satisfies \[ \int_0^{\infty}e^{-\alpha x} x^2 c(dx) \lt \infty \] for every \( \alpha \gt \alpha_0 \), we associate an “Esscher family” of infinitely divisible Wald couples. We give many examples of such Esscher families and we prove that the particular ones which are associated with the gamma and the zeta functions enjoy remarkable properties.

[393] B. Roynette, P. Vallois, and M. Yor: “Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III,” Period. Math. Hung. 50 : 1–2 (August 2005), pp. 247–280. In homage to Professors E. Csáki and P. Révész. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2162812 Zbl 1150.60308 ArXiv math/0511102 article

Pál Révész	Related
Bernard Roynette	Related
Endre Csáki	Related

@article {key2162812m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Limiting laws for long {B}rownian bridges
		 perturbed by their one-sided maximum,
		 {III}},
	JOURNAL = {Period. Math. Hung.},
	FJOURNAL = {Periodica Mathematica Hungarica. Journal
		 of the J\'anos Bolyai Mathematical Society},
	VOLUME = {50},
	NUMBER = {1--2},
	MONTH = {August},
	YEAR = {2005},
	PAGES = {247--280},
	DOI = {10.1007/s10998-005-0015-7},
	NOTE = {In homage to Professors E. Cs\'aki and
		 P. R\'ev\'esz. Parts I--X have very
		 different titles. I was published in
		 \textit{Studia Sci. Math. Hung.} \textbf{46}:2
		 (2003); II in \textit{Studia Sci. Math.
		 Hung.} \textbf{43}:3 (2006); IV in \textit{Studia
		 Sci. Math. Hung.} \textbf{44}:4 (2007);
		 V in \textit{Studia Sci. Math. Hung.}
		 \textbf{45}:1 (2008); VI in \textit{ESAIM
		 Probab. Stat.} \textbf{13} (2009); VII
		 in \textit{Ann. Inst. Henri Poincar\'e,
		 Probab. Stat.} \textbf{45}:2 (2009);
		 VIII in \textit{J. Funct. Anal.} \textbf{255}:9
		 (2008); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 ArXiv:math/0511102. MR:2162812. Zbl:1150.60308.},
	ISSN = {0031-5303},
}

[394] A. Nikeghbali and M. Yor: “A definition and some characteristic properties of pseudo-stopping times,” Ann. Probab. 33 : 5 (2005), pp. 1804–1824. MR 2165580 Zbl 1083.60035 ArXiv math/0406459 article

[395] J. Bertoin and M. Yor: “Exponential functionals of Lévy processes,” Probability Surveys 2 (2005), pp. 191–212. MR 2178044 Zbl 1189.60096 ArXiv math/0511265 article

[396] H. Matsumoto and M. Yor: “Exponential functionals of Brownian motion, I: Probability laws at fixed time,” Probab. Surv. 2 (2005), pp. 312–347. MR 2203675 Zbl 1189.60150 ArXiv math/0511517 article

[397] H. Matsumoto and M. Yor: “Exponential functionals of Brownian motion, II: Some related diffusion processes,” Probab. Surv. 2 (2005), pp. 348–384. MR 2203676 Zbl 1189.91232 ArXiv math/0511519 article

[398] P. Carr, H. Geman, D. B. Madan, and M. Yor: “Pricing options on realized variance,” Finance Stoch. 9 : 4 (October 2005), pp. 453–475. MR 2213777 Zbl 1096.91022 article

Hélyette Geman	Related
Dilip B. Madan	Related

[399] M. Yor: Basic facts about Brownian motion, stochastic integration and stochastic differential equations, July 2005. unpublished

[400] M. Yor: Introduction aux probabilités [Introduction to probability], June 2005. unpublished

[401] J.-M. Bony, P. G. Ciarlet, and M. Yor: “Les Comptes Rendus en mathématique: Passé, présent, futur” [Comptes Rendus in mathematics: Past, present, future], C. R., Math., Acad. Sci. Paris 340 : 11 (June 2005), pp. 785–786. Zbl 1075.01502 article

[402] D. Madan and M. Yor: CGMY and Meixner subordinators are absolutely continuous with respect to one sided stable subordinators. Preprint, January 2006. ArXiv math/0601173 techreport

[403] E. Gobet, G. Pagès, and M. Yor: “Mathématiques et finance” [Mathematics and finance], pp. 77–94 in Aspects des mathématiques financiéres [Aspects of financial mathematics] (Paris, 1 February 2005). Edited by M. Yor. Lavoisier/TEC et DOC (Paris), 2006. Translated and reprinted in Aspects of mathematical finance (2008). incollection

[404] P. Deheuvels, G. Peccati, and M. Yor: “On quadratic functionals of the Brownian sheet and related processes,” Stochastic Process. Appl. 116 : 3 (March 2006), pp. 493–538. MR 2199561 Zbl 1090.60020 article

Motivated by asymptotic problems in the theory of empirical processes, and specifically by tests of independence, we study the law of quadratic functionals of the (weighted) Brownian sheet and of the bivariate Brownian bridge on \( [0,1]^2 \). In particular:

we use Fubini-type techniques to establish identities in law with quadratic functionals of other Gaussian processes,
we explicitly calculate the Laplace transform of such functionals by means of Karhunen–Loève expansions,
we prove central and non-central limit theorems in the spirit of Peccati and Yor [Four limit theorems involving quadratic functionals of Brownian motion and Brownian bridge, Asymptotic Methods in Stochastics, American Mathematical Society, Fields Institute Communication Series, 2004, pp. 75–87] and Nualart and Peccati [Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33(1) (2005) 177–193].

Our results extend some classical computations due to Lévy [Wiener’s random function and other Laplacian random functions, in: Second Berkeley Symposium in Probability and Statistics, 1950, pp. 171–186], as well as the formulae recently obtained by Deheuvels and Martynov [Karhunen–Loève expansions for weighted Wiener processes and Brownian bridges via Bessel functions, Progress in Probability, vol. 55, Birkhäuser Verlag, Basel, 2003, pp. 57–93].

[405] R. Mansuy and M. Yor: Random times and enlargements of filtrations in a Brownian setting. Lecture Notes in Mathematics 1873. Springer (Berlin), 2006. MR 2200733 Zbl 1103.60003 book

[406] A. Gnedin, J. Pitman, and M. Yor: “Asymptotic laws for compositions derived from transformed subordinators,” Ann. Probab. 34 : 2 (2006), pp. 468–492. MR 2223948 Zbl 1142.60327 ArXiv math/0403438 article

A random composition of \( n \) appears when the points of a random closed set \( \tilde{\mathscr{R}}\subset [0,1] \) are used to separate into blocks \( n \) points sampled from the uniform distribution. We study the number of parts \( K_n \) of this composition and other related functionals under the assumption that \[ \tilde{\mathscr{R}} = \phi(S_{\bullet}) ,\] where \( (S_t \), \( t\geq 0) \) is a subordinator and \( \phi:[0,\infty]\to [0,1] \) is a diffeomorphism. We derive the asymptotics of \( K_n \) when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function \( \phi(x) = 1-e^{-x} \), we establish a connection between the asymptotics of \( K_n \) and the exponential functional of the subordinator.

[407] D. Khoshnevisan, P. Salminen, and M. Yor: “A note on a.s. finiteness of perpetual integral functionals of diffusions,” Electron. Commun. Probab. 11 (2006), pp. 108–117. MR 2231738 Zbl 1111.60061 ArXiv math/0511336 article

[408] J. Obłój and M. Yor: “On local martingale and its supremum: Harmonic functions and beyond,” pp. 517–533 in From stochastic calculus to mathematical finance: The Shiryaev Festschrift (Meatbief, France, 9–15 January 2005). Edited by Y. Kabanov, R. Lipster, and J. Stoyanov. Springer (Berlin), 2006. MR 2234288 Zbl 1120.60045 ArXiv math/0412196 incollection

[409] A. Gnedin, J. Pitman, and M. Yor: “Asymptotic laws for regenerative compositions: Gamma subordinators and the like,” Probab. Theory Relat. Fields 135 : 4 (August 2006), pp. 576–602. MR 2240701 Zbl 1099.60023 ArXiv math.PR/0405440 article

For \( \tilde{\mathcal{R}} = 1 - \exp(-\mathcal{R}) \) a random closed set obtained by exponential transformation of the closed range \( \mathcal{R} \) of a subordinator, a regenerative composition of generic positive integer \( n \) is defined by recording the sizes of clusters of \( n \) uniform random points as they are separated by the points of \( \tilde{\mathcal{R}} \). We focus on the number of parts \( K_n \) of the composition when \( \tilde{\mathcal{R}} \) is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for \( K_n \) and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at \( 0+ \).

[410] B. Roynette, P. Vallois, and M. Yor: “Asymptotics for the distribution of lengths of excursions of a \( d \)-dimensional Bessel process \( (0\lt d\lt 2) \),” C. R., Math., Acad. Sci. Paris 343 : 3 (August 2006), pp. 201–208. MR 2246339 Zbl 1155.60329 article

Let \( (R_t \), \( t\geq 0) \) denote a \( d \)-dimensional Bessel process (\( 0\lt d\lt 2 \)). For every \( t\geq 0 \), we consider the times \[ g_t = \sup\{s\leq t\mid R_s = 0\} \quad\text{and}\quad d_t = \inf\{s\gt t\mid R_s = 0\} ,\] as well as the three sequences: \( (V^n_{g_t} \), \( n\geq 1) \), \( (V^n_t \), \( n\geq 2) \), and \( (V^n_{d_t} \), \( n\geq 2) \), which consist of the lengths of excursions of \( R \) away from 0 before \( g_t \), before \( t \), and before \( d_t \), respectively, each one being ranked by decreasing order.

We obtain a limit theorem concerning each of the laws of these three sequences, as \( t\to\infty \). The result is expressed in terms of a positive, \( \sigma \)-finite measure \( \Pi \) on the set \( \mathscr{S}^{\downarrow} \) of decreasing sequences. \( Pi \) is closely related with the Poisson–Dirichlet laws on \( \mathscr{S}^{\downarrow} \).

[411] A. Nikeghbali and M. Yor: “Doob’s maximal identity, multiplicative decompositions and enlargements of filtrations,” pp. 791–814 in Joseph Doob: A collection of mathematical articles in his memory, published as Ill. J. Math. 50 : 1–4. Issue edited by J. Burkholder. University of Illinois at Urbana-Champaign, 2006. MR 2247846 Zbl 1101.60059 incollection

In the theory of progressive enlargements of filtrations, the supermartingale \[ Z_t = \mathbf{P}(g\gt t \mid \mathcal{F}_t) \] associated with an honest time \( g \), and its additive (Doob–Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale \( Z_t \), based on Doob’s maximal identity. We thus give new examples of progressive enlargements. Moreover, we give, in our setting, a proof of the decomposition formula for martingales, using initial enlargement techniques, and use it to obtain some path decompositions given the maximum or minimum of some processes.

Joseph L. Doob	Related	Volume
Ashkan Nikeghbali	Related

[412] T. Funaki, Y. Hariya, and M. Yor: “Wiener integrals for centered Bessel and related processes, II,” ALEA, Lat. Am. J. Probab. Math. Stat. 1 (2006), pp. 225–240. Electronic only. Part I was published (with a slightly different title) in Markov Process. Relat. Fields 13:1 (2007). MR 2249656 Zbl 1112.60042 article

This is the second part of a series of papers on the construction of stochastic integrals of Wiener’s type for the centered \( \delta \)-dimensional Bessel processes (BES(\( \delta \))-processes in short) and their variants. The approach adopted in the present paper is via the Brascamp–Lieb inequality. This method works well for the BES(\( \delta \))-processes, BES(\( \delta \))-bridges with \( \delta \geq 3 \), the Brownian meander and their extensions described by a class of stochastic differential equations, but not for their powers. As we have seen in the first part, another approach via Hardy’s \( L^2 \) inequality is effective for BES(\( \delta \))-processes with \( \delta\geq 1 \) and their powers. The method used in this paper is powerful to establish a family of accurate bounds on the distributions of these Wiener integrals.

[413] B. Roynette, P. Vallois, and M. Yor: “Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II,” Studia Sci. Math. Hung. 43 : 3 (2006), pp. 295–360. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2253307 Zbl 1121.60004 ArXiv math/0510575 article

Let \( P_0 \) denote the Wiener measure defined on the canonical space \[ \bigl( \Omega = \mathcal{C}(\mathbb{R}_+,\mathbb{R}), \,(X_t)_{t\geq 0}, \,(\mathcal{F}_t)_{t\geq 0} \bigr) ,\] and \( (S_t) \), resp. \( (I_t) \), be the one sided-maximum, resp. minimum, \( (L_t^0) \) the local time at 0, and \( (D_t) \) the number of down-crossings from \( b \) to \( a \) (with \( b \gt a \)). Let \[ f:\mathbb{R}\times\mathbb{R}^d\to (0,+\infty) \] be a Borel function, and \( (A_t) \) be a process chosen within the set: \[ \bigl\{(S_t), \,(S_t,t), \,(L_t^0), \,(S_t,I_t,L_t^0), \,(D_t) \bigr\} ,\] which consists of 5 elements. We prove a penalization result: under some suitable assumptions on \( f \), there exists a positive \( ((\mathcal{F}_t) \), \( P_0) \)-martingale \( (M_t^f) \), starting at 1, such that: \begin{equation*}\tag{1} \lim_{t\to\infty}\frac{E_0[1_{\Gamma_s}f(X_t,A_t)]}{E_0[f(X_t,A_t)]} = Q_0^f(\Gamma_s) := E_0[1_{\Gamma_s}M_s^f],\quad \forall\,\Gamma_s\in\mathcal{F}_s,\,s\geq 0. \end{equation*} We determine the law of \( (X_t) \) under the p.m. \( Q_0^f \) defined on \( (\Omega \), \( \mathcal{F}_{\infty}) \) by (1). For the 1st, 3rd and 5th elements of the set, we prove first that \[ Q_0^f(A_{\infty} \lt \infty) = 1 ,\] and more generally \[ Q_0^f(0 \lt g \lt \infty) = 1 \] where \( g = \sup\{s \gt 0 \mid A_s = A_{\infty}\} \) (with the convention \( \sup\emptyset = 0 \)). Secondly, we split the trajectory of \( (X_t) \) in two parts: \( (X_t)_{0\leq t\leq g} \) and \( (X_{t+g})_{t\geq 0} \), and we describe their laws under \( Q_0^f \), conditionally on \( A_{\infty} \). For the 2nd and 4th elements, a similar result holds replacing \( A_{\infty} \) by resp. \( S_{\infty} \), \( S_{\infty} \vee I \).

@article {key2253307m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Limiting laws associated with {B}rownian
		 motion perturbed by its maximum, minimum
		 and local time, {II}},
	JOURNAL = {Studia Sci. Math. Hung.},
	FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
		 A Quarterly of the Hungarian Academy
		 of Sciences},
	VOLUME = {43},
	NUMBER = {3},
	YEAR = {2006},
	PAGES = {295--360},
	DOI = {10.1556/SScMath.43.2006.3.3},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); III
		 in \textit{Period. Math. Hung.} \textbf{50}:1--2
		 (2005); IV in \textit{Studia Sci. Math.
		 Hung.} \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VI in \textit{ESAIM Probab. Stat.} \textbf{13}
		 (2009); VII in \textit{Ann. Inst. Henri
		 Poincar\'e Probab. Stat.} \textbf{45}:2
		 (2009); VIII in \textit{J. Funct. Anal.}
		 \textbf{255}:9 (2008); IX in \textit{ESAIM
		 Probab. Stat.} \textbf{14} (2010); X
		 in \textit{Theory Stoch. Process.} \textbf{14}:2
		 (2008). ArXiv:math/0510575. MR:2253307.
		 Zbl:1121.60004.},
	ISSN = {0081-6906},
}

[414] L. Gallardo and M. Yor: “A chaotic representation property of the multidimensional Dunkl processes,” Ann. Probab. 34 : 4 (2006), pp. 1530–1549. MR 2257654 Zbl 1107.60015 ArXiv math/0609679 article

[415] B. Roynette, P. Vallois, and M. Yor: “Some penalisations of the Wiener measure,” Jpn. J. Math. (3) 1 : 1 (April 2006), pp. 263–290. MR 2261065 Zbl 1160.60315 article

[416] In memoriam Paul-André Meyer: Séminaire de probabilités XXXIX [In memoriam Paul-André Meyer: Thirty-ninth probability seminar]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics 1874. Springer (Berlin), 2006. MR 2265371 Zbl 1092.60003 book

Paul-André Meyer	Related
Michel Emery	Related

[417] M. Yor: “The life and scientific work of Paul-André Meyer (August 21st, 1934–January 30th, 2003): ‘Un modèle pour nous tous’” [The life and scientific work of Paul-André Meyer (August 21st, 1934–January 30th, 2003): ‘A model for us all’], pp. 13–26 in In memoriam Paul-André Meyer: Séminaire de probabilités, XXXIX [In memoriam Paul-André Meyer: Thirty-ninth probability seminar]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics 1874. Springer (Berlin), 2006. MR 2276885 Zbl 1216.01025 incollection

Paul-André Meyer	Related
Michel Emery	Related

@incollection {key2276885m,
	AUTHOR = {Yor, Marc},
	TITLE = {The life and scientific work of {P}aul-{A}ndr\'e
		 {M}eyer ({A}ugust 21st, 1934--{J}anuary
		 30th, 2003): ``{U}n mod\`ele pour nous
		 tous'' [The life and scientific work
		 of {P}aul-{A}ndr\'e {M}eyer ({A}ugust
		 21st, 1934--{J}anuary 30th, 2003): ``{A}
		 model for us all'']},
	BOOKTITLE = {In memoriam {P}aul-{A}ndr\'e {M}eyer:
		 {S}\'eminaire de probabilit\'es, {XXXIX}
		 [In memoriam {P}aul-{A}ndr\'e {M}eyer:
		 {T}hirty-ninth probability seminar]},
	EDITOR = {\'Emery, M. and Yor, M.},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1874},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {2006},
	PAGES = {13--26},
	DOI = {10.1007/978-3-540-35513-7_2},
	NOTE = {MR:2276885. Zbl:1216.01025.},
	ISSN = {0075-8434},
	ISBN = {9783540309949},
}

[418] D. B. Madan and M. Yor: “Itô’s integrated formula for strict local martingales,” pp. 157–170 in In memoriam Paul-André Meyer: Séminaire de probabilités, XXXIX [In memoriam Paul-André Meyer: Thirty-ninth probability seminar]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics. Springer (Berlin), 2006. MR 2276895 Zbl 1133.60025 incollection

Paul-André Meyer	Related
Michel Emery	Related
Dilip B. Madan	Related

[419] B. Roynette, P. Vallois, and M. Yor: “Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement Brownien et à son maximum unilatère” [Penalizations and some extensions of Pitman’s theorem relative to Brownian motion and its one-sided maximum], pp. 305–336 in In memoriam Paul-André Meyer: Séminaire de probabilités XXXIX [In memoriam Paul-André Meyer: Thirty-ninth probability seminar]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics 1874. Springer (Berlin), 2006. MR 2276902 Zbl 1124.60034 incollection

Paul-André Meyer	Related
Michel Emery	Related
Bernard Roynette	Related

@incollection {key2276902m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {P\'enalisations et quelques extensions
		 du th\'eor\`eme de {P}itman, relatives
		 au mouvement {B}rownien et \`a son maximum
		 unilat\`ere [Penalizations and some
		 extensions of {P}itman's theorem relative
		 to {B}rownian motion and its one-sided
		 maximum]},
	BOOKTITLE = {In memoriam {P}aul-{A}ndr\'e {M}eyer:
		 {S}\'eminaire de probabilit\'es {XXXIX}
		 [In memoriam {P}aul-{A}ndr\'e {M}eyer:
		 {T}hirty-ninth probability seminar]},
	EDITOR = {\'Emery, M. and Yor, M.},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {1874},
	PUBLISHER = {Springer},
	ADDRESS = {Berlin},
	YEAR = {2006},
	PAGES = {305--336},
	DOI = {10.1007/978-3-540-35513-7_20},
	NOTE = {MR:2276902. Zbl:1124.60034.},
	ISSN = {0075-8434},
	ISBN = {9783540309949},
}

[420] L. Gallardo and M. Yor: “Some remarkable properties of the Dunkl martingales,” pp. 337–356 in In memoriam Paul-André Meyer: Séminaire de probabilités, XXXIX [In memoriam Paul-André Meyer: Thirty-ninth probability seminar]. Edited by M. Émery and M. Yor. Lecture Notes in Mathematics. Springer (Berlin), 2006. MR 2276903 Zbl 1128.60027 incollection

In this paper, we study a class, depending on a parameter \( k\geq 0 \), of real valued Feller processes \[ X^{(k)} = (X_t^{(k)})_{t\gt 0} ,\] the so called Dunkl processes which are martingales satisfying the Brownian scaling property. These processes are the only martingales whose absolute value is a Bessel process. Moreover, the absolute continuity and intertwining relations valid for some pairs of Bessel processes may be generalized to Dunkl processes. The main result of the paper is a mixed chaotic representation property for the \( L^2 \) space of the martingale \( X^{(k)} \) in terms of its continuous part (which is a Brownian motion) and its purely discontinuous part, a martingale \( \gamma \) with bracket \( \langle\gamma\rangle_t = 2kt \).

Paul-André Meyer	Related
Michel Emery	Related

[421] C. Donati-Martin and M. Yor: “Some explicit Krein representations of certain subordinators, including the gamma process,” Publ. Res. Inst. Math. Sci. 42 : 4 (December 2006), pp. 879–895. MR 2289152 Zbl 1123.60028 ArXiv math/0503254 article

[422] T. Funaki, Y. Hariya, F. Hirsch, and M. Yor: “On the construction of Wiener integrals with respect to certain pseudo-Bessel processes,” Stochastic Process. Appl. 116 : 12 (December 2006), pp. 1690–1711. MR 2307055 Zbl 1110.60055 article

In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals \[ \int_0^1 h(s)\,d\overline{R}_s ,\] for every \( h\in L^2([0,1] \), \( ds) \) and \( (\overline{R}_s) \) any centered Bessel process with dimension \( d\gt 1 \). In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form \[ \sqrt{t}f\left(\frac{R_t}{\sqrt{t}}\right) .\]

[423] S. È. Graversen, A. N. Shiryaev, and M. Yor: “On stochastic integral representations of functionals of Brownian motion, II,” Teor. Veroyatn. Primen. 51 : 1 (2006), pp. 64–77. An English translation was published in Theory Probab. Appl. 51:1 (2007). MR 2324166 article

[424] J. Bertoin, T. Fujita, B. Roynette, and M. Yor: “On a particular class of self-decomposable random variables: The durations of Bessel excursions straddling independent exponential times,” Probab. Math. Stat. 26 : 2 (2006), pp. 315–366. MR 2325310 Zbl 1123.60063 article

Jean Bertoin	Related
Bernard Roynette	Related

[425] G. Peccati and M. Yor: “Identities in law between quadratic functionals of bivariate Gaussian processes, through Fubini theorems and symmetric projections,” pp. 235–250 in Approximation and probability: Papers of the conference held on the occasion of the 70th anniversary of Prof. Zbigniew Ciesielski (Będlewo, Poland, 20–24 September 2004). Edited by T. Figiel and A. Kamont. Banach Center Publications 72. Polish Academy of Sciences (Warsaw), 2006. MR 2325748 Zbl 1116.60013 ArXiv math/0501506 incollection

[426] M. Yor: Vingt thèmes de recherches [Twenty research themes], May 2006. unpublished

[427] Aspects des mathématiques financiéres [Aspects of mathematical finance] (Paris, 1 Februrary 2005). Edited by M. Yor. Lavoisier/TEC et DOC (Paris), 2006. An English translation was published in 2008. Zbl 1090.91003 book

[428] L. F. James and M. Yor: Tilted stable subordinators, Gamma time changes and occupation time of rays by Bessel spiders. Preprint, January 2007. ArXiv math/0701049 techreport

[429] P. Carr, H. Geman, D. B. Madan, and M. Yor: “Self-decomposability and option pricing,” Math. Finance 17 : 1 (2007), pp. 31–57. MR 2281791 Zbl 1278.91157 article

The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with 6–10 free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices.

Hélyette Geman	Related
Dilip B. Madan	Related

[430] T. Funaki, Y. Hariya, F. Hirsch, and M. Yor: “On some Fourier aspects of the construction of certain Wiener integrals,” Stochastic Process. Appl. 117 : 1 (2007), pp. 1–22. MR 2287100 Zbl 1113.60055 article

The existence and best \( L^2 \)-bounds for the Wiener type integrals \[ \int_0^1 f(s)\,dX_s ,\] where \( X \) ranges through a wide class of Bessel-like centered processes, and \( f \) belongs to \( L^2([0,1]) \), are discussed in terms of Fourier transforms associated with some characteristics of \( X \), thus providing some unification of previous results on this topic obtained by the authors [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes and their Related Fields (2006), T. Funaki, Y. Hariya, F. Hirsch, M. Yor, On the construction of Wiener integrals with respect to certain pseudo-Bessel processes, Stochastic Processes and their Applications (2006)] as well as yielding new results.

[431] M. Yor: “How K. Itô revolutionized the study of stochastic processes,” Gaz. Math., Soc. Math. Fr. 111 (January 2007), pp. 51–55. An English translation was published in Jpn. J. Math. 2:1 (2007). MR 2289679 Zbl 1149.60302 article

[432] J. Pitman and M. Yor: “Itô’s excursion theory and its applications,” Japan. J. Math. (3) 2 : 1 (March 2007), pp. 83–96. MR 2295611 Zbl 1156.60066 article

[433] P. Salminen, P. Vallois, and M. Yor: “On the excursion theory for linear diffusions,” Japan. J. Math. (3) 2 : 1 (March 2007), pp. 97–127. MR 2295612 Zbl 1160.60024 ArXiv math/0612687 article

We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g., the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.

[434] M. Yor: “How K. Itô revolutionized the study of stochastic processes,” Jpn. J. Math. (3) 2 : 1 (March 2007), pp. 137–143. English translation of French original published in Gaz. Math. 111 (2007). MR 2295615 Zbl 1157.60325 article

[435] P. Bourgade, T. Fujita, and M. Yor: “Euler’s formulae for \( \zeta(2n) \) and products of Cauchy variables,” Electron. Commun. Probab. 12 (2007), pp. 73–80. Article no. 9. MR 2300217 Zbl 1129.60088 article

[436] T. Funaki, Y. Hariya, and M. Yor: “Wiener integrals for centered powers of Bessel processes, I,” Markov Process. Relat. Fields 13 : 1 (2007), pp. 21–56. Part II was published (with a slightly different title) in Lat. Am. J. Probab. Math. Stat. 1 (2006). MR 2321750 Zbl 1123.60035 article

[437] C. Donati-Martin and M. Yor: “Further examples of explicit Krein representations of certain subordinators,” Publ. Res. Inst. Math. Sci. 43 : 2 (2007), pp. 315–328. MR 2341013 Zbl 1129.60042 ArXiv math/0509041 article

[438] T. Fujita and M. Yor: “On the remarkable distributions of maxima of some fragments of the standard reflecting random walk and Brownian motion,” Probab. Math. Stat. 27 : 1 (2007), pp. 89–104. MR 2353273 Zbl 1130.60051 article

[439] T. Fujita and M. Yor: “A warning about an independence property related to random Brownian scaling,” Probab. Math. Stat. 27 : 1 (2007), pp. 105–108. MR 2353274 Zbl 1131.60075 article

[440] M. Yor: “Some remarkable properties of gamma processes,” pp. 37–47 in Advances in mathematical finance (College Park, MD, 29 September–1 October 1 2006). Edited by M. C. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott. Applied Numerical Harmonic Analysis. Birkhäuser (Boston, MA), 2007. Paper presented at the mathematical finance conference in honor of the 60th birthday of Dilip B. Madan. MR 2359361 Zbl 1156.60030 incollection

Ju-Yi J. Yen	Related
Dilip B. Madan	Related

[441] M. Yor: “A note about Selberg’s integrals in relation with the beta-gamma algebra,” pp. 49–58 in Advances in mathematical finance (College Park, MD, 29 September–1 October 1 2006). Edited by M. C. Fu, R. A. Jarrow, J.-Y. Yen, and R. J. Elliott. Applied Numerical Harmonic Analysis. Birkhäuser (Boston, MA), 2007. Paper presented at the mathematical finance conference in honor of the 60th birthday of Dilip B. Madan. MR 2359362 Zbl 1160.33002 incollection

To prove their formulae for the moments of the characteristic polynomial of the generic matrix of \( U(N) \), Keating and Snaith [2000] (see also Keating [2004]) use Selberg’s integrals as a ‘black box’. In this note, we point out some identities in law which are equivalent to the expressions of Selberg’s integrals and which involve beta, gamma, and normal variables. However, this is a mere probabilistic translation of Selberg’s results, and does not provide an independent proof of them. An outcome of some of these translations is that certain logarithms of (Vandermonde) random discriminants are self-decomposable, which hinges on the self-decomposability of the logarithms of the beta \( (a,b) \) (\( 2a + b \geq 1 \)) and gamma (\( a \gt 0 \)) variables. Such selfdecomposability properties have been of interest in some joint papers with D. Madan.

Ju-Yi J. Yen	Related
Dilip B. Madan	Related

[442] B. Roynette, P. Vallois, and M. Yor: “Some extensions of Pitman and Ray–Knight theorems for penalized Brownian motions and their local times, IV,” Studia Sci. Math. Hung. 44 : 4 (2007), pp. 469–516. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2361439 Zbl 1164.60355 article

@article {key2361439m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Some extensions of {P}itman and {R}ay--{K}night
		 theorems for penalized {B}rownian motions
		 and their local times, {IV}},
	JOURNAL = {Studia Sci. Math. Hung.},
	FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
		 A Quarterly of the Hungarian Academy
		 of Sciences},
	VOLUME = {44},
	NUMBER = {4},
	YEAR = {2007},
	PAGES = {469--516},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 V in \textit{Studia Sci. Math. Hung.}
		 \textbf{45}:1 (2008); VI in \textit{ESAIM
		 Probab. Stat.} \textbf{13} (2009); VII
		 in \textit{Ann. Inst. Henri Poincar\'e,
		 Probab. Stat.} \textbf{45}:2 (2009);
		 VIII in \textit{J. Funct. Anal.} \textbf{255}:9
		 (2008); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 MR:2361439. Zbl:1164.60355.},
	ISSN = {0081-6906},
}

[443] M. Yor: “Joseph Leo Doob (27 février 1910–7 juin 2004)” [Joseph Leo Doob (27 Feburary 1910–7 June 2004)], Gaz. Math. 114 (2007), pp. 33–39. MR 2361707 Zbl 1221.01089 article

[444] H. Geman, D. B. Madan, and M. Yor: “Probing option prices for information,” Methodol. Comput. Appl. Probab. 9 : 1 (March 2007), pp. 115–131. MR 2364984 Zbl 1157.60067 article

We present a methodology for extracting information from option prices when the market is viewed as knowledgeable. By expanding the information filtration judiciously and determining conditional characteristic functions for the log of the stock price, we obtain option pricing formulae which when fit to market data may reveal this information. In particular, we consider probing option prices for knowledge of the future stock price, instantaneous volatility, and the asymptotic dividend stream. Additionally the bridge laws developed are also useful for simulation based on stratified sampling that conditions on the terminal values of paths.

Hélyette Geman	Related
Dilip B. Madan	Related

[445] J. Najnudel, B. Roynette, and M. Yor: “A remarkable \( \sigma \)-finite measure on \( \mathcal{C}(\mathbf{R}_+,\mathbf{R}) \) related to many Brownian penalisations,” C. R. Math. Acad. Sci. Paris 345 : 8 (2007), pp. 459–466. MR 2367926 Zbl 1221.60003 article

Joseph Najnudel	Related
Bernard Roynette	Related

[446] L. Chaumont, L. Mazliak, and M. Yor: “Some aspects of the probabilistic work,” Chapter 3, pp. 41–66 in Kolmogorov’s heritage in mathematics. Edited by É. Charpentier, A. Lesne, and N. Nikolski. Springer (Berlin), 2007. English translation of chapter 3 of L’Héritage de Kolmogorov en mathématiques (2004). MR 2376738 Zbl 1369.60003 incollection

[447] P. Salminen and M. Yor: “Tanaka formula for symmetric Lévy processes,” pp. 265–285 in Séminaire de probabilités XL [Fortieth probability seminar]. Edited by C. Donati-Martin, M. Émery, A. Rouault, and C. Stricker. Lecture Notes in Mathematics 1899. Springer (Berlin), 2007. MR 2409011 Zbl 1129.60043 ArXiv math/0501182 incollection

[448] M. Yor: “Le mouvement Brownien: Une martingale exceptionnelle et néanmoins générique” [Brownian motion: An exceptional yet generic martingale], Chapter 4, pp. 103–138 in Leçons de mathématiques d’aujourd’hui [Today’s mathematics lessons], vol. 3. Edited by É. Charpentier and N. Nikolski. Le Sel et le Fer 17. Cassini (Paris), 2007. incollection

[449] S. È. Graversen, A. N. Shiryaev, and M. Yor: “On stochastic integral representations of functionals of Brownian motion, II,” Theory Probab. Appl. 51 : 1 (2007), pp. 65–77. English translation of Russian original published in Teor. Veroyatn. Primen. 51:1 (2007). Zbl 1116.60022 article

[450] M. Atlan, H. Geman, D. B. Madan, and M. Yor: “Correlation and the pricing of risks,” Ann. Finance 3 : 4 (October 2007), pp. 411–453. Zbl 1233.91320 article

Given a pricing kernel we investigate the class of risks that are not priced by this kernel. Risks are random payoffs written on underlying uncertainties that may themselves either be random variables, processes, events or information filtrations. A risk is said to be not priced by a kernel if all derivatives on this risk always earn a zero excess return, or equivalently the derivatives may be priced without a change of measure. We say that such risks are not kernel priced. It is shown that reliance on direct correlation between the risk and the pricing kernel as an indicator for the kernel pricing of a risk can be misleading. Examples are given of risks that are uncorrelated with the pricing kernel but are kernel priced. These examples lead to new definitions for risks that are not kernel priced in correlation terms. Additionally we show that the pricing kernel itself viewed as a random variable is strongly negatively kernel priced implying in particular that all monotone increasing functions of the kernel receive a negative risk premium. Moreover the equivalence class of the kernel under increasing monotone transformations is unique in possessing this property.

Marc Atlan	Related
Hélyette Geman	Related
Dilip B. Madan	Related

[451] A. Bentata and M. Yor: From Black–Scholes and Dupire formulae to last passage times of local martingales, Part A: The infinite time horizon. Preprint, June 2008. ArXiv 0806.0239 techreport

[452] A. Bentata and M. Yor: From Black–Scholes and Dupire formulae to last passage times of local martingales, Part B: The finite time horizon. Preprint, July 2008. ArXiv 0807.0788 techreport

[453] C. P. Hughes, A. Nikeghbali, and M. Yor: “An arithmetic model for the total disorder process,” Probab. Theory Relat. Fields 141 : 1–2 (May 2008), pp. 47–59. MR 2372965 Zbl 1144.60020 article

[454] B. Roynette, P. Vallois, and M. Yor: “Penalizing a \( \mathrm{BES}(d) \) process \( (0\lt d\lt 2) \) with a function of its local time, V,” Studia Sci. Math. Hung. 45 : 1 (2008), pp. 67–124. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4(2007); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2401169 Zbl 1164.60307 article

@article {key2401169m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Penalizing a \$\mathrm{BES}(d)\$ process
		 \$(0\lt d\lt 2)\$ with a function of its
		 local time, {V}},
	JOURNAL = {Studia Sci. Math. Hung.},
	FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
		 A Quarterly of the Hungarian Academy
		 of Sciences},
	VOLUME = {45},
	NUMBER = {1},
	YEAR = {2008},
	PAGES = {67--124},
	DOI = {10.1556/SScMath.2007.1042},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4(2007); VI in \textit{ESAIM
		 Probab. Stat.} \textbf{13} (2009); VII
		 in \textit{Ann. Inst. Henri Poincar\'e,
		 Probab. Stat.} \textbf{45}:2 (2009);
		 VIII in \textit{J. Funct. Anal.} \textbf{255}:9
		 (2008); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 MR:2401169. Zbl:1164.60307.},
	ISSN = {0081-6906},
}

[455] Aspects of mathematical finance (Paris, 1 February 2005). Edited by M. Yor. Springer (Berlin), 2008. English translation of 2006 French original. MR 2404094 Zbl 1132.91001 book

[456] E. Gobet, G. Pagès, and M. Yor: “Mathematics and finance,” pp. 63–76 in Aspects of mathematical finance (Paris, 1 February 2005). Edited by M. Yor. Springer (Berlin), 2008. English translation of French original from Mathématiques et finance (2006). MR 2409696 Zbl 1153.91300 incollection

[457] C. Donati-Martin, B. Roynette, P. Vallois, and M. Yor: “On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension \( d = 2(1-\alpha) \), \( 0 \lt \alpha \lt 1 \),” Studia Sci. Math. Hung. 45 : 2 (2008), pp. 207–221. MR 2417969 Zbl 1212.60070 article

[458] S. N. Majumdar, J. Randon-Furling, M. J. Kearney, and M. Yor: “On the time to reach maximum for a variety of constrained Brownian motions,” J. Phys. A 41 : 36 (2008). Article no. 365005, 18 pp. MR 2447861 Zbl 1195.82020 ArXiv 0802.2619 article

We derive \( P(M,t_m) \), the joint probability density of the maximum \( M \) and the time \( t_m \) at which this maximum is achieved, for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over \( M \), the marginal density \( P(t_m) \) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of \( P(t_m) \) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path-integral technique; however, an alternative approach is also outlined which is founded on certain ‘agreement formulae’ that are encountered more generally in probabilistic studies of Brownian motion processes.

[459] P. Bourgade, C. P. Hughes, A. Nikeghbali, and M. Yor: “The characteristic polynomial of a random unitary matrix: A probabilistic approach,” Duke Math. J. 145 : 1 (2008), pp. 45–69. MR 2451289 Zbl 1155.15025 ArXiv 0706.0333 article

In this article, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin–Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [2000] using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [2000] is now obtained from the classical central limit theorems of probability theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm-type results.

[460] R. Mansuy and M. Yor: Aspects of Brownian motion. Universitext. Springer (Berlin), 2008. MR 2454984 Zbl 1162.60022 book

[461] M.-K. von Renesse, M. Yor, and L. Zambotti: “Quasi-invariance properties of a class of subordinators,” Stochastic Process. Appl. 118 : 11 (November 2008), pp. 2038–2057. MR 2462287 Zbl 1180.60044 ArXiv 0706.3010 article

[462] N. Enriquez, C. Sabot, and M. Yor: “Renewal series and square-root boundaries for Bessel processes,” Electron. Commun. Probab. 13 (2008), pp. 649–652. Article no. 59. MR 2466192 Zbl 1190.60031 ArXiv 0806.3197 article

[463] B. Roynette and M. Yor: “Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII,” J. Funct. Anal. 255 : 9 (November 2008), pp. 2606–2640. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2473270 Zbl 1180.60070 article

@article {key2473270m,
	AUTHOR = {Roynette, B. and Yor, M.},
	TITLE = {Ten penalisation results of {B}rownian
		 motion involving its one-sided supremum
		 until first and last passage times,
		 {VIII}},
	JOURNAL = {J. Funct. Anal.},
	FJOURNAL = {Journal of Functional Analysis},
	VOLUME = {255},
	NUMBER = {9},
	MONTH = {November},
	YEAR = {2008},
	PAGES = {2606--2640},
	DOI = {10.1016/j.jfa.2008.04.024},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VI in \textit{ESAIM Probab. Stat.} \textbf{13}
		 (2009); VII in \textit{Ann. Inst. Henri
		 Poincar\'e, Probab. Stat.} \textbf{45}:2
		 (2009); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 MR:2473270. Zbl:1180.60070.},
	ISSN = {0022-1236},
}

[464] L. F. James, B. Roynette, and M. Yor: “Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples,” Probab. Surv. 5 (2008), pp. 346–415. MR 2476736 Zbl 1189.60035 ArXiv 0708.3932 article

In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (GGC) variables, their Wiener–Gamma representations, and relation with the Dirichlet processes.

To a GGC variable, one may associate a unique Thorin measure. Let \( G \) a positive r.v. and \( \Gamma_t(G) \) (resp. \( \Gamma_t(1/G) \) the Generalized Gamma Convolution with Thorin measure \( t \)-times the law of \( G \) (resp. the law of \( 1/G \)). In Section 2, we compare the laws of \( \Gamma_t(G) \) and \( \Gamma_t(1/G) \).

In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.

Lancelot Fitzgerald James	Related
Bernard Roynette	Related

[465] B. Roynette, P. Vallois, and M. Yor: “Penalisations of Brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding, X,” Theory Stoch. Process. 14 : 2 (2008), pp. 116–138. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010). MR 2479739 Zbl 1224.60076 article

@article {key2479739m,
	AUTHOR = {Roynette, B. and Vallois, P. and Yor,
		 M.},
	TITLE = {Penalisations of {B}rownian motion with
		 its maximum and minimum processes as
		 weak forms of {S}korokhod embedding,
		 {X}},
	JOURNAL = {Theory Stoch. Process.},
	FJOURNAL = {Theory of Stochastic Processes},
	VOLUME = {14},
	NUMBER = {2},
	YEAR = {2008},
	PAGES = {116--138},
	URL = {https://hal.archives-ouvertes.fr/hal-00275179},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VI in \textit{ESAIM Probab. Stat.} \textbf{13}
		 (2009); VII in \textit{Ann. Inst. Henri
		 Poincar\'e, Probab. Stat.} \textbf{45}:2
		 (2009); VIII in \textit{J. Funct. Anal.}
		 \textbf{255}:9 (2008); IX in \textit{ESAIM
		 Probab. Stat.} \textbf{14} (2010). MR:2479739.
		 Zbl:1224.60076.},
	ISSN = {0321-3900},
}

[466] G. Peccati and M. Yor: “Burkholder’s submartingales from a stochastic calculus perspective,” Ill. J. Math. 52 : 3 (2008), pp. 815–824. MR 2546009 Zbl 1176.60033 ArXiv 0705.3633 article

[467] D. Madan, B. Roynette, and M. Yor: “Option prices as probabilities,” Financ. Res. Lett. 5 : 2 (June 2008), pp. 79–87. article

Four distribution functions are associated with call and put prices seen as functions of their strike and maturity. The random variables associated with these distributions are identified when the process for moneyness defined as the stock price relative to the forward price is a positive local martingale with no positive jumps that tends to zero at infinity. Results on calls require moneyness to be a continuous martingale as well. It is shown that for puts the distributions in the strike are those for the remaining supremum while for calls, they relate to the remaining infimum. In maturity we see the distribution functions for the last passage times of moneyness to strike.

Bernard Roynette	Related
Dilip B. Madan	Related

[468] B. Roynette and M. Yor: Existence and properties of pseudo-inverses for Bessel and related processes. Preprint, December 2008. techreport

[469] M. Yor: “Faut-il avoir peur des Mathématiques Financieres?” [Should we be afraid of financial mathematics?], Matapli 87 (2008), pp. 47–52. article

[470] M. Yor, J.-P. Kahane, S. Jaffard, and D. Talay: “Mathématiques financières et industrie bancaire: Le point actuel; quelques perspectives” [Financial mathematics and the banking industry: The current point; some perspectives], Matapli 86 (June 2008), pp. 21–34. article

Stéphane Jaffard	Related
Jean-Pierre Kahane	Related

[471] D. Madan, B. Roynette, and M. Yor: “Unifying Black–Scholes type formulae which involve Brownian last passage times up to a finite horizon,” Asia-Pac. Financ. Mark. 15 : 2 (June 2008), pp. 97–115. Zbl 1163.91414 article

Bernard Roynette	Related
Dilip B. Madan	Related

[472] Harmonic and stochastic analysis of Dunkl processes. Edited by P. Graczyk, M. Rösler, and M. Yor. Travaux en Cours: Mathematique 71. Lavoisier Hermann (Paris), 2008. Zbl 1205.60005 book

[473] M. Yor: “Kiyosi Itô (1915–2008),” Gaz. Math., Soc. Math. Fr. 119 (January 2009), pp. 115. MR 2482837 article

[474] B. Roynette and M. Yor: Penalising Brownian paths. Lecture Notes in Mathematics 1969. Springer (Berlin), 2009. Dedicated to Frank Knight (1933–2007). MR 2504013 Zbl 1190.60002 book

[475] B. Roynette, P. Vallois, and M. Yor: “Penalisations of multidimensional Brownian motion, VI,” ESAIM Probab. Stat. 13 (January 2009), pp. 152–180. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2518544 Zbl 1189.60069 article

As in preceding papers in which we studied the limits of penalized 1-dimensional Wiener measures with certain functionals \( \Gamma_t \), we obtain here the existence of the limit, as \( t\to\infty \), of \( d \)-dimensional Wiener measures penalized by a function of the maximum up to time \( t \) of the Brownian winding process (for \( d = 2 \)), or in \( d\geq 2 \) dimensions for Brownian motion prevented to exit a cone before time \( t \). Various extensions of these multidimensional penalisations are studied, and the limit laws are described. Throughout this paper, the skew-product decomposition of \( d \)-dimensional Brownian motion plays an important role.

@article {key2518544m,
	AUTHOR = {Roynette, Bernard and Vallois, Pierre
		 and Yor, Marc},
	TITLE = {Penalisations of multidimensional {B}rownian
		 motion, {VI}},
	JOURNAL = {ESAIM Probab. Stat.},
	FJOURNAL = {European Series in Applied and Industrial
		 Mathematics (ESAIM): Probability and
		 Statistics},
	VOLUME = {13},
	MONTH = {January},
	YEAR = {2009},
	PAGES = {152--180},
	DOI = {10.1051/ps:2008003},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VII in \textit{Ann. Inst. Henri Poincar\'e,
		 Probab. Stat.} \textbf{45}:2 (2009);
		 VIII in \textit{J. Funct. Anal.} \textbf{255}:9
		 (2008); IX in \textit{ESAIM Probab.
		 Stat.} \textbf{14} (2010); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 MR:2518544. Zbl:1189.60069.},
	ISSN = {1292-8100},
}

[476] D. Baker and M. Yor: “A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion,” Electron. J. Probab. 14 : 52 (2009), pp. 1532–1540. Article no. 52. MR 2519530 Zbl 1201.60033 article

[477] F. Hirsch and M. Yor: “Fractional intertwinings between two Markov semigroups,” Potential Anal. 31 : 2 (August 2009), pp. 133–146. MR 2520721 Zbl 1175.26010 article

[478] B. Roynette, P. Vallois, and M. Yor: “Brownian penalisations related to excursion lengths, VII,” Ann. Inst. Henri Poincaré, Probab. Stat. 45 : 2 (2009), pp. 421–452. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VIII in J. Funct. Anal. 255:9 (2008); IX in ESAIM Probab. Stat. 14 (2010); X in Theory Stoch. Process. 14:2 (2008). MR 2521408 Zbl 1181.60046 article

@article {key2521408m,
	AUTHOR = {Roynette, B. and Vallois, P. and Yor,
		 M.},
	TITLE = {Brownian penalisations related to excursion
		 lengths, {VII}},
	JOURNAL = {Ann. Inst. Henri Poincar\'e, Probab.
		 Stat.},
	FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
		 Probabilit\'es et Statistiques},
	VOLUME = {45},
	NUMBER = {2},
	YEAR = {2009},
	PAGES = {421--452},
	DOI = {10.1214/08-AIHP177},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VI in \textit{ESAIM Probab. Stat.} \textbf{13}
		 (2009); VIII in \textit{J. Funct. Anal.}
		 \textbf{255}:9 (2008); IX in \textit{ESAIM
		 Probab. Stat.} \textbf{14} (2010); X
		 in \textit{Theory Stoch. Process.} \textbf{14}:2
		 (2008). MR:2521408. Zbl:1181.60046.},
	ISSN = {0246-0203},
}

[479] J. Najnudel, B. Roynette, and M. Yor: A global view of Brownian penalisations. MSJ Memoirs 19. Mathematical Society of Japan (Tokyo), 2009. MR 2528440 Zbl 1180.60004 ArXiv 0905.2220 book

Joseph Najnudel	Related
Bernard Roynette	Related

[480] A. Nikeghbali and M. Yor: “The Barnes \( G \) function and its relations with sums and products of generalized gamma convolution variables,” Electron. Commun. Probab. 14 (2009), pp. 396–411. Article no. 39. MR 2545290 Zbl 1189.60073 ArXiv 0707.3187 article

We give a probabilistic interpretation for the Barnes \( G \)-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating–Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes \( G \)-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.

[481] K. Yano, Y. Yano, and M. Yor: “Penalising symmetric stable Lévy paths,” J. Math. Soc. Japan 61 : 3 (2009), pp. 757–798. MR 2552915 Zbl 1180.60008 ArXiv 0807.4336 article

[482] M. Yor: “J. L. Doob (27 February 1910–7 June 2004),” Ann. Probab. 37 : 5 (2009), pp. 1664–1670. Translation of French original published in Gaz. Math. 114 (2007). MR 2561429 Zbl 1176.60004 article

[483] M. Jeanblanc, M. Yor, and M. Chesney: Mathematical methods for financial markets. Springer Finance. Springer (London), 2009. MR 2568861 Zbl 1205.91003 book

[484] M. Yor: “Some aspects of K. Itô’s works,” Eur. Math. Soc. Newsl. 72 (June 2009), pp. 11–12. Reprinted in Stochastic Process. Appl. 120:5 (2010). MR 2571601 Zbl 1189.01067 article

[485] F. Hirsch and M. Yor: “A construction of processes with one dimensional martingale marginals, based upon path-space Ornstein–Uhlenbeck processes and the Brownian sheet,” J. Math. Kyoto Univ. 49 : 2 (2009), pp. 389–417. MR 2571849 Zbl 1203.60122 article

[486] F. Hirsch and M. Yor: “A construction of processes with one-dimensional martingale marginals, associated with a Lévy process, via its Lévy sheet,” J. Math. Kyoto Univ. 49 : 4 (2009), pp. 785–815. MR 2591117 Zbl 1191.60040 article

[487] D. Madan, B. Roynette, and M. Yor: “Put option prices as joint distribution functions in strike and maturity: The Black–Scholes case,” Int. J. Theor. Appl. Finance 12 : 8 (2009), pp. 1075–1090. MR 2598932 Zbl 1183.91179 article

Bernard Roynette	Related
Dilip B. Madan	Related

[488] K. Yano, Y. Yano, and M. Yor: “On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes,” pp. 187–227 in Séminaire de probabilités XLII [Forty-second probability seminar]. Edited by C. Donati-Martin, M. Émery, A. Rouault, and C. Stricker. Lecture Notes in Mathematics 1979. Springer (Berln), 2009. MR 2599211 Zbl 1201.60043 ArXiv 0811.2046 incollection

[489] P. Bourgade and M. Yor: “Random matrices and the Riemann zeta function,” pp. 25–40 in Journées Élie Cartan 2006, 2007 et 2008 [Élie Cartan Days 2006, 2007 and 2008]. Edited by J. Sokolowski. Institut Élie Cartan 19. Institut Élie Cartan (Vandoeuvre-lès-Nancy, France), 2009. MR 2792032 incollection

[490] B. Roynette, P. Vallois, and M. Yor: “A family of generalized gamma convoluted variables,” Probab. Math. Statist. 29 : 2 (2009), pp. 181–204. MR 2792539 Zbl 1197.60012 article

[491] M. Yor: “Ébauche de réponse à M. Michel Rocard” [Draft response to Mr. Michel Rocard], Gaz. Math., Soc. Math. Fr. 119 (January 2009), pp. 75–76. article

[492] L. Chaumont and M. Yor: Exercises in probability: A guided tour from measure theory to random processes, via conditioning, Reprint edition. Cambridge University Press, 2009. Reprint of 2003 original. Zbl 1180.60002 book

[493] M. Yor and M. E. Vares: “A tribute to Professor Kiyosi Itô,” Stochastic Process. Appl. 120 : 1 (January 2010), pp. 104. This is a brief announcement of the special issue dedicated to Itô, Stochastic Process. Appl. 120:5 (2010). MR 2565848 Zbl 1178.01066 article

Maria Eulália Vares	Related
Kiyoshi Itō	Related

[494] C. Profeta, B. Roynette, and M. Yor: Option prices as probabilities: A new look at generalized Black–Scholes formulae. Springer Finance. Springer (Berlin), 2010. MR 2582990 Zbl 1188.91004 book

To the best of our knowledge this book discusses in a unique way last passage times. The Black–Scholes formula plays a central role in Mathematical Finance; it gives the right price at which buyer and seller can agree with, in the geometric Brownian framework, when strike \( K \) and maturity \( T \) are given. This yields an explicit well-known formula, obtained by Black and Scholes in 1973. The present volume gives another representation of this formula in terms of Brownian last passages times, which, to our knowledge, has never been made in this sense. The volume is devoted to various extensions and discussions of features and quantities stemming from the last passages times representation in the Brownian case such as: past-future martingales, last passage times up to a finite horizon, pseudo-inverses of processes.. They are developed in eight chapters, with complements, appendices and exercises.

Bernard Roynette	Related
Christophe Profeta	Related

[495] M. Yor: “Some aspects of K. Itô’s works,” Stochastic Process. Appl. 120 : 5 (May 2010), pp. 577–579. Reprinted from Eur. Math. Soc. Newsl. 72 (2009). MR 2603052 Zbl 1194.60002 article

[496] O. Kella and M. Yor: “A new formula for some linear stochastic equations with applications,” Ann. Appl. Probab. 20 : 2 (2010), pp. 367–381. MR 2650036 Zbl 1196.60122 ArXiv 1009.3373 article

We give a representation of the solution for a stochastic linear equation of the form \[ X_t = Y_t + \int_{(0,t]}X_{s^-} \,dZ_s \] where \( Z \) is a càdlàg semimartingale and \( Y \) is a càdlàg adapted process with bounded variation on finite intervals. As an application we study the case where \( Y \) and \( -Z \) are nondecreasing, jointly have stationary increments and the jumps of \( -Z \) are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When \( Y \) and \( Z \) are, in addition, independent Lévy processes, the resulting \( X \) is called a generalized Ornstein–Uhlenbeck process.

[497] B. Roynette and M. Yor: “Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX,” ESAIM Probab. Stat. 14 (March 2010), pp. 65–92. Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46:2 (2003); II in Studia Sci. Math. Hung. 43:3 (2006); III in Period. Math. Hung. 50:1–2 (2005); IV in Studia Sci. Math. Hung. 44:4 (2007); V in Studia Sci. Math. Hung. 45:1 (2008); VI in ESAIM Probab. Stat. 13 (2009); VII in Ann. Inst. Henri Poincaré, Probab. Stat. 45:2 (2009); VIII in J. Funct. Anal. 255:9 (2008); X in Theory Stoch. Process. 14:2 (2008). MR 2654548 Zbl 1219.60036 article

@article {key2654548m,
	AUTHOR = {Roynette, Bernard and Yor, Marc},
	TITLE = {Local limit theorems for {B}rownian
		 additive functionals and penalisation
		 of {B}rownian paths, {IX}},
	JOURNAL = {ESAIM Probab. Stat.},
	FJOURNAL = {European Series in Applied and Industrial
		 Mathematics (ESAIM): Probability and
		 Statistics},
	VOLUME = {14},
	MONTH = {March},
	YEAR = {2010},
	PAGES = {65--92},
	DOI = {10.1051/ps:2008028},
	NOTE = {Parts I--X have very different titles.
		 I was published in \textit{Studia Sci.
		 Math. Hung.} \textbf{46}:2 (2003); II
		 in \textit{Studia Sci. Math. Hung.}
		 \textbf{43}:3 (2006); III in \textit{Period.
		 Math. Hung.} \textbf{50}:1--2 (2005);
		 IV in \textit{Studia Sci. Math. Hung.}
		 \textbf{44}:4 (2007); V in \textit{Studia
		 Sci. Math. Hung.} \textbf{45}:1 (2008);
		 VI in \textit{ESAIM Probab. Stat.} \textbf{13}
		 (2009); VII in \textit{Ann. Inst. Henri
		 Poincar\'e, Probab. Stat.} \textbf{45}:2
		 (2009); VIII in \textit{J. Funct. Anal.}
		 \textbf{255}:9 (2008); X in \textit{Theory
		 Stoch. Process.} \textbf{14}:2 (2008).
		 MR:2654548. Zbl:1219.60036.},
	ISSN = {1292-8100},
}

[498] F. Hirsch and M. Yor: “Looking for martingales associated to a self-decomposable law,” Electron. J. Probab. 15 (2010), pp. 932–961. Article no. 29. MR 2659753 Zbl 1225.60131 article

[499] M. Yor: “Paul Malliavin (1925–2010) et son calcul (1976–…)” [Paul Malliavin (1925–2010) and his calculus (1976–…)], Gaz. Math., Soc. Math. Fr. 126 (2010), pp. 114–115. MR 2722013 Zbl 1226.01032 article

[500] F. Hirsch, B. Roynette, and M. Yor: “Applying Itô’s motto: ‘Look at the infinite dimensional picture’ by constructing sheets to obtain processes increasing in the convex order,” Period. Math. Hung. 61 : 1–2 (2010), pp. 195–211. MR 2728438 Zbl 1274.60052 article

Bernard Roynette	Related
Kiyoshi Itō	Related

[501] F. Hirsch, B. Roynette, and M. Yor: “Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets,” Expo. Math. 28 : 4 (2010), pp. 299–324. MR 2734446 Zbl 1223.60027 article

[502] K. Yano, Y. Yano, and M. Yor: “Penalisation of a stable Lévy process involving its one-sided supremum,” Ann. Inst. Henri Poincaré, Probab. Stat. 46 : 4 (2010), pp. 1042–1054. MR 2744885 Zbl 1208.60046 article

[503] J. Yen and M. Yor: Moments thoughts about an integration by parts in distribution for Brownian quadratic functionals, 2010. unpublished

[504] M. Yor: Dix autre thèmes de recherche sur les processus stochastiques, II [Ten other research themes on stochastic processes, II], August 2010. Part I, with a different title, was also unpublished, 2011. unpublished

[505] M. Yor: Du mouvement brownien aux processus de Lévy, via les semi-martingales et les diffusions [From Brownian motions to Lévy processes, via semi-martingales and diffusions], February 2010. unpublished

[506] M. Yor: “Squared Bessel processes,” pp. 1678–1679 in Encyclopedia of quantitative finance, vol. 4. Edited by R. Cont and P. Tankov. Wiley (Hoboken, NJ), 2010. incollection

[507] A tribute to Kiyosi Itô, published as Stochastic Process. Appl. 120 : 5. Issue edited by M. Yor and M. E. Vares. Elsevier (Amsterdam), May 2010. A brief announcement of this special issue was published in Stochastic Process. Appl. 120:1 (2010). book

Maria Eulália Vares	Related
Kiyoshi Itō	Related

[508] M. Yor and M. E. Vares: “Introducing the volume,” pp. 585–589 in A tribute to Kiyosi Itô, published as Stochastic Processes Appl. 120 : 5 (special issue). Issue edited by M. Yor and M. E. Vares. Elsevier (Amsterdam), May 2010. Zbl 1203.60003 incollection

Maria Eulália Vares	Related
Kiyoshi Itō	Related

[509] D. Dufresne and M. Yor: A two-dimensional extension of Bougerol’s identity in law for the exponential functional of Brownian motion. Preprint, November 2011. techreport

[510] D. Dufresne and M. Yor: A two-dimensional extension of Bougerol’s identity in law for the exponential functional of Brownian motion, II: The story so far…, 2011. unpublished

[511] J.-Y. Yen and M. Yor: “Truncation functions and Laplace transform,” Stat. Probab. Lett. 81 : 3 (March 2011), pp. 417–419. MR 2748950 Zbl 1209.60012 article

[512] P. Salminen and M. Yor: “On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes,” Period. Math. Hung. 62 : 1 (March 2011), pp. 75–101. Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays. MR 2772384 Zbl 1274.60251 ArXiv 1012.2038 article

Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary \( t\mapsto a + bt \), \( a\geq 0 \), \( b\in\mathbb{R} \), by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter \( \delta \gt 0 \) and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.

Pál Révész	Related
Endre Csáki	Related

[513] J.-Y. Yen and M. Yor: “Call option prices based on Bessel processes,” Methodol. Comput. Appl. Probab. 13 : 2 (June 2011), pp. 329–347. MR 2788861 Zbl 1217.60071 ArXiv 0808.3402 article

[514] D. Baker and M. Yor: “On martingales with given marginals and the scaling property,” pp. 437–439 in Séminaire de probabilités XLIII [Forty third probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2006. Springer (Berlin), 2011. MR 2790385 Zbl 1216.60040 incollection

[515] D. Baker, C. Donati-Martin, and M. Yor: “A sequence of Albin type continuous martingales with Brownian marginals and scaling,” pp. 441–449 in Séminaire de probabilités XLIII [Forty-third probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2006. Springer (Berlin), 2011. MR 2790386 Zbl 1216.60039 incollection

[516] F. Hirsch, C. Profeta, B. Roynette, and M. Yor: “Constructing self-similar martingales via two Skorokhod embeddings,” pp. 451–503 in Séminaire de probabilités XLIII [Forty-third probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2006. Springer (Berlin), 2011. MR 2790387 Zbl 1234.60047 incollection

Bernard Roynette	Related
Christophe Profeta	Related

[517]D. W. Stroock, M. Yor, J.-P. Kahane, R. Gundy, L. Gross, and M. Vergne: “Remembering Paul Malliavin,” Notices Amer. Math. Soc. 58 : 4 (2011), pp. 568–573. MR 2807523

[518] F. Hirsch, C. Profeta, B. Roynette, and M. Yor: Peacocks and associated martingales, with explicit constructions. Bocconi & Springer Series 3. Springer (New York), 2011. MR 2808243 Zbl 1227.60001 book

Bernard Roynette	Related
Christophe Profeta	Related

[519] S. Vakeroudis, M. Yor, and D. Holcman: “The mean first rotation time of a planar polymer,” J. Stat. Phys. 143 : 6 (June 2011), pp. 1074–1095. MR 2813786 Zbl 1221.82159 ArXiv 1101.1737 article

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of \( n \) rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on \( \sqrt{n} \) and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.

[520] M. Hoffmann and D. Lamberton: “Preface.” Edited by M. Hoffmann and D. Lamberton. ESAIM, Probab. Stat. 15 : supplement (2011), pp. S1. MR 2817340 article

[521] D. B. Madan and M. Yor: “The S&P 500 index as a Sato process travelling at the speed of the VIX,” Appl. Math. Finance 18 : 3–4 (2011), pp. 227–244. MR 2818165 Zbl 1239.91186 article

The logarithm of SPX is modeled as a Sato process running at a speed proportional to the current level of the VIX. When the logarithm of the VIX is an exponential compound Poisson process with drift one may obtain exact expressions for the prices of equity options taken at an independent exponential maturity. The parameters for the Lévy process are calibrated from VIX options while the parameters for the Sato process driving the stock may be calibrated from market option prices taken at an independent exponential maturity. Results confirm that both the option surface and the VIX time changed Sato process volatilities, skews and term volatility spreads are responsive to the VIX level and the VIX option surface.

[522] F. Hirsch, B. Roynette, and M. Yor: “From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order,” J. Math. Soc. Japan 63 : 3 (2011), pp. 887–917. MR 2836749 Zbl 1233.60008 article

[523] P. Carr, H. Geman, D. B. Madan, and M. Yor: “Options on realized variance and convex orders,” Quant. Finance 11 : 11 (2011), pp. 1685–1694. MR 2850996 Zbl 1277.91164 article

Realized variance option and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk neutral densities are said to be increasing in the convex order. For Lévy processes such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability then the resulting risk neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally we consider modelling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.

Hélyette Geman	Related
Dilip B. Madan	Related

[524] M. Yor: “Small and big probability worlds,” pp. 261–271 in Marcinkiewicz centenary volume (28 June–2 July 2010). Edited by M. Nawrocki and W. Wnuk. Banach Center Publications 95. Polish Academy of Sciences (Warsaw), 2011. MR 2918098 Zbl 1238.60001 incollection

[525] F. Petit and M. Yor: A propos de certaines extensions de l’algèbre beta-gamma [On certain extensions of the beta-gamma algebra], September 2011. unpublished

[526] M. Yor: Dix thèmes de recherche sur les processus stochastiques qui me tiennent à coeur et m’ont longtemps occupé, I [Ten research themes on stochastic processes that are important to me and have long occupied me, I], 2011. Part II, with a different title, was also unpublished, 2010. unpublished

[527] M. Yor: “Un modèle stochastique peut en cacher un autre” [A stochastic model can hide another], La Recherche 451 (April 2011), pp. 2–21. article

[528] M. Yor: “Why I became especially interested to work from F. Spitzer’s paper about the asymptotics of planar Brownian windings,” pp. 341–346 in All that math: Portraits of mathematicians as young readers. Edited by A. Córdoba, J. L. Fernández, and P. Ferández. Revista Matemática Iberoamericana (Madrid), 2011. incollection

Antonio Córdoba	Related
Frank Ludvig Spitzer	Related

[529] S. Vakeroudis and M. Yor: “A central limit theorem for a sequence of Brownian motions in the unit sphere in \( \mathbb{R}^n \),” Stat. Probab. Lett. 82 : 3 (March 2012), pp. 599–605. MR 2887477 Zbl 1239.60022 ArXiv 1107.3230 article

[530] F. T. Bruss and M. Yor: “Stochastic processes with proportional increments and the last-arrival problem,” Stochastic Process. Appl. 122 : 9 (September 2012), pp. 3239–3261. MR 2946441 Zbl 1255.60166 article

The notion of stochastic processes with proportional increments is introduced. This notion is of general interest as indicated by its relationship with several stochastic processes, as counting processes, Lévy processes, and others, as well as martingales related with these processes. The focus of this article is on the motivation to introduce processes with proportional increments, as instigated by certain characteristics of stopping problems under weak information. We also study some general properties of such processes. These lead to new insights into the mechanism and characterization of Pascal processes. This again will motivate the introduction of more general f-increment processes as well as the analysis of their link with martingales. As a major application we solve the no-information version of the last-arrival problem which was an open problem. Further applications deal with the impact of proportional increments on modelling investment problems, with a new proof of the \( 1/e \)-law of best choice, and with other optimal stopping problems.

[531] F. Hirsch and M. Yor: “On temporally completely monotone functions for Markov processes,” Probab. Surv. 9 (2012), pp. 253–286. MR 2947802 Zbl 1245.60071 article

Any negative moment of an increasing Lamperti process \( (Y_t \), \( t\geq 0) \) is a completely monotone function of \( t \). This property enticed us to study systematically, for a given Markov process \( (Y_t \), \( t\geq 0) \), the functions \( f \) such that the expectation of \( f(Y_t) \) is a completely monotone function of \( t \). We call these functions temporally completely monotone (for \( Y \)). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when \( Y \) is a Lévy process, a Bessel process, or an increasing Lamperti process.

[532] S. Vakeroudis and M. Yor: “Some infinite divisibility properties of the reciprocal of planar Brownian motion exit time from a cone,” Electron. Commun. Probab. 17 (2012). Article no. 23, 9 pp. MR 2950189 Zbl 1259.60097 ArXiv 1201.2718 article

[533] M. Émery and M. Yor: “Erratum to Séminaire XXVII: ‘On the Lévy transformation of Brownian motions and continuous martingales’,” pp. 467 in Séminaire de probabilités XLIV [Forty-fourth probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2046. Springer (Berlin), 2012. Erratum to an article published in Séminaire de probabilités XXVII (1993). MR 2953360 incollection

[534] L. Chaumont and M. Yor: Exercises in probability: A guided tour from measure theory to random processes, via conditioning, 2nd edition. Cambridge Series in Statistical and Probabilistic Mathematics 35. Cambridge University Press, 2012. Republication of 2003 original. MR 2964501 Zbl 1246.60001 book

[535] C.-T. Wu, J.-Y. Yen, and M. Yor: “Measuring the ‘non-stopping timeness’ of ends of previsible sets,” Taiwanese J. Math. 16 : 5 (October 2012), pp. 1589–1599. MR 2970673 Zbl 1273.60049 ArXiv 0810.1059 article

[536] D. B. Madan and M. Yor: “Moments of Wiener integrals for subordinators,” Electron. Commun. Probab. 17 (2012). Article no. 55, 8 pp. MR 2999983 Zbl 1329.60165 article

[537] S. Vakeroudis and M. Yor: “A scaling proof for Walsh’s Brownian motion extended arc-sine law,” Electron. Commun. Probab. 17 (2012). Article no. 63, 9 pp. MR 3007207 Zbl 1323.60115 ArXiv 1206.3688 article

[538] N. El Karoui, E. Pardoux, and M. Yor: Stochastic filtering at Saint-Flour (Saint-Flour, France, 1979). Probability at Saint-Flour. Springer (Heidelberg), 2012. Reprint of lectures originally published in Ecole d’eté de probabilités de Saint-Flour IX–1979 (1981), including Yor’s contributions. MR 3075395 Zbl 05993935 book

[539] J. Najnudel, D. Stroock, and M. Yor: “On a flow of transformations of a Wiener space,” Chapter 5, pp. 119–131 in Stochastic analysis and related topics: In honour of Ali Süleyman Üstünel (Paris, 14–15 June 2010). Edited by L. Decreusefond and J. Najim. Springer Proceedings in Mathematics & Statistics 22. Springer (Berlin), 2012. MR 3236089 Zbl 1338.60176 incollection

Daniel W. Stroock	Related
Joseph Najnudel	Related

[540] P. Salminen and M. Yor: On some probabilistic occurrences of the Poisson summation formula, 2012. unpublished

[541] A. P. C. Lim, J.-Y. Yen, and M. Yor: “Some examples of Skorokhod embeddings obtained from the Azéma–Yor algorithm,” Stochastic Process. Appl. 123 : 2 (February 2013), pp. 329–346. MR 3003354 Zbl 1258.60032 article

[542] J. Bertoin and M. Yor: “Retrieving information from subordination,” pp. 97–106 in Prokhorov and contemporary probability theory. Edited by A. N. Shiryaev, S. R. S. Varadhan, and E. L. Presman. Springer Proceedings in Mathematics & Statistics 33. Springer (Heidelberg), 2013. MR 3070468 Zbl 1284.60084 ArXiv 1005.3187 incollection

S. R. S. Varadhan	Related	Volume
Albert N. Shiryaev	Related
Jean Bertoin	Related

[543] J. Bertoin and M. Yor: “Pure jump increasing processes and the change of variables formula,” Electron. Commun. Probab. 18 (2013). Article no. 41, 7 pp. MR 3070907 Zbl 1306.60120 ArXiv 1303.6452 article

[544] J.-Y. Yen and M. Yor: “Illustration of various methods for solving partly Skorokhod’s embedding problem,” Electron. Commun. Probab. 18 : 48 (2013). Article no. 48, 5 pp. MR 3078011 Zbl 1326.60056 article

[545] J.-Y. Yen and M. Yor: “On an identity in law between Brownian quadratic functionals,” Statist. Probab. Lett. 83 : 9 (September 2013), pp. 2015–2018. MR 3079037 Zbl 1290.60081 article

[546] F. Hirsch and M. Yor: “On the Mellin transforms of the perpetuity and the remainder variables associated to a subordinator,” Bernoulli 19 : 4 (2013), pp. 1350–1377. MR 3102555 Zbl 1287.60096 ArXiv 1309.7801 article

[547] S. Vakeroudis and M. Yor: “Integrability properties and limit theorems for the exit time from a cone of planar Brownian motion,” Bernoulli 19 : 5A (2013), pp. 2000–2009. MR 3127946 Zbl 1294.60103 ArXiv 1201.2716 article

[548] F. Hirsch and M. Yor: “On the remarkable Lamperti representation of the inverse local time of a radial Ornstein–Uhlenbeck process,” Bull. Belg. Math. Soc. Simon Stevin 20 : 3 (2013), pp. 435–449. MR 3129051 Zbl 1287.60048 article

[549] J.-Y. Yen and M. Yor: Local times and excursion theory for Brownian motion: A tale of Wiener and Itô measures. Lecture Notes in Mathematics 2088. Springer (Cham, Switzerland), 2013. MR 3134857 Zbl 1364.60003 book

[550] M. Yor: “Sur l’œuvre de Paul Lévy” [On the works of Paul Lévy], ESAIM, Probab. Stat. 17 (2013), pp. 789. MR 3141783 Zbl 1285.01021 article

[551] J. Bertoin, D. Dufresne, and M. Yor: “Some two-dimensional extensions of Bougerol’s identity in law for the exponential functional of linear Brownian motion,” Rev. Mat. Iberoam. 29 : 4 (2013), pp. 1307–1324. MR 3148605 Zbl 1303.60073 ArXiv 1201.1495 article

Jean Bertoin	Related
Daniel Dufresne	Related

[552] E. Eberlein, D. Madan, M. Pistorius, and M. Yor: “A simple stochastic rate model for rate equity hybrid products,” Appl. Math. Finance 20 : 5–6 (2013), pp. 461–488. MR 3169863 Zbl 1396.91780 article

A positive spot rate model driven by a gamma process and correlated with equity is introduced and calibrated via closed forms for the joint characteristic function for the rate \( r \), its integral \( y \) and the logarithm of the stock price \( s \) under the \( T \)-forward measure. The law of the triple \( (r,y,s) \) is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics, 46(1), doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described by Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk, 14(1), pp. 95–122).

[553] M. Yor: “On weak and strong Brownian filtrations: Definitions and examples,” pp. 115–121 in Self-similar processes and their applications (Angers, France, 20–24 July 2009). Edited by L. Chaumont, P. Graczyk, and L. Vostrikova. Séminaires et Congrès 28. Société Mathématique de France (Paris), 2013. MR 3203521 Zbl 1311.60090 incollection

[554] J.-Y. Yen and M. Yor: (Planar) Brownian motion as a key stochastic process, 2013. Conference paper delivered at the seminar “Ars Conjectandi: A celebration of 300 years of stochastics,” Freiburg-Basel, Germany, 21–24 May 2013. unpublished

[555] J.-F. Le Gall and J. Pitman: “Obituary: Marc Yor 1949–2014,” Notices Am. Math. Soc. 61 : 5 (May 2014), pp. 508–509. article

James W. Pitman	Related
Jean-François Le Gall	Related

[556] E. Eberlein, D. Madan, M. Pistorius, W. Schoutens, and M. Yor: “Two price economies in continuous time,” Ann. Finance 10 : 1 (February 2014), pp. 71–100. MR 3159206 Zbl 1298.91086 article

Static and discrete time pricing operators for two price economies are reviewed and then generalized to the continuous time setting of an underlying Hunt process. The continuous time operators define nonlinear partial integro-differential equations that are solved numerically for the three valuations of bid, ask and expectation. The operators employ concave distortions by inducing a probability into the infinitesimal generator of a Hunt process. This probability is then distorted. Two nonlinear operators based on different approaches to truncating small jumps are developed and termed \( QV \) for quadratic variation and \( NL \) for normalized Lévy. Examples illustrate the resulting valuations. A sample book of derivatives on a single underlier is employed to display the gap between the bid and ask values for the book and the sum of comparable values for the components of the book.

[557] H. Geman and M. Jeanblanc: “Marc Yor: A beautiful mind has disappeared,” Stochastic Process. Appl. 124 : 6 (June 2014), pp. v–vii. MR 3188345 Zbl 1294.01043 article

[558] R. Elie, M. Rosenbaum, and M. Yor: “On the expectation of normalized Brownian functionals up to first hitting times,” Electron. J. Probab. 19 (2014), pp. Article no. 37, 23 pp. MR 3194736 Zbl 1291.60164 ArXiv 1310.1181 article

[559] J.-F. Le Gall: “Marc Yor (1949–2014),” Gaz. Math., Soc. Math. Fr. 140 (2014), pp. 73–74. MR 3201605 Zbl 1358.01120 article

[560] J.-F. Le Gall and J. Pitman: “Marc Yor (1949–2014),” Notices Am. Math. Soc. 61 : 5 (May 2014), pp. 508–509. MR 3203242 Zbl 1338.01050 article

James W. Pitman	Related
Jean-François Le Gall	Related

[561] J. Bertoin and M. Yor: “Local times for functions with finite variation: Two versions of Stieltjes change-of-variables formula,” Bull. Lond. Math. Soc. 46 : 3 (2014), pp. 553–560. MR 3210711 Zbl 1295.26011 ArXiv 1307.1288 article

[562] E. Eberlein, D. B. Madan, M. Pistorius, and M. Yor: “Bid and ask prices as non-linear continuous time G-expectations based on distortions,” Math. Financ. Econ. 8 : 3 (June 2014), pp. 265–289. MR 3212643 Zbl 1307.91086 article

[563] P. K. Friz, S. Gerhold, and M. Yor: “How to make Dupire’s local volatility work with jumps,” Quant. Finance 14 : 8 (2014), pp. 1327–1331. MR 3231348 Zbl 06425903 ArXiv 1302.5548 article

[564] M. Jeanblanc and F. Hirsch: “Marc Yor 1949–2014,” Matapli 103 (2014), pp. 51–55. With remembrances of Yor by Bernard Roynette. MR 3235376 Zbl 1365.01056 article

[565] F. Hirsch and M. Yor: “Comparing Brownian stochastic integrals for the convex order,” pp. 3–19 in Modern stochastics and applications (Kiev, 10–14 September 2012). Edited by V. Korolyuk, N. Limnios, Y. Mishura, L. Sakhno, and G. Shevchenko. Springer Optimization and its Applications 90. Springer (Cham, Switzerland), 2014. Dedicated to B. V. Gnedenko on the occasion of his 100th birthday and to M. I. Yadrenko on the occasion of his 80th birthday. MR 3236065 Zbl 1322.60079 incollection

@incollection {key3236065m,
	AUTHOR = {Hirsch, Francis and Yor, Marc},
	TITLE = {Comparing {B}rownian stochastic integrals
		 for the convex order},
	BOOKTITLE = {Modern stochastics and applications},
	EDITOR = {Korolyuk, Volodymyr and Limnios, Nikolaos
		 and Mishura, Yuliya and Sakhno, Lyudmyla
		 and Shevchenko, Georgiy},
	SERIES = {Springer Optimization and its Applications},
	NUMBER = {90},
	PUBLISHER = {Springer},
	ADDRESS = {Cham, Switzerland},
	YEAR = {2014},
	PAGES = {3--19},
	DOI = {10.1007/978-3-319-03512-3_1},
	NOTE = {(Kiev, 10--14 September 2012). Dedicated
		 to B. V. Gnedenko on the occasion of
		 his 100th birthday and to M. I. Yadrenko
		 on the occasion of his 80th birthday.
		 MR:3236065. Zbl:1322.60079.},
	ISSN = {1931-6828},
	ISBN = {9783319035116},
}

[566] M. Rosenbaum and M. Yor: “On the law of a triplet associated with the pseudo-Brownian bridge,” pp. 359–375 in Séminaire de probabilités, XLVI [Forty-sixth probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2123. Springer (Cham, Switzerland), 2014. MR 3330825 Zbl 1390.60298 ArXiv 1310.7164 incollection

[567] T. Fujita, Y. Kawanishi, and M. Yor: “On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called ‘meander option’,” Pac. J. Math. Ind. 6 (December 2014), pp. 65–71. Article no. 2. MR 3404143 Zbl 1386.91140 article

[568] R. Mansuy: Une excursion avec Marc Yor [An excursion with Marc Yor], February 2014. Online, on CNRS “Images des Mathématiques” website. misc

[569] Z. Shi: “Obituary: Marc Yor (1949–2014),” Bernoulli News 21 : 1 (May 2014), pp. 6. article

[570] M. Atlan, D. Madan, and H. Geman: “Marc Yor and mathematical finance,” pp. 79–90 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Hélyette Geman	Related
Dilip B. Madan	Related
Jean Bertoin	Related
Jean-François Le Gall	Related
Marc Atlan	Related

[571] J. Bertoin: “Marc Yor et les temps locaux” [Marc Yor and local time], pp. 25–38 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[572] P. Bourgade: “Marc Yor et les matrices aléatoires” [Marc Yor and random matrices], pp. 91–102 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[573] C. Donati and F. Petit: “Marc Yor et les identités en loi” [Marc Yor and identities in law], pp. 67–78 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[574] M. Émery: “Un Balzac des probabilité” [A Balzac of probability], pp. 15–24 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Michel Emery	Related
Jean Bertoin	Related
Jean-François Le Gall	Related

[575] F. Hirsch and B. Roynette: “Marc Yor et les peacocks” [Marc Yor and peacocks], pp. 111–120 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Bernard Roynette	Related
Jean Bertoin	Related
Jean-François Le Gall	Related

[576] M. Jeanblanc: “Quelques souvenirs de Marc” [Some memories of Marc], pp. 9–12 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[577] M. Jeanblanc-Picqué and A. N. Shiryaev: “In memory of Marc Yor,” Teor. Veroyatnost. i Primenen. 59 : 1 (2015), pp. 205–206. English translation of Russian original published in Theory Probab. Appl. 59:1 (2005). article

[578] J.-F. Le Gall: “Marc Yor et les nombres de tours du mouvement brownien” [Marc Yor and the winding numbers of Brownian motion], pp. 39–53 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[579] J.-Y. Yen and M. Yor: “A variant of Pitman’s theorem on \( (2J_s-R_s \), \( s\geq 0) \) for a general transient Bessel process \( R_{(+)} \) and its implications for the corresponding Ito’s measure \( \mathbf{n}_{(-)} \),” J. Theor. Probab. 28 : 1 (March 2015), pp. 223–230. MR 3320966 Zbl 1327.60160 article

Projection properties of the future infimum of a transient Bessel process \( R_{(+)} \) with dimension \( d_{(+)} = 2(1+ \alpha) \) (\( \alpha \in (0,1) \)) as well as the definition of the local time \( (L_t) \) of a recurrent Bessel process \( R_{(+)} \) with dimension \( d_{(+)} = 2(1 - \alpha) \) as the compensator of \( (R_{(+)})^{2 \alpha} \) may be seen to play some hidden but quite efficient role to obtain several integral representation formulae for the excursion theory of the \( R_{(+)} \) process away from 0. The precise formulae, which involve simple universal constants, are quite useful when dealing with the whole family of Bessel processes for dimensions between 0 and 2 (i.e., the reflecting case) and between 2 and 4 (i.e., the transient case).

[580] F. T. Bruss and M. Yor: “A new proof of Williams’ decomposition of the Bessel process of dimension three with a look at last-hitting times,” Bull. Belg. Math. Soc. Simon Stevin 22 : 2 (2015), pp. 319–330. MR 3351045 Zbl 1329.60275 ArXiv 1301.2527 article

[581] Y. Hu, Z. Shi, and M. Yor: “The maximal drawdown of the Brownian meander,” Electron. Commun. Probab. 20 (2015). Article no. 39, 6 pp. MR 3352334 Zbl 1325.60134 ArXiv 1604.04765 article

[582] K. Yano and M. Yor: “Around Tsirelson’s equation, or: The evolution process may not explain everything,” Probab. Surv. 12 (2015), pp. 1–12. MR 3374628 Zbl 1328.60170 ArXiv 0906.3442 article

[583] J.-Y. Yen and M. Yor: “On two results of P. Deheuvels,” pp. 305–308 in Mathematical statistics and limit theorems: Festschrift in honour of Paul Deheuvels (Paris, 20–21 June 2013). Edited by M. Hallin, D. Mason, D. Pfeifer, and J. Steinebach. Springer (Cham, Switzerland), 2015. MR 3380743 Zbl 1317.60108 incollection

[584] J.-Y. Yen and M. Yor: “Some topics in probability theory,” pp. 309–314 in Mathematical statistics and limit theorems: Festschrift in honour of Paul Deheuvels (Paris, 20–21 June 2013). Edited by M. Hallin, D. Mason, D. Pfeifer, and J. G. Steinebach. Springer (Cham, Switzerland), 2015. MR 3380744 Zbl 1328.60002 incollection

[585] In memoriam Marc Yor: Séminaire de probabilités XLVII [In memoriam Marc Yor: Forty-seventh probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2137. Springer (Cham, Switzerland), 2015. MR 3381858 Zbl 1327.60016 book

[586] C.-O. Ewald and M. Yor: “On increasing risk, inequality and poverty measures: Peacocks, lyrebirds and exotic options,” J. Econom. Dynam. Control 59 (October 2015), pp. 22–36. MR 3396306 Zbl 06917012 article

We extend the Rothschild and Stiglitz [1970] notion of increasing risk to families of random variables and in this way link their approach to the concept of stochastic processes which are increasing in the convex order. These processes have been introduced in seminal work by Strassen [1965], Doob [1968] and Kellerer [1972], who showed that such processes have the same marginals as a martingale. In fact, we demonstrate that their results include the results of Rothschild and Stiglitz as a special case. Further, we show that it makes sense to look at a larger class of processes, which we refer to as lyrebirds. We also show how these processes link up with the concept of second order stochastic dominance and are helpful in studying the dynamics of inequality and poverty measures. Further applications discussed include geometric and hyperbolic discounting, exotic derivatives and real options.

[587] M. Yor: “A Gaussian martingale which is the sum of two independent Gaussian non-semimartingales,” Electron. Commun. Probab. 20 (2015). Article no. 70, 5 pp. MR 3407214 Zbl 1329.60099 article

[588] F. Hirsch, B. Roynette, and M. Yor: “Kellerer’s theorem revisited,” pp. 347–363 in Asymptotic laws and methods in stochastics: A volume in honour of Miklós Csörgő on the occasion of his 80th birthday (Ottawa, 3–6 July 2012). Edited by D. Dawson, R. Kulik, M. Ould Haye, B. Szyszkowicz, and Y. Zhao. Fields Institute Communications 76. Fields Institute (Toronto), 2015. MR 3409839 Zbl 1368.60045 incollection

Bernard Roynette	Related
Rafał Kulik	Related
Miklós Csörgő	Related
Donald Dawson	Related

[589] M. Jeanblanc and A. Shiryaev: “In memory of Marc Yor,” Theory Probab. Appl. 59 : 1 (2015), pp. 180. English translation of Russian original published in Teor. Veroyatnost. i Primenen. 59:1 (2005). MR 3416074 Zbl 1314.01019 article

[590] M. Rosenbaum and M. Yor: “Random scaling and sampling of Brownian motion,” J. Math. Soc. Japan 67 : 4 (2015), pp. 1771–1784. MR 3417513 Zbl 1335.60157 article

[591] M. Rosenbaum and M. Yor: “Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling,” ESAIM, Probab. Stat. 19 (December 2015), pp. 578–589. MR 3433427 Zbl 1333.60181 ArXiv 1311.1900 article

We show that simple explicit formulas can be obtained for several relevant quantities related to the laws of the uniformly sampled Brownian bridge, Brownian meander and three dimensional Bessel process. To prove such results, we use the distribution of a triplet of random variables associated to the pseudo-Brownian bridge given in [M. Rosenbaum and M. Yor, Séminaire de Probabilités XLVI (2014) 359–375], together with various relationships between the laws of these four processes. Finally, we consider the variable \[ B_{UT_1}/\sqrt{T_1} ,\] where \( B \) is a Brownian motion, \( T_1 \) its first hitting time of level one and \( U \) a uniform random variable independent of \( B \). This variable is shown to be centered in [R. Elie, M. Rosenbaum and M. Yor, Electron. J. Probab. 37 (2014) 1–23; M. Rosenbaum and M. Yor, Séminaire de Probabilités XLVI (2014) 359–375]. The results obtained here enable us to revisit this intriguing property through an enlargement of filtration formula.

[592] J. Azéma, P. Barrieu, J. Bertoin, M. E. Caballero, C. Donati-Martin, M. Émery, F. Hirsch, Y. Hu, M. Ledoux, J. Najnudel, R. Mansuy, L. Miclo, Z. Shi, and D. Williams: “Témoignages” [Testimonials], pp. xi–xxx in In memoriam Marc Yor: Séminaire de probabilités XLVII [In memoriam Marc Yor: Forty-seventh probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2137. Springer (Cham, Switzerland), 2015. MR 3444289 incollection

Joseph Najnudel	Related
Jacques Azéma	Related
Michel Emery	Related
David Williams	Related
Jean Bertoin	Related

@incollection {key3444289m,
	AUTHOR = {Az\'ema, Jacques and Barrieu, Pauline
		 and Bertoin, Jean and Caballero, Maria
		 Emilia and Donati-Martin, Catherine
		 and \'Emery, Michel and Hirsch, Francis
		 and Hu, Yueyun and Ledoux, Michel and
		 Najnudel, Joseph and Mansuy, Roger and
		 Miclo, Laurent and Shi, Zhan and Williams,
		 David},
	TITLE = {T\'emoignages [Testimonials]},
	BOOKTITLE = {In memoriam {M}arc {Y}or: {S}\'eminaire
		 de probabilit\'es {XLVII} [In memoriam
		 {M}arc {Y}or: {F}orty-seventh probability
		 seminar]},
	EDITOR = {Donati-Martin, Catherine and Lejay,
		 Antoine and Rouault, Alain},
	SERIES = {Lecture Notes in Mathematics},
	NUMBER = {2137},
	PUBLISHER = {Springer},
	ADDRESS = {Cham, Switzerland},
	YEAR = {2015},
	PAGES = {xi--xxx},
	NOTE = {MR:3444289.},
	ISSN = {0075-8434},
	ISBN = {9783319185842},
}

[593] B. Bru: “Marc et le dossier Doeblin” [Marc and the Doeblin dossier], pp. xxxi–l in In memoriam Marc Yor: Séminaire de probabilités XLVII [In memoriam Marc Yor: Forty-seventh probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2137. Springer (Cham, Switzerland), 2015. MR 3444290 incollection

Wolfgang Doeblin	Related
Bernard Bru	Related

[594] P. Salminen, J.-Y. Yen, and M. Yor: “Integral representations of certain measures in the one-dimensional diffusions excursion theory,” pp. 1–15 in In memoriam Marc Yor: Séminaire de probabilités XLVII [In memoriam Marc Yor: Forty-seventh probability seminar]. Edited by C. Donati-Martin, A. Lejay, and A. Rouault. Lecture Notes in Mathematics 2137. Springer (Cham, Switzerland), 2015. MR 3444291 Zbl 1334.60165 incollection

In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion \( X \). These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, \( \mathrm{BES}(3) \), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair \( (\mathrm{RBM} \), \( \mathrm{BES}(3)) \) generalize to pair \( (X,X^{\uparrow}) \), where \( X^{\uparrow} \) denotes the diffusion obtained from \( X \) by conditioning \( X \) not to hit 0. We illustrate the results for the pair \( (R_-,R_+) \) consisting of a recurrent Bessel process with dimension \( d_- = 2(1-\alpha) \), \( \alpha\in (0,1) \), and a transient Bessel process with dimension \( d_+ = 2(1 + \alpha) \). Pair \( (\mathrm{RBM} \), \( \mathrm{BES}(3)) \) is, clearly, obtained by choosing \( \alpha=1/2 \).

[595] J. Pitman: “Marc Yor and Brownian excursions,” pp. 55–66 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

James W. Pitman	Related
Jean Bertoin	Related
Jean-François Le Gall	Related

[596] B. Roynette: “Le travail de Marc sur les pénalisations” [Marc’s work on penalizations], pp. 103–110 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Bernard Roynette	Related
Jean Bertoin	Related
Jean-François Le Gall	Related

[597] Z. Shi: “Des points et des lignes: Souvenirs de Marc Yor” [Points and lines: Memories of Marc Yor], pp. 13–14 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[598] M. Yor: “Dix thèmes de recherche” [Ten research themes], pp. 121–152 in Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. incollection

Jean Bertoin	Related
Jean-François Le Gall	Related

[599] Marc Yor: La passion du mouvement brownien [Marc Yor: The passion of Brownian motion]. Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi. Société Mathématique de France (Paris), 2015. Gazette des Mathématiciens and Matapli special issue. Zbl 1314.60009 book

Jean Bertoin	Related
Jean-François Le Gall	Related

[600] D. B. Madan and M. Yor: “On valuing stochastic perpetuities using new long horizon stock price models distinguishing booms, busts, and balanced markets,” Math. Finance 26 : 2 (2016), pp. 296–328. MR 3481306 Zbl 1348.91272 article

For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.

[601] B. Malleinum and M. Yor: Exercices sur les temps locaux de semi-martingales continues et les excursions browniennes [Exercises on local times of continuous semi-martingales and Brownian excursions]. Preprint, June 2016. ArXiv 1606.07118 techreport

Depuis le tout début du XX\( {}^\textrm{e} \) siècle, l’étude des processus stochastiques est un domaine très actif de la recherche en mathématiques. Parmi ces processus, le mouvement brownien–dont l’étude mathématique a été initiée dès 1900, avec la thèse de Bachelier, entre autres travaux–a joué, et joue encore, un rôle primordial. Ceci peut s’expliquer par le fait que le mouvement brownien est l’objet limite quasi-universel qui apparaît dans le théorème central limite, lorsqu’on fait agir le temps. Depuis la fin de la seconde guerre mondiale et les travaux d’Itô, Meyer, Tanaka et bien d’autres, les temps locaux et les excursions sont devenus des outils essentiels pour étudier ce processus.

Les exercices de ce volume ont été élaborés, année après année, par le second auteur, soit à la suite de lectures d’articles présentant, parfois avec des méthodes très différentes, telle ou telle propriété brownienne, soit simplement pour illustrer le contenu de son cours de DEA (anciennement), de M2 aujourd’hui. Le premier auteur en a organisé la synthèse, de façon économique et néanmoins–espérons-le–très lisible. Les chapitres ont été conçus pour créer un aller-retour permanent entre les principaux résultats du cours et les exercices corrigés, afin que la compréhension des uns renforce celle des autres. C’est ainsi que de nombreuses solutions d’exercices données ici offrent un aperçu de la façon de prouver certains des théorèmes rappelés plus haut.

[602] O. Kella and M. Yor: “Unifying the Dynkin and Lebesgue–Stieltjes formulae,” J. Appl. Probab. 54 : 1 (March 2017), pp. 252–266. MR 3632617 Zbl 06943558 ArXiv 1308.5795 article

We establish a local martingale \( M \) associate with \( f(X,Y) \) under some restrictions on \( f \), where \( Y \) is a process of bounded variation (on compact intervals) and either \( X \) is a jump diffusion (a special case being a Lévy process) or \( X \) is some general (càdlàg metric-space valued) Markov process. In the latter case, \( f \) is restricted to the form \[ f(x,y) = \sum_{k=1}^K \xi_k(x)\eta_k(y) .\] This local martingale unifies both Dynkin’s formula for Markov processes and the Lebesgue–Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes an \( L^2 \)-martingale. Convergence of the product of this Martingale with some deterministic function (of time) to 0 both in \( L^2 \) and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

[603] M. Yor: “Grossissements de filtrations: Grossissements initiaux et progressifs” [Enlargements of filtrations: Initial and progressive enlargements], ESAIM, Proc. Surv. 56 (June 2017), pp. 139–143. Zbl 06847886 article

[604] C.-O. Ewald and M. Yor: “On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales,” Math. Finance 28 : 2 (2018), pp. 536–549. MR 3780966 Zbl 1390.91299 article

[605] J. Pitman and M. Yor: A guide to Brownian motion stochastic processes. Preprint, February 2018. ArXiv 1802.09679 techreport

This is a guide to the mathematical theory of Brownian motion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley and Durrett , and who wants a broader perspective on the theory of Brownian motion and related stochastic processes than can be found in these texts.

[606] W. Doeblin: Œuvres complètes: Collected works. Edited by M. Yor and B. Bru. Springer (Berlin), 2018. Zbl 06625546 book

Wolfgang Doeblin	Related
Bernard Bru	Related

[607]B. Malleinu and M. Yor: Temps locaux de semi-martingales continues et les excursions browniennes. Enseignement des mathématiques. Cassini, 2023. Forthcoming. book