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
Abstract
People
BibTeX
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.
@incollection {key580109m,
AUTHOR = {Barlow, M. T. and Yor, M.},
TITLE = {Sur la construction d'une martingale
continue, de valeur absolue donn\'ee
[On the construction of a continuous
martingale of given absolute value]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XIV}
[Fourteenth probability seminar]},
EDITOR = {Az\'ema, Jacques and Yor, Marc},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {784},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1980},
PAGES = {62--75},
DOI = {10.1007/BFb0089470},
URL = {http://www.numdam.org/item?id=SPS_1980__14__62_0},
NOTE = {MR:580109. Zbl:0426.60047.},
ISSN = {0075-8434},
ISBN = {9783540097600},
}
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
People
BibTeX
@article {key608019m,
AUTHOR = {Barlow, M. T. and Yor, M.},
TITLE = {({S}emi-) martingale inequalities and
local times},
JOURNAL = {Z. Wahrscheinlichkeitstheor. Verw. Geb.},
FJOURNAL = {Zeitschrift f\"ur Wahrscheinlichkeitstheorie
und Verwandte Gebiete},
VOLUME = {55},
NUMBER = {3},
YEAR = {1981},
PAGES = {237--254},
DOI = {10.1007/BF00532117},
NOTE = {MR:608019. Zbl:0451.60050.},
ISSN = {0044-3719},
}
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
People
BibTeX
@article {key664645m,
AUTHOR = {Barlow, Martin T. and Yor, Marc},
TITLE = {Application du lemme de {G}arsia--{R}odemich--{R}umsey
\`a certaines in\'egalit\'es de (semi)
martingales continues [Application of
the lemma of {G}arsia--{R}odemich--{R}umsey
to certain inequalities for continuous
(semi) martingales]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus des S\'eances de l'Acad\'emie
des Sciences. S\'erie I. Math\'ematique},
VOLUME = {294},
NUMBER = {19},
YEAR = {1982},
PAGES = {665--668},
NOTE = {MR:664645. Zbl:0488.60060.},
ISSN = {0249-6321},
}
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
Abstract
People
BibTeX
With the help of the Garsia–Rodemich–Rumsey lemma, some improvements of the Burkholder–Davis–Gundy inequalities for continuous martingales are obtained, as well as certain estimates of the \( L^p \) norm of the supremum of
the local times of a continuous semi-martingale and
the difference of the local times of two continuous martingales.
@article {key680660m,
AUTHOR = {Barlow, M. T. and Yor, M.},
TITLE = {Semimartingale inequalities via the
{G}arsia--{R}odemich--{R}umsey lemma,
and applications to local times},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {49},
NUMBER = {2},
MONTH = {November},
YEAR = {1982},
PAGES = {198--229},
DOI = {10.1016/0022-1236(82)90080-5},
NOTE = {MR:680660. Zbl:0505.60054.},
ISSN = {0022-1236},
}
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
People
BibTeX
@article {key761264m,
AUTHOR = {Barlow, Martin T. and Jacka, Saul D.
and Yor, Marc},
TITLE = {In\'egalit\'es pour un couple de processus
arr\^et\'es \`a un temps quelconque
[Inequalities for a pair of processes
stopped at a random time]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus des S\'eances de l'Acad\'emie
des Sciences. S\'erie I. Math\'ematique},
VOLUME = {299},
NUMBER = {8},
YEAR = {1984},
PAGES = {351--354},
NOTE = {MR:761264. Zbl:0574.60022.},
ISSN = {0249-6291},
}
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
People
BibTeX
@article {key812449m,
AUTHOR = {Barlow, M. T. and Jacka, S. D. and Yor,
M.},
TITLE = {Inequalities for a pair of processes
stopped at a random time},
JOURNAL = {Proc. Lond. Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {52},
NUMBER = {1},
YEAR = {1986},
PAGES = {142--172},
DOI = {10.1112/plms/s3-52.1.142},
NOTE = {MR:812449. Zbl:0585.60055.},
ISSN = {0024-6115},
}
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
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},
}
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
People
BibTeX
@incollection {key1022918m,
AUTHOR = {Barlow, Martin and Pitman, Jim and Yor,
Marc},
TITLE = {Une extension multidimensionnelle de
la loi de l'arc sinus [A multidimensional
extension of the arcsine law]},
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 = {294--314},
DOI = {10.1007/BFb0083980},
URL = {http://www.numdam.org/item?id=SPS_1989__23__294_0},
NOTE = {MR:1022918. Zbl:0738.60072.},
ISSN = {0075-8434},
ISBN = {9783540511915},
}
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
Abstract
People
BibTeX
Tsirelson has shown that no Walsh’s Brownian motion with three rays or more can live in a Brownian filtration [1997]. Using his methods, the result is extended to spider martingales. A conjecture of M. Barlow is also proved: if \( L \) is an honest time in a (possibly multidimensional) Brownian filtration, then \( \mathcal{F}_{L+} \) is generated by \( \mathcal{F}_L \) and at most one event. Last, it is shown that a Walsh’s Brownian motion can live in the filtration generated by another Walsh’s Brownian motion only if the former is obtained from the latter by aggregating rays.
@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]},
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},
YEAR = {1998},
PAGES = {264--305},
DOI = {10.1007/BFb0101763},
URL = {http://www.numdam.org/item?id=SPS_1998__32__264_0},
NOTE = {MR:1655299. Zbl:0914.60064.},
ISSN = {0075-8434},
ISBN = {9783540643760},
}