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},
}
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},
}
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
Abstract
BibTeX
Using Fubini’s theorem for double Wiener integrals, it is possible to show that certain quadratic functionals of Brownian motion have the same law. This is applied to the variance of the Brownian path, which has the same law as the integreal of the square of the Brownian bridge.
@article {key1118933m,
AUTHOR = {Donati-Martin, C. and Yor, M.},
TITLE = {Fubini's theorem for double {W}iener
integrals and the variance of the {B}rownian
path},
JOURNAL = {Ann. Inst. Henri Poincar\'e, Probab.
Stat.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
Probabilit\'es et Statistiques},
VOLUME = {27},
NUMBER = {2},
YEAR = {1991},
PAGES = {181--200},
URL = {http://www.numdam.org/item?id=AIHPB_1991__27_2_181_0},
NOTE = {MR:1118933. Zbl:0738.60074.},
ISSN = {0246-0203},
}
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
BibTeX
@book {key1193919m,
AUTHOR = {Yor, Marc},
TITLE = {Some aspects of {B}rownian motion},
VOLUME = {1: Some special functionals},
SERIES = {Lectures in Mathematics ETH Z\"urich},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Basel},
YEAR = {1992},
PAGES = {x+136},
NOTE = {MR:1193919. Zbl:0779.60070.},
ISBN = {9783764328078},
}
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
Abstract
BibTeX
@article {key1416415m,
AUTHOR = {Shi, Zhan and Yor, Marc},
TITLE = {On an identity in law for the variance
of the {B}rownian bridge},
JOURNAL = {Bull. London Math. Soc.},
FJOURNAL = {Bulletin of the London Mathematical
Society},
VOLUME = {29},
NUMBER = {1},
YEAR = {1997},
PAGES = {103--108},
DOI = {10.1112/S0024609396001920},
NOTE = {MR:1416415. Zbl:0956.60085.},
ISSN = {0024-6093},
}
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
People
BibTeX
@incollection {key1648654m,
AUTHOR = {Alili, Larbi and Dufresne, Daniel and
Yor, Marc},
TITLE = {Sur l'identit\'e de {B}ougerol pour
les fonctionnelles exponentielles du
mouvement brownien avec drift [On the
{B}ougerol identity for the exponential
functionals of {B}rownian motion with
drift]},
BOOKTITLE = {Exponential functionals and principal
values related to {B}rownian motion},
EDITOR = {Yor, Marc},
SERIES = {Biblioteca de la Revista Matem\'atica
Iberoamericana},
PUBLISHER = {Universidad Aut\'onoma de Madrid},
YEAR = {1997},
PAGES = {3--14},
NOTE = {MR:1648654. Zbl:0905.60059.},
ISSN = {0213-2230},
ISBN = {9788460094616},
}
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
Abstract
BibTeX
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.
@incollection {key1648657m,
AUTHOR = {Carmona, Philippe and Petit, Fr\'ed\'erique
and Yor, Marc},
TITLE = {On the distribution and asymptotic results
for exponential functionals of {L}\'evy
processes},
BOOKTITLE = {Exponential functionals and principal
values related to {B}rownian motion},
EDITOR = {Yor, Marc},
SERIES = {Biblioteca de la Revista Matem\'atica
Iberoamericana},
PUBLISHER = {Universidad Aut\'onoma de Madrid},
YEAR = {1997},
PAGES = {73--130},
NOTE = {MR:1648657. Zbl:0905.60056.},
ISSN = {0213-2230},
ISBN = {9788460094616},
}
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
Abstract
BibTeX
The distribution of the terminal value \( A_{\infty} \) of the exponential functional
\[ A_t(\xi) = \int_0^t \exp(\xi_s) \,ds \]
of a Lévy process \( (\xi_t)_{t\geq 0} \) plays an important role in Mathematical Physics and Mathematical Finance. We show how this distribution can be computed by means of Lamperti’s transformation and generalized Ornstein–Uhlenbeck processes.
@incollection {key1833691m,
AUTHOR = {Carmona, Philippe and Petit, Fr\'ed\'erique
and Yor, Marc},
TITLE = {Exponential functionals of {L}\'evy
processes},
BOOKTITLE = {L\'evy processes: {T}heory and applications},
EDITOR = {Barndorff-Nielsen, Ole E. and Resnick,
Sidney I. and Mikosch, Thomas},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {2001},
PAGES = {41--55},
DOI = {10.1007/978-1-4612-0197-7_2},
NOTE = {MR:1833691. Zbl:0979.60038.},
ISBN = {9781461266570},
}
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
Abstract
People
BibTeX
We present three generic constructions of martingales that all have the Markov property with known and prespecified marginal densities. These constructions are further investigated for the special case when the prespecified marginals satisfy the scaling property and hence the only datum needed for the construction is the density at unit time. Interesting relations with stochastic orders are presented, along with numerous examples of the resulting martingales.
@article {key1914701m,
AUTHOR = {Madan, Dilip B. and Yor, Marc},
TITLE = {Making {M}arkov martingales meet marginals:
{W}ith explicit constructions},
JOURNAL = {Bernoulli},
FJOURNAL = {Bernoulli. Official Journal of the Bernoulli
Society for Mathematical Statistics
and Probability},
VOLUME = {8},
NUMBER = {4},
YEAR = {2002},
PAGES = {509--536},
URL = {https://projecteuclid.org/euclid.bj/1078681382},
NOTE = {MR:1914701. Zbl:1009.60037.},
ISSN = {1350-7265},
}
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
Abstract
BibTeX
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.
@article {key1981630m,
AUTHOR = {Jacobsen, Martin and Yor, Marc},
TITLE = {Multi-self-similar {M}arkov processes
on \$\mathbb{R}_+^n\$ and their {L}amperti
representations},
JOURNAL = {Probab. Theory Relat. Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {126},
NUMBER = {1},
MONTH = {May},
YEAR = {2003},
PAGES = {1--28},
DOI = {10.1007/s00440-003-0263-5},
NOTE = {MR:1981630. Zbl:1031.60029.},
ISSN = {0178-8051},
}
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
Abstract
People
BibTeX
@article {key2178044m,
AUTHOR = {Bertoin, J. and Yor, M.},
TITLE = {Exponential functionals of {L}\'evy
processes},
JOURNAL = {Probability Surveys},
FJOURNAL = {Probab. Surv.},
VOLUME = {2},
YEAR = {2005},
PAGES = {191--212},
DOI = {10.1214/154957805100000122},
NOTE = {ArXiv:math/0511265. MR:2178044. Zbl:1189.60096.},
ISSN = {1549-5787},
}
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
Abstract
People
BibTeX
With the help of two Skorokhod embeddings, we construct martingales which enjoy the Brownian scaling property and the (inhomogeneous) Markov property. The second method necessitates randomization, but allows to reach any law with finite moment of order 1, centered, as the distribution of such a martingale at unit time. The first method does not necessitate randomization, but an additional restriction on the distribution at unit time is needed.
@incollection {key2790387m,
AUTHOR = {Hirsch, Francis and Profeta, Christophe
and Roynette, Bernard and Yor, Marc},
TITLE = {Constructing self-similar martingales
via two {S}korokhod embeddings},
BOOKTITLE = {S\'eminaire de probabilit\'es {XLIII}
[Forty-third probability seminar]},
EDITOR = {Donati-Martin, Catherine and Lejay,
Antoine and Rouault, Alain},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {2006},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2011},
PAGES = {451--503},
DOI = {10.1007/978-3-642-15217-7_21},
NOTE = {MR:2790387. Zbl:1234.60047.},
ISSN = {0075-8434},
ISBN = {9783642152160},
}
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
People
BibTeX
@book {key2808243m,
AUTHOR = {Hirsch, Francis and Profeta, Christophe
and Roynette, Bernard and Yor, Marc},
TITLE = {Peacocks and associated martingales,
with explicit constructions},
SERIES = {Bocconi \& Springer Series},
NUMBER = {3},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {2011},
PAGES = {xxxii+384},
DOI = {10.1007/978-88-470-1908-9},
NOTE = {MR:2808243. Zbl:1227.60001.},
ISSN = {2039-1471},
ISBN = {9788847019072},
}
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
Abstract
People
BibTeX
We present a two-dimensional extension of an identity in distribution due to Bougerol [1983] that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend to the level of processes, we point out further striking relations in this direction.
@article {key3148605m,
AUTHOR = {Bertoin, Jean and Dufresne, Daniel and
Yor, Marc},
TITLE = {Some two-dimensional extensions of {B}ougerol's
identity in law for the exponential
functional of linear {B}rownian motion},
JOURNAL = {Rev. Mat. Iberoam.},
FJOURNAL = {Revista Matem\'atica Iberoamericana},
VOLUME = {29},
NUMBER = {4},
YEAR = {2013},
PAGES = {1307--1324},
DOI = {10.4171/RMI/758},
NOTE = {ArXiv:1201.1495. MR:3148605. Zbl:1303.60073.},
ISSN = {0213-2230},
}
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
Abstract
BibTeX
Let \( B \) be a Brownian motion and \( T_1 \) its first hitting time of the level 1. For \( U \) a uniform random variable independent of \( B \) , we study in depth the distribution of
\[ B_{UT_1}/\sqrt{T_1} ,\]
that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered.
@article {key3194736m,
AUTHOR = {Elie, Romuald and Rosenbaum, Mathieu
and Yor, Marc},
TITLE = {On the expectation of normalized {B}rownian
functionals up to first hitting times},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {19},
YEAR = {2014},
PAGES = {Article no. 37, 23 pp.},
DOI = {10.1214/EJP.v19-3049},
NOTE = {ArXiv:1310.1181. MR:3194736. Zbl:1291.60164.},
ISSN = {1083-6489},
}
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
Abstract
BibTeX
We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for \( B \) a Brownian motion and \( T_1 \) its first hitting time of the level one, this remarkable law allows us to understand some properties of the process
\[ \bigl( B_{uT_1}/\sqrt{T_1}, \,u\leq 1\bigr) \]
under uniform random sampling, a study started in (Elie, Rosenbaum, and Yor, On the expectation of normalized Brownian functionals up to first hitting times, Preprint, arXiv:1310.1181, 2013).
@incollection {key3330825m,
AUTHOR = {Rosenbaum, Mathieu and Yor, Marc},
TITLE = {On the law of a triplet associated with
the pseudo-{B}rownian bridge},
BOOKTITLE = {S\'eminaire de probabilit\'es, {XLVI}
[Forty-sixth probability seminar]},
EDITOR = {Donati-Martin, Catherine and Lejay,
Antoine and Rouault, Alain},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {2123},
PUBLISHER = {Springer},
ADDRESS = {Cham, Switzerland},
YEAR = {2014},
PAGES = {359--375},
DOI = {10.1007/978-3-319-11970-0_14},
NOTE = {ArXiv:1310.7164. MR:3330825. Zbl:1390.60298.},
ISSN = {0075-8434},
ISBN = {9783319119694},
}
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
Abstract
BibTeX
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.
@article {key3433427m,
AUTHOR = {Rosenbaum, Mathieu and Yor, Marc},
TITLE = {Some explicit formulas for the {B}rownian
bridge, {B}rownian meander and {B}essel
process under uniform sampling},
JOURNAL = {ESAIM, Probab. Stat.},
FJOURNAL = {ESAIM. Probability and Statistics},
VOLUME = {19},
MONTH = {December},
YEAR = {2015},
PAGES = {578--589},
DOI = {10.1051/ps/2015009},
NOTE = {ArXiv:1311.1900. MR:3433427. Zbl:1333.60181.},
ISSN = {1292-8100},
}