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},
}
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
Abstract
BibTeX
@article {key2519530m,
AUTHOR = {Baker, D. and Yor, M.},
TITLE = {A {B}rownian sheet martingale with the
same marginals as the arithmetic average
of geometric {B}rownian motion},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {14},
NUMBER = {52},
YEAR = {2009},
PAGES = {1532--1540},
DOI = {10.1214/EJP.v14-674},
NOTE = {Article no. 52. MR:2519530. Zbl:1201.60033.},
ISSN = {1083-6489},
}
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
Abstract
BibTeX
Using a variation from the construction of the Ornstein–Uhlenbeck process on canonical path-space \( C([0,1] \) ; \( \mathbb{R}) \) in terms of the Brownian sheet, we obtain a large class of processes, adapted to the Brownian filtration, which admit the one dimensional marginals of a martingale.
@article {key2571849m,
AUTHOR = {Hirsch, Francis and Yor, Marc},
TITLE = {A construction of processes with one
dimensional martingale marginals, based
upon path-space {O}rnstein--{U}hlenbeck
processes and the {B}rownian sheet},
JOURNAL = {J. Math. Kyoto Univ.},
FJOURNAL = {Journal of Mathematics of Kyoto University},
VOLUME = {49},
NUMBER = {2},
YEAR = {2009},
PAGES = {389--417},
URL = {https://projecteuclid.org/euclid.kjm/1256219164},
NOTE = {MR:2571849. Zbl:1203.60122.},
ISSN = {0023-608X},
}
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
Abstract
BibTeX
We give some adequate extension, in the framework of a general Lévy process, of our previous construction of processes with one-dimensional martingale marginals, done originally in the set-up of Brownian motion. The Lévy process framework allows us to streamline our previous arguments, as well as to reach a larger class of such processes, even in the Brownian case. We give some illustrations of our construction when the Lévy process is either a Gamma process, or a Poisson process. We also work in the fractional Brownian and stable frameworks.
@article {key2591117m,
AUTHOR = {Hirsch, Francis and Yor, Marc},
TITLE = {A construction of processes with one-dimensional
martingale marginals, associated with
a {L}\'evy process, via its {L}\'evy
sheet},
JOURNAL = {J. Math. Kyoto Univ.},
FJOURNAL = {Journal of Mathematics of Kyoto University},
VOLUME = {49},
NUMBER = {4},
YEAR = {2009},
PAGES = {785--815},
DOI = {10.1215/kjm/1265899483},
NOTE = {MR:2591117. Zbl:1191.60040.},
ISSN = {0023-608X},
}
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
Abstract
BibTeX
@article {key2659753m,
AUTHOR = {Hirsch, F. and Yor, M.},
TITLE = {Looking for martingales associated to
a self-decomposable law},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {15},
YEAR = {2010},
PAGES = {932--961},
DOI = {10.1214/EJP.v15-786},
URL = {http://ejp.ejpecp.org/article/view/786},
NOTE = {Article no. 29. MR:2659753. Zbl:1225.60131.},
ISSN = {1083-6489},
}
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
Abstract
People
BibTeX
Strongly inspired by the result due to Carr–Ewald–Xiao that the arithmetic average of geometric Brownian motion is an increasing process in the convex order, we extend this result to integrals of Lévy processes and Gaussian processes. Our method consists in finding an appropriate sheet associated to the original Lévy or Gaussian process, from which the one-dimensional marginals of the integrals will appear to be those of a martingale, thus proving the increase in the convex order property.
@article {key2728438m,
AUTHOR = {Hirsch, Francis and Roynette, Bernard
and Yor, Marc},
TITLE = {Applying {I}t\^o's motto: ``{L}ook at
the infinite dimensional picture'' by
constructing sheets to obtain processes
increasing in the convex order},
JOURNAL = {Period. Math. Hung.},
FJOURNAL = {Periodica Mathematica Hungarica},
VOLUME = {61},
NUMBER = {1--2},
YEAR = {2010},
PAGES = {195--211},
DOI = {10.1007/s10998-010-3195-8},
NOTE = {MR:2728438. Zbl:1274.60052.},
ISSN = {0031-5303},
}
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
Abstract
People
BibTeX
In this paper, we present a unified framework for our previous constructions of martingales with the same one-dimensional marginals as particular cases of processes increasing in the convex order. This framework encompasses our former uses of Lévy sheets, Sato sheets and self-decomposable laws. New examples of processes increasing in the convex order are also exhibited, but we do not know how to associate to them martingales with the same one-dimensional marginals.
@article {key2734446m,
AUTHOR = {Hirsch, Francis and Roynette, Bernard
and Yor, Marc},
TITLE = {Unifying constructions of martingales
associated with processes increasing
in the convex order, via {L}\'evy and
{S}ato sheets},
JOURNAL = {Expo. Math.},
FJOURNAL = {Expositiones Mathematicae},
VOLUME = {28},
NUMBER = {4},
YEAR = {2010},
PAGES = {299--324},
DOI = {10.1016/j.exmath.2010.04.001},
NOTE = {MR:2734446. Zbl:1223.60027.},
ISSN = {0723-0869},
}
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
Abstract
BibTeX
@incollection {key2790386m,
AUTHOR = {Baker, David and Donati-Martin, Catherine
and Yor, Marc},
TITLE = {A sequence of {A}lbin type continuous
martingales with {B}rownian marginals
and scaling},
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 = {441--449},
DOI = {10.1007/978-3-642-15217-7_20},
NOTE = {MR:2790386. Zbl:1216.60039.},
ISSN = {0075-8434},
ISBN = {9783642152160},
}
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},
}
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
Abstract
People
BibTeX
We present an Itô type formula for a Gaussian process, in which only the one-marginals of the Gaussian process are involved. Thus, this formula is well adapted to the study of processes increasing in the convex order, in a Gaussian framework. In particular, we give conditions ensuring that processes defined as integrals, with respect to one parameter, of exponentials of two-parameter Gaussian processes, are increasing in the convex order with respect to the other parameter. Finally, we construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the previously defined processes.
@article {key2836749m,
AUTHOR = {Hirsch, Francis and Roynette, Bernard
and Yor, Marc},
TITLE = {From an {I}t\^o type calculus for {G}aussian
processes to integrals of log-normal
processes increasing in the convex order},
JOURNAL = {J. Math. Soc. Japan},
FJOURNAL = {Journal of the Mathematical Society
of Japan},
VOLUME = {63},
NUMBER = {3},
YEAR = {2011},
PAGES = {887--917},
DOI = {10.2969/jmsj/06330887},
NOTE = {MR:2836749. Zbl:1233.60008.},
ISSN = {0025-5645},
}
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
Abstract
People
BibTeX
We show that, in general, inequalities between integrands with respect to Brownian motion do not lead to majorization in the convex order for the corresponding stochastic integrals. Particular examples and counterexamples are discussed.
@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},
}
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
Abstract
People
BibTeX
Kellerer’s theorem asserts the existence of a Markov martingale with given marginals, assumed to increase in the convex order. It is revisited here, in the light of previous papers by Hirsch–Roynette and by G. Lowther.
@incollection {key3409839m,
AUTHOR = {Hirsch, Francis and Roynette, Bernard
and Yor, Marc},
TITLE = {Kellerer's theorem revisited},
BOOKTITLE = {Asymptotic laws and methods in stochastics:
{A} volume in honour of {M}ikl\'os {C}s\"org\H{o}
on the occasion of his 80th birthday},
EDITOR = {Dawson, D. and Kulik, R. and Ould Haye,
M. and Szyszkowicz, B. and Zhao, Y.},
SERIES = {Fields Institute Communications},
NUMBER = {76},
PUBLISHER = {Fields Institute},
ADDRESS = {Toronto},
YEAR = {2015},
PAGES = {347--363},
DOI = {10.1007/978-1-4939-3076-0_18},
NOTE = {(Ottawa, 3--6 July 2012). MR:3409839.
Zbl:1368.60045.},
ISSN = {1069-5265},
ISBN = {9781493930753},
}