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
Abstract
People
BibTeX
The investigation concerns the limit of several families of Riemann sums, converging to the Itô stochastic integral of a continuous process with respect to a continuous semimartingale, to the Stratonovich stochastic integral, or to the Stieltjes integral with respect to the bracket of two continuous semimartingales. The last section concerns the stochastic integral of a differential form.
@incollection {key448556m,
AUTHOR = {Yor, Marc},
TITLE = {Sur quelques approximations d'int\'egrales
stochastiques [On some approximations
of stochastic integrals]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XI} [Eleventh
probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {581},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {257--297},
DOI = {10.1007/BFb0087222},
URL = {http://www.numdam.org/item?id=SPS_1977__11__518_0},
NOTE = {MR:448556. Zbl:0367.60058.},
ISSN = {0075-8434},
ISBN = {9783540081456},
}
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
Abstract
People
BibTeX
Yoeurp’s lemma is the following: if \( A \) is a previsible process of bounded variation, its square bracket \( [A,L] \) with any local martingale \( L \) is a local martingale. This useful result was not easily accessible, thus a complete proof is given, with several new applications — in particular, this characterizes previsible processes of bounded variation among semimartingales.
@incollection {key451396m,
AUTHOR = {Yor, Marc},
TITLE = {\`{A} propos d'un lemme de {C}h.~{Y}oeurp
[On a lemma of {C}h. {Y}oeurp]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XI} [Eleventh
probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {581},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {493--501},
DOI = {10.1007/BFb0087220},
URL = {http://www.numdam.org/item?id=SPS_1977__11__493_0},
NOTE = {MR:451396. Zbl:0373.60047.},
ISSN = {0075-8434},
ISBN = {9783540081456},
}
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
Abstract
People
BibTeX
The main result on the relation between the previsible representation property of a set of local martingales and the extremality of their joint law appeared in a celebrated paper of Jacod and Yor Z. für W-theorie , 38 , 1977. Several concrete applications are given here, in particular a complete proof of a “folklore” result on the representation of local martingales of a Lévy process, and a discussion of the commutation problem of [Dellacherie and Stricker 1977].
@incollection {key458580m,
AUTHOR = {Yor, Marc},
TITLE = {Remarques sur la repr\'esentation des
martingales comme int\'egrales stochastiques
[Remarks on the the representation of
martingales as stochastic integrals]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XI} [Eleventh
probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {581},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {502--517},
DOI = {10.1007/BFb0087221},
URL = {https://eudml.org/doc/113134},
NOTE = {MR:458580. Zbl:0367.60046.},
ISSN = {0075-8434},
ISBN = {9783540081456},
}
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
People
BibTeX
@incollection {key471060m,
AUTHOR = {Yor, Marc},
TITLE = {Sur les th\'eories du filtrage et de
la pr\'ediction [On the theories of
filtration and prediction]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XI} [Eleventh
probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {581},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1977},
PAGES = {257--297},
DOI = {10.1007/BFb0087195},
URL = {http://www.numdam.org/item?id=SPS_1977__11__257_0},
NOTE = {MR:471060. Zbl:0367.60041.},
ISSN = {0075-8434},
ISBN = {9783540081456},
}
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
Abstract
People
BibTeX
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.
@incollection {key519996m,
AUTHOR = {Yor, Marc},
TITLE = {Grossissement d'une filtration et semi-martingales:
{T}h\'eor\`emes g\'en\'eraux [Enlargement
of a filtration and semi-martingales:
{G}eneral theorems]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}
[Twelfth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {61--69},
DOI = {10.1007/BFb0064595},
URL = {http://www.numdam.org/item?id=SPS_1978__12__61_0},
NOTE = {MR:519996. Zbl:0411.60044.},
ISSN = {0075-8434},
ISBN = {9783540087618},
}
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
Abstract
People
BibTeX
@incollection {key519998m,
AUTHOR = {Jeulin, T. and Yor, M.},
TITLE = {Grossissement d'une filtration et semi-martingales:
{F}ormules explicites [Enlargement of
a filtration and semi-martingales: {E}xplicit
formulas]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}
[Twelfth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {78--97},
DOI = {10.1007/BFb0064597},
URL = {http://www.numdam.org/item?id=SPS_1978__12__78_0},
NOTE = {MR:519998. Zbl:0411.60045.},
ISSN = {0075-8434},
ISBN = {9783540087618},
}
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
Abstract
People
BibTeX
The general subject is the weak topology \( \sigma(H^1,BMO) \) on the space \( H^1 \) . Its relatively compact sets are characterized by a uniform integrability property of the maximal functions. A sequential completeness a result (a Vitali–Hahn–Saks like theorem) is proved. Finally, a separate section is devoted to the denseness of \( L^{\infty} \) in \( BMO \) , a subject which has greatly progressed since (the Garnett–Jones theorem, see [Émery 1981]; see also [Schachermayer 1996] and [Grandits 1999]).
@incollection {key519999m,
AUTHOR = {Dellacherie, C. and Meyer, P.-A. and
Yor, M.},
TITLE = {Sur certaines propri\'et\'es des espaces
de {B}anach \$H^1\$ et {BMO} [On certain
properties of \$H^1\$ {B}anach and {BMO}
spaces]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}
[Twelfth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {98--113},
DOI = {10.1007/BFb0064598},
URL = {http://www.numdam.org/item?id=SPS_1978__12__98_0},
NOTE = {MR:519999. Zbl:0392.60009.},
ISSN = {0075-8434},
ISBN = {9783540087618},
}
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
Abstract
People
BibTeX
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.
@incollection {key520008m,
AUTHOR = {Yor, Marc},
TITLE = {Sous-espaces denses dans \$L^1\$ ou \$H^1\$
et repr\'esentation des martingales
[Dense subspaces in \$L^1\$ or \$H^1\$ and
representation of martingales]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}
[Twelfth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {265--309},
DOI = {10.1007/BFb0064607},
URL = {http://www.numdam.org/item?id=SPS_1978__12__265_0},
NOTE = {With an appendix by the author and J.
de Sam Lazaro. MR:520008. Zbl:0391.60046.},
ISSN = {0075-8434},
ISBN = {9783540087618},
}
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
Abstract
People
BibTeX
Given a kernel \( K(x,dy) \) consisting of probability measures, all of them absolutely continuous with respect to a measure \( \mu \) , Doob proved long ago using martingale theory that
\[ K(x,dy)=k(x,y)\,\mu(dy) \]
with a jointly measurable density \( k(x,y) \) . What happens if the measures \( K(x,dy) \) are \( \sigma \) -finite? The answer is that Doob’s result remains valid if \( K \) , considered as a mapping
\[ x\mapsto K(x,\,\cdot\,) \]
taking values in the set of all \( \sigma \) -finite measures absolutely continuous w.r.t. \( \mu \) (i.e., of classes of a.s. finite measurable functions), is Borel with respect to the topology of convergence in probability.
@incollection {key520022m,
AUTHOR = {Yor, Marc and Meyer, P.-A.},
TITLE = {Sur l'extension d'un th\'eor\`eme de
{D}oob \`a un noyau \$\sigma\$-fini, d'apr\`es
{G}. {M}okobodzki [On the extension
of a theorem of {D}oob to a \$\sigma\$-finite
kernel, after {G}. {M}okobodzki]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XII}
[Twelfth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {649},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {482--488},
DOI = {10.1007/BFb0064621},
URL = {http://www.numdam.org/item?id=SPS_1978__12__482_0},
NOTE = {MR:520022. Zbl:0391.60047.},
ISSN = {0075-8434},
ISBN = {9783540087618},
}
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
Abstract
People
BibTeX
An explicit solution is given to Skorohod’s problem: given a distribution \( \mu \) with mean 0 and finite second moment \( \sigma^2 \) , find a (non randomized) stopping time \( T \) of a Brownian motion \( (X_t) \) such that \( X_T \) has the distribution \( \mu \) and \( \mathbb{E}[T]=\sigma^2 \) . It is shown that if \( S_t \) is the one-sided supremum of \( X \) at time \( t \) ,
\[ T=\inf\{t:S_t\geq \psi(X_t)\} \]
solves the problem, where \( \psi(x) \) is the barycenter of \( \mu \) restricted to \( [x,\infty) \) . The paper has several interesting side results, like explicit families of Brownian martingales, and a proof of the Ray–Knight theorem on local times.
@incollection {key544782m,
AUTHOR = {Az\'ema, Jacques and Yor, Marc},
TITLE = {Une solution simple au probl\`eme de
{S}korokhod [A simple solution to a
problem of {S}korokhod]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {90--115},
DOI = {10.1007/BFb0070852},
URL = {http://www.numdam.org/item?id=SPS_1979__13__90_0},
NOTE = {MR:544782. Zbl:0414.60055.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
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.
@incollection {key544805m,
AUTHOR = {Jeulin, T. and Yor, M.},
TITLE = {In\'egalit\'e de {H}ardy, semimartingales,
et faux-amis [Hardy's inequality, semimartingales
and false friends]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {332--359},
DOI = {10.1007/BFb0070874},
URL = {http://www.numdam.org/item?id=SPS_1979__13__332_0},
NOTE = {MR:544805. Zbl:0419.60049.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
@incollection {key544806m,
AUTHOR = {Jeulin, T. and Yor, M.},
TITLE = {Sur l'expression de la dualit\'e entre
\$H^1\$ et \$BMO\$ [On the expression of
the duality between \$H^1\$ and \$BMO\$]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {360--370},
DOI = {10.1007/BFb0070875},
URL = {http://www.numdam.org/item?id=SPS_1979__13__360_0},
NOTE = {MR:544806. Zbl:0426.60046.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
This is an account of current folklore, i.e., small remarks which settle natural questions, possibly published elsewhere but difficult to locate. Among the quotable results, one may mention that if a sequence of martingales converges in \( L^1 \) , one can stop them at arbitrary large stopping times so that the stopped processes converge in \( H^1 \) .
@incollection {key544810m,
AUTHOR = {Yor, Marc},
TITLE = {Quelques \'epilogues [Some conclusions]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {400--406},
DOI = {10.1007/BFb0070879},
URL = {http://www.numdam.org/item?id=SPS_1979__13__400_0},
NOTE = {MR:544810. Zbl:0427.60040.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
M. Yor :
“In search of a natural definition of optional stochastic integrals 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
Abstract
People
BibTeX
While the stochastic integral of a previsible process is a very natural object, the optional (compensated) stochastic integral is somewhat puzzling: it concerns martingales only, and depends on the probability law. This paper sketches a “pedagogical” approach, using a version of Fefferman’s inequality for thin processes to characterize those thin processes which are jump processes of local martingales. The results of [Chou 1977], [Lépingle 1977] are easily recovered. Then an attempt is made to extend the optional integral to semimartingales.
@incollection {key544811m,
AUTHOR = {Yor, Marc},
TITLE = {In search of a natural definition of
optional stochastic integrals},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {407--426},
DOI = {10.1007/BFb0070880},
URL = {http://www.numdam.org/item?id=SPS_1979__13__407_0},
NOTE = {MR:544811. Zbl:0439.60041.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
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.
@incollection {key544812m,
AUTHOR = {Yor, Marc},
TITLE = {Les filtrations de certaines martingales
du mouvement brownien dans \$\mathbb{R}^n\$
[The filtrations of certain martingales
of {B}rownian motion on \$\mathbb{R}^n\$]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {427--440},
DOI = {10.1007/BFb0070881},
URL = {http://www.numdam.org/item?id=SPS_1979__13__427_0},
NOTE = {MR:544812. Zbl:0418.60057.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
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 \( X_0=0 \) ,
\[ \mathbb{E}[X_{\infty}|\mathcal{F}_L^o]=0\neq \mathbb{E}[X_{\infty}|\mathcal{F}_L^{\pi}] .\]
@incollection {key544815m,
AUTHOR = {Yor, Marc},
TITLE = {Sur le balayage des semi-martingales
continues [On the balayage of continuous
semi-martingales]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {453--471},
DOI = {10.1007/BFb0070884},
URL = {http://www.numdam.org/item?id=SPS_1979__13__453_0},
NOTE = {MR:544815. Zbl:0409.60042.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
It is shown here that under the same hypotheses, the semimartingale \( Z_{g_t}X_t \) is a sum of three terms: the stochastic integral \( \int_0^t \zeta_s dX_s \) , where \( \zeta \) is the previsible projection of \( Z \) , an explicit sum of jumps involving \( Z-\zeta \) , and a mysterious continuous process with finite variation \( (R_t) \) such that \( dR_t \) is carried by \( H \) , equal to 0 if \( Z \) was optional.
@incollection {key544817m,
AUTHOR = {Meyer, P.-A. and Stricker, C. and Yor,
M.},
TITLE = {Sur une formule de la th\'eorie du balayage
[On a formula from balayage theory]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {478--487},
DOI = {10.1007/BFb0070886},
URL = {http://www.numdam.org/item?id=SPS_1979__13__478_0},
NOTE = {MR:544817. Zbl:0415.60031.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
People
BibTeX
@incollection {key544821m,
AUTHOR = {Yor, Marc},
TITLE = {Sur le supremum du mouvement brownien:
{L}es th\'eor\`emes de {P}. {L}\'evy
et {J}. {P}itman [On the supremum of
{B}rownian motion: {T}he theorems of
{P}. {L}\'evy and {J}. {P}itman]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {528--532},
DOI = {10.1007/BFb0070890},
URL = {http://www.numdam.org/item?id=SPS_1979__13__521_0},
NOTE = {This is an appendix of T. Jeunlin's
article ``Un th\'eor\`eme de J.W. Pitman''.
MR:544821. Zbl:0422.60028.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
The balayage formula allows the construction of many martingales vanishing on the zeros of a given continuous martingale \( X \) , namely martingales of the form \( Z_{g_t}X_t \) where \( Z \) is previsible. Taking \( X \) to be Brownian motion, an example is given of a martingale vanishing on its zeros which is not of the above form.
@incollection {key544831m,
AUTHOR = {Yor, M.},
TITLE = {Un exemple de {J}. {P}itman [An example
of {J}. {P}itman]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIII}
[Thirteenth probability seminar]},
EDITOR = {Dellacherie, C. and Meyer, P. A. and
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {624},
DOI = {10.1007/BFb0070900},
URL = {http://www.numdam.org/item?id=SPS_1979__13__624_0},
NOTE = {MR:544831. Zbl:0422.60029.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
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
Abstract
People
BibTeX
@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
Weil, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {721},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1979},
PAGES = {625--633},
DOI = {10.1007/BFb0070901},
URL = {http://www.numdam.org/item?id=SPS_1979__13__625_0},
NOTE = {This is a supplement to the preceding
article in \textit{S\'eminaire de probabilit\'es
XIII} \textbf{721} (1979). MR:544832.
Zbl:0414.60056.},
ISSN = {0075-8434},
ISBN = {9783540095057},
}
