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
Abstract
People
BibTeX
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.
@incollection {key440699m,
AUTHOR = {Yor, Marc},
TITLE = {Sur les int\'egrales stochastiques optionnelles
et une suite remarquable de formules
exponentielles [On optional stochastic
integrals and a remarkable series of
exponential formulas]},
BOOKTITLE = {S\'eminaire de probabilit\'es {X} [Tenth
probability seminar]},
EDITOR = {Meyer, P. A.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {511},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1976},
PAGES = {481--500},
DOI = {10.1007/BFb0101123},
URL = {http://www.numdam.org/item?id=SPS_1976__10__481_0},
NOTE = {MR:440699. Zbl:0393.60057.},
ISSN = {0075-8434},
ISBN = {9783540076810},
}
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},
}
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},
}
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 :
“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},
}
M. Barlow, J. Pitman, and M. Yor :
“On Walsh’s Brownian motions 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
People
BibTeX
@incollection {key1022917m,
AUTHOR = {Barlow, Martin and Pitman, Jim and Yor,
Marc},
TITLE = {On {W}alsh's {B}rownian motions},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXIII}
[Twenty-third probability seminar]},
EDITOR = {Az\'ema, J. and Meyer, P. A. and Yor,
M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1372},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1989},
PAGES = {275--293},
DOI = {10.1007/BFb0083979},
URL = {http://www.numdam.org/item?id=SPS_1989__23__275_0},
NOTE = {MR:1022917. Zbl:0747.60072.},
ISSN = {0075-8434},
ISBN = {9783540511915},
}
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
People
BibTeX
@incollection {key1071543m,
AUTHOR = {Jeulin, T. and Yor, M.},
TITLE = {Filtration des ponts browniens et \'equations
diff\'erentielles stochastiques lin\'eaires
[Filtering of {B}rownian bridges and
linear stochastic differential equations]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXIV}
[Twenty-fourth probability seminar]},
EDITOR = {Az\'ema, Jacques and Meyer, Paul Andr\'e
and Yor, Marc},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1426},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1990},
PAGES = {227--265},
DOI = {10.1007/BFb0083768},
URL = {http://www.numdam.org/item?id=SPS_1990__24__227_0},
NOTE = {MR:1071543. Zbl:0699.60075.},
ISSN = {0075-8434},
ISBN = {9783540526940},
}
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
People
BibTeX
@incollection {key1231999m,
AUTHOR = {Az\'ema, J. and Yor, M.},
TITLE = {Sur les z\'eros des martingales continues
[On the zeros of continuous martingales]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXVI}
[Twenty-sixth probability seminar]},
EDITOR = {Az\'ema, J. and Meyer, P.-A. and Yor,
M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1526},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1992},
PAGES = {248--306},
DOI = {10.1007/BFb0084326},
URL = {http://www.numdam.org/item?id=SPS_1992__26__248_0},
NOTE = {MR:1231999. Zbl:0765.60038.},
ISSN = {0075-8434},
ISBN = {9783540560210},
}
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
People
BibTeX
@incollection {key1232000m,
AUTHOR = {Az\'ema, J. and Meyer, P.-A. and Yor,
M.},
TITLE = {Martingales relatives [Relative martingales]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXVI}
[Twenty-sixth probability seminar]},
EDITOR = {Az\'ema, J. and Meyer, P.-A. and Yor,
M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1526},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1992},
PAGES = {307--321},
DOI = {10.1007/BFb0084327},
URL = {http://www.numdam.org/item?id=SPS_1992__26__307_0},
NOTE = {MR:1232000. Zbl:0765.60037.},
ISSN = {0075-8434},
ISBN = {9783540560210},
}
