M. Gradinaru, B. Roynette, P. Vallois, and M. Yor :
“Abel transform and integrals of Bessel local times Ann. Inst. Henri Poincaré, Probab. Stat.
35 : 4
(July 1999 ),
pp. 531–572 .
MR
1702241 Zbl
0937.60080 article
Abstract
People
BibTeX
We study integrals of the type \( \int_0^t\phi(s)\,dL_s \) , where \( \phi \) is a positive locally bounded Borel function and \( L_t \) denotes the local time at level 0 of a Bessel process of dimension \( d \) , \( 0 \lt d \lt 2 \) .
@article {key1702241m,
AUTHOR = {Gradinaru, Mihai and Roynette, Bernard
and Vallois, Pierre and Yor, Marc},
TITLE = {Abel transform and integrals of {B}essel
local times},
JOURNAL = {Ann. Inst. Henri Poincar\'e, Probab.
Stat.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
Probabilit\'es et Statistiques},
VOLUME = {35},
NUMBER = {4},
MONTH = {July},
YEAR = {1999},
PAGES = {531--572},
DOI = {10.1016/S0246-0203(99)00105-3},
NOTE = {MR:1702241. Zbl:0937.60080.},
ISSN = {0246-0203},
}
M. Gradinaru, B. Roynette, P. Vallois, and M. Yor :
“The laws of Brownian local time integrals Comput. Appl. Math.
18 : 3
(1999 ),
pp. 259–331 .
MR
1993869 Zbl
1123.60065 article
Abstract
People
BibTeX
We obtain some identities in law and some limit theorems for integrals of the type
\[ \int_0^t\phi(s) \,dL_s .\]
Here \( \phi \) is a positive locally bounded Borel function and \( L_t \) denotes the local time at 0 of processes such as Brownian motion, Brownian bridge, Ornstein–Uhlenbeck process, Bessel process or Bessel bridge of dimension \( d \) , \( 0\lt d\lt 2 \) .
@article {key1993869m,
AUTHOR = {Gradinaru, Mihai and Roynette, Bernard
and Vallois, Pierre and Yor, Marc},
TITLE = {The laws of {B}rownian local time integrals},
JOURNAL = {Comput. Appl. Math.},
FJOURNAL = {Computational and Applied Mathematics},
VOLUME = {18},
NUMBER = {3},
YEAR = {1999},
PAGES = {259--331},
URL = {https://hal.inria.fr/hal-00091335/},
NOTE = {MR:1993869. Zbl:1123.60065.},
ISSN = {0101-8205},
}
B. De Meyer, B. Roynette, P. Vallois, and M. Yor :
“Sur l’indépendance d’un temps d’arrêt \( T \) et de la position \( B_T \) d’un mouvement brownien \( (B_u \) , \( u\geqq 0) \) On the independence of a stopping time \( T \) and the position \( B_T \) of a Brownian motion \( (B_u \) , \( u\geqq 0) \) ],
C. R. Acad. Sci., Paris, Sér. I
333 : 11
(December 2001 ),
pp. 1017–1022 .
MR
1872465 Zbl
0995.60077 article
Abstract
People
BibTeX
In this note, we describe many examples of two-dimensional random variables \( \{B_T \) , \( T\} \) obtained from the position of a Brownian motion \( (B_t \) ; \( t\geqq 0) \) at a stopping time \( T \) such that \( T \) and \( B_T \) are independent. For such pairs, the law of \( T \) determines that of \( B_T \) and vice versa; we study the constraints on these laws induced by the independence assumption.
@article {key1872465m,
AUTHOR = {De Meyer, Bernard and Roynette, Bernard
and Vallois, Pierre and Yor, Marc},
TITLE = {Sur l'ind\'ependance d'un temps d'arr\^et
\$T\$ et de la position \$B_T\$ d'un mouvement
brownien \$(B_u\$, \$u\geqq 0)\$ [On the
independence of a stopping time \$T\$
and the position \$B_T\$ of a {B}rownian
motion \$(B_u\$, \$u\geqq 0)\$]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {333},
NUMBER = {11},
MONTH = {December},
YEAR = {2001},
PAGES = {1017--1022},
DOI = {10.1016/S0764-4442(01)02168-1},
NOTE = {MR:1872465. Zbl:0995.60077.},
ISSN = {0764-4442},
}
B. Roynette, P. Vallois, and M. Yor :
“A solution to Skorokhod’s embedding for linear Brownian motion and its local time Studia Sci. Math. Hung.
39 : 1–2
(May 2002 ),
pp. 97–127 .
MR
1909150 Zbl
1026.60047 article
People
BibTeX
@article {key1909150m,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {A solution to {S}korokhod's embedding
for linear {B}rownian motion and its
local time},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {39},
NUMBER = {1--2},
MONTH = {May},
YEAR = {2002},
PAGES = {97--127},
DOI = {10.1556/SScMath.39.2002.1-2.6},
NOTE = {MR:1909150. Zbl:1026.60047.},
ISSN = {0081-6906},
}
B. De Meyer, B. Roynette, P. Vallois, and M. Yor :
“On independent times and positions for Brownian motions Rev. Mat. Iberoamericana
18 : 3
(2002 ),
pp. 541–586 .
MR
1954864 Zbl
1055.60078 article
Abstract
People
BibTeX
Let \( (B_t \) ; \( t\geq 0) \) , resp. \( ((X_t,Y_t) \) ; \( t\geq 0) \) , be a one, resp. two, dimensional Brownian motion started at 0. Let \( T \) be a stopping time such that \( (B_{t\wedge T} \) ; \( t\geq 0) \) , resp. \( (X_{t \wedge T} \) ; \( t\geq 0) \) , \( (Y_{t\wedge T} \) ; \( t \geq 0)) \) , is uniformly integrable. The main results obtained in the paper are:
if \( T \) and \( B_T \) are independent and \( T \) has all exponential moments, then \( T \) is constant.
If \( X_T \) and \( Y_T \) are independent and have all exponential moments, then \( X_T \) and \( Y_T \) are Gaussian.
We also give a number of examples of stopping times \( T \) , with only some exponential moments, such that \( T \) and \( B_T \) are independent, and similarly for \( X_T \) and \( Y_T \) . We also exhibit bounded non-constant stopping times \( T \) such that \( X_T \) and \( Y_T \) are independent and Gaussian.
@article {key1954864m,
AUTHOR = {De Meyer, Bernard and Roynette, Bernard
and Vallois, Pierre and Yor, Marc},
TITLE = {On independent times and positions for
{B}rownian motions},
JOURNAL = {Rev. Mat. Iberoamericana},
FJOURNAL = {Revista Matem\'atica Iberoamericana},
VOLUME = {18},
NUMBER = {3},
YEAR = {2002},
PAGES = {541--586},
DOI = {10.4171/RMI/328},
NOTE = {MR:1954864. Zbl:1055.60078.},
ISSN = {0213-2230},
}
B. Roynette, P. Vallois, and M. Yor :
“Limiting laws associated with Brownian motion perturbated by normalized exponential weights C. R., Math., Acad. Sci. Paris
337 : 10
(November 2003 ),
pp. 667–673 .
With French summary.
MR
2030109 Zbl
1031.60021 ArXiv
math/0510550 article
Abstract
People
BibTeX
@article {key2030109m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Limiting laws associated with {B}rownian
motion perturbated by normalized exponential
weights},
JOURNAL = {C. R., Math., Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus. Math\'ematique. Acad\'emie
des Sciences, Paris},
VOLUME = {337},
NUMBER = {10},
MONTH = {November},
YEAR = {2003},
PAGES = {667--673},
DOI = {10.1016/j.crma.2003.09.025},
NOTE = {With French summary. ArXiv:math/0510550.
MR:2030109. Zbl:1031.60021.},
ISSN = {1631-073X},
}
J. Bertoin, B. Roynette, and M. Yor :
Some connections between (sub)critical branching mechanisms and Bernstein functions .
Preprint ,
December 2004 .
ArXiv
math/0412322 techreport
Abstract
People
BibTeX
We describe some connections, via composition, between two functional spaces: the space of (sub)critical branching mechanisms and the space of Bernstein functions. The functions
\[ \mathbf{e}_{\alpha}: x\to x^{\alpha} \]
where \( x\geq 0 \) and \( 0\lt\alpha\leq 1/2 \) , and in particular the critical parameter \( \alpha = 1/2 \) , play a distinguished role.
@techreport {keymath/0412322a,
AUTHOR = {Bertoin, Jean and Roynette, Bernard
and Yor, Marc},
TITLE = {Some connections between (sub)critical
branching mechanisms and {B}ernstein
functions},
TYPE = {preprint},
MONTH = {December},
YEAR = {2004},
NOTE = {ArXiv:math/0412322.},
}
B. Roynette, P. Vallois, and M. Yor :
Feynman–Kac asymptotics and the corresponding renormalizations of the Wiener measure ,
February 2004 .
Submitted to Studia Sci. Math. Hungarica.
unpublished
People
BibTeX
@unpublished {key91757973,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {Feynman--{K}ac asymptotics and the corresponding
renormalizations of the {W}iener measure},
MONTH = {February},
YEAR = {2004},
NOTE = {Submitted to Studia Sci. Math. Hungarica.},
}
B. Roynette, P. Vallois, and M. Yor :
Examples of \( BM-BES(3) \) concatenations related to renormalization of Brownian motion induced by its maximum and local time processes ,
May 2004 .
unpublished
People
BibTeX
@unpublished {key63804398,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {Examples of \$BM-BES(3)\$ concatenations
related to renormalization of {B}rownian
motion induced by its maximum and local
time processes},
MONTH = {May},
YEAR = {2004},
}
B. Roynette and M. Yor :
“Couples de Wald indéfiniment divisibles: Exemples liés à la fonction gamma d’Euler et à la fonction zeta de Riemann Infinitely divisible Wald couples: Examples linked with the Euler gamma function and the Riemann zeta function ],
Ann. Inst. Fourier (Grenoble)
55 : 4
(2005 ),
pp. 1219–1283 .
MR
2157168 Zbl
1083.60012 article
Abstract
People
BibTeX
To any positive measure \( c \) on \( \mathbb{R}_+ \) , such that:
\[ \int^{\infty}_0 (x \wedge x^2) \,c(dx) \lt \infty \]
we associate an infinitely divisible Wald couple, i.e., a couple of random variables \( (X,H) \) such that \( X \) and \( H \) are infinitely divisible, \( H \geq 0 \) , and for any \( \lambda \geq 0 \) ,
\[ E\bigl(e^{\lambda X}\bigr) \cdot E\bigl(e^{-\frac{\lambda^2}{2} H}\bigr)=1 .\]
More generally, to a positive measure \( c \) on \( \mathbb{R}_+ \) which satisfies
\[ \int_0^{\infty}e^{-\alpha x} x^2 c(dx) \lt \infty \]
for every \( \alpha \gt \alpha_0 \) , we associate an “Esscher family” of infinitely divisible Wald couples. We give many examples of such Esscher families and we prove that the particular ones which are associated with the gamma and the zeta functions enjoy remarkable properties.
@article {key2157168m,
AUTHOR = {Roynette, Bernard and Yor, Marc},
TITLE = {Couples de {W}ald ind\'efiniment divisibles:
{E}xemples li\'es \`a la fonction gamma
d'{E}uler et \`a la fonction zeta de
{R}iemann [Infinitely divisible {W}ald
couples: {E}xamples linked with the
{E}uler gamma function and the {R}iemann
zeta function]},
JOURNAL = {Ann. Inst. Fourier (Grenoble)},
FJOURNAL = {Annales de l'Institut Fourier},
VOLUME = {55},
NUMBER = {4},
YEAR = {2005},
PAGES = {1219--1283},
DOI = {10.5802/aif.2125},
URL = {http://aif.cedram.org/item?id=AIF_2005__55_4_1219_0},
NOTE = {MR:2157168. Zbl:1083.60012.},
ISSN = {0373-0956},
}
B. Roynette, P. Vallois, and M. Yor :
“Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III Period. Math. Hung.
50 : 1–2
(August 2005 ),
pp. 247–280 .
In homage to Professors E. Csáki and P. Révész.
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2162812 Zbl
1150.60308 ArXiv
math/0511102 article
Abstract
People
BibTeX
Results of penalization of a one-dimensional Brownian motion \( (X_t) \) , by its one-sided maximum
\[ S_t =\( \) \sup_{0\leq u\leq t}X_u ,\]
which were recently obtained by the authors are improved with the consideration–in the present paper–of the asymptotic behaviour of the likewise penalized Brownian bridges of length \( t \) , as \( t\to\infty \) , or penalizations by functions of \( (S_t \) , \( X_t) \) , and also the study of the speed of convergence, as \( t\to\infty \) , of the penalized distributions at time \( t \) .
@article {key2162812m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Limiting laws for long {B}rownian bridges
perturbed by their one-sided maximum,
{III}},
JOURNAL = {Period. Math. Hung.},
FJOURNAL = {Periodica Mathematica Hungarica. Journal
of the J\'anos Bolyai Mathematical Society},
VOLUME = {50},
NUMBER = {1--2},
MONTH = {August},
YEAR = {2005},
PAGES = {247--280},
DOI = {10.1007/s10998-005-0015-7},
NOTE = {In homage to Professors E. Cs\'aki and
P. R\'ev\'esz. Parts I--X have very
different titles. I was published in
\textit{Studia Sci. Math. Hung.} \textbf{46}:2
(2003); II in \textit{Studia Sci. Math.
Hung.} \textbf{43}:3 (2006); IV in \textit{Studia
Sci. Math. Hung.} \textbf{44}:4 (2007);
V in \textit{Studia Sci. Math. Hung.}
\textbf{45}:1 (2008); VI in \textit{ESAIM
Probab. Stat.} \textbf{13} (2009); VII
in \textit{Ann. Inst. Henri Poincar\'e,
Probab. Stat.} \textbf{45}:2 (2009);
VIII in \textit{J. Funct. Anal.} \textbf{255}:9
(2008); IX in \textit{ESAIM Probab.
Stat.} \textbf{14} (2010); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
ArXiv:math/0511102. MR:2162812. Zbl:1150.60308.},
ISSN = {0031-5303},
}
B. Roynette, P. Vallois, and M. Yor :
“Asymptotics for the distribution of lengths of excursions of a \( d \) -dimensional Bessel process \( (0\lt d\lt 2) \) C. R., Math., Acad. Sci. Paris
343 : 3
(August 2006 ),
pp. 201–208 .
MR
2246339 Zbl
1155.60329 article
Abstract
People
BibTeX
Let \( (R_t \) , \( t\geq 0) \) denote a \( d \) -dimensional Bessel process (\( 0\lt d\lt 2 \) ). For every \( t\geq 0 \) , we consider the times
\[ g_t = \sup\{s\leq t\mid R_s = 0\} \quad\text{and}\quad d_t = \inf\{s\gt t\mid R_s = 0\} ,\]
as well as the three sequences: \( (V^n_{g_t} \) , \( n\geq 1) \) , \( (V^n_t \) , \( n\geq 2) \) , and \( (V^n_{d_t} \) , \( n\geq 2) \) , which consist of the lengths of excursions of \( R \) away from 0 before \( g_t \) , before \( t \) , and before \( d_t \) , respectively, each one being ranked by decreasing order.
We obtain a limit theorem concerning each of the laws of these three sequences, as \( t\to\infty \) . The result is expressed in terms of a positive, \( \sigma \) -finite measure \( \Pi \) on the set \( \mathscr{S}^{\downarrow} \) of decreasing sequences. \( Pi \) is closely related with the Poisson–Dirichlet laws on \( \mathscr{S}^{\downarrow} \) .
@article {key2246339m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Asymptotics for the distribution of
lengths of excursions of a \$d\$-dimensional
{B}essel process \$(0\lt d\lt 2)\$},
JOURNAL = {C. R., Math., Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie
des Sciences. Paris},
VOLUME = {343},
NUMBER = {3},
MONTH = {August},
YEAR = {2006},
PAGES = {201--208},
DOI = {10.1016/j.crma.2006.06.010},
NOTE = {MR:2246339. Zbl:1155.60329.},
ISSN = {1631-073X},
}
B. Roynette, P. Vallois, and M. Yor :
“Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II Studia Sci. Math. Hung.
43 : 3
(2006 ),
pp. 295–360 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2253307 Zbl
1121.60004 ArXiv
math/0510575 article
Abstract
People
BibTeX
Let \( P_0 \) denote the Wiener measure defined on the canonical space
\[ \bigl( \Omega = \mathcal{C}(\mathbb{R}_+,\mathbb{R}), \,(X_t)_{t\geq 0}, \,(\mathcal{F}_t)_{t\geq 0} \bigr) ,\]
and \( (S_t) \) , resp. \( (I_t) \) , be the one sided-maximum, resp. minimum, \( (L_t^0) \) the local time at 0, and \( (D_t) \) the number of down-crossings from \( b \) to \( a \) (with \( b \gt a \) ). Let
\[ f:\mathbb{R}\times\mathbb{R}^d\to (0,+\infty) \]
be a Borel function, and \( (A_t) \) be a process chosen within the set:
\[ \bigl\{(S_t), \,(S_t,t), \,(L_t^0), \,(S_t,I_t,L_t^0), \,(D_t) \bigr\} ,\]
which consists of 5 elements. We prove a penalization result: under some suitable assumptions on \( f \) , there exists a positive \( ((\mathcal{F}_t) \) , \( P_0) \) -martingale \( (M_t^f) \) , starting at 1, such that:
\begin{equation*}\tag{1} \lim_{t\to\infty}\frac{E_0[1_{\Gamma_s}f(X_t,A_t)]}{E_0[f(X_t,A_t)]} = Q_0^f(\Gamma_s) := E_0[1_{\Gamma_s}M_s^f],\quad \forall\,\Gamma_s\in\mathcal{F}_s,\,s\geq 0. \end{equation*}
We determine the law of \( (X_t) \) under the p.m. \( Q_0^f \) defined on \( (\Omega \) , \( \mathcal{F}_{\infty}) \) by (1). For the 1st, 3rd and 5th elements of the set, we prove first that
\[ Q_0^f(A_{\infty} \lt \infty) = 1 ,\]
and more generally
\[ Q_0^f(0 \lt g \lt \infty) = 1 \]
where \( g = \sup\{s \gt 0 \mid A_s = A_{\infty}\} \) (with the convention \( \sup\emptyset = 0 \) ). Secondly, we split the trajectory of \( (X_t) \) in two parts: \( (X_t)_{0\leq t\leq g} \) and \( (X_{t+g})_{t\geq 0} \) , and we describe their laws under \( Q_0^f \) , conditionally on \( A_{\infty} \) . For the 2nd and 4th elements, a similar result holds replacing \( A_{\infty} \) by resp. \( S_{\infty} \) , \( S_{\infty} \vee I \) .
@article {key2253307m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Limiting laws associated with {B}rownian
motion perturbed by its maximum, minimum
and local time, {II}},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {43},
NUMBER = {3},
YEAR = {2006},
PAGES = {295--360},
DOI = {10.1556/SScMath.43.2006.3.3},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); III
in \textit{Period. Math. Hung.} \textbf{50}:1--2
(2005); IV in \textit{Studia Sci. Math.
Hung.} \textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VI in \textit{ESAIM Probab. Stat.} \textbf{13}
(2009); VII in \textit{Ann. Inst. Henri
Poincar\'e Probab. Stat.} \textbf{45}:2
(2009); VIII in \textit{J. Funct. Anal.}
\textbf{255}:9 (2008); IX in \textit{ESAIM
Probab. Stat.} \textbf{14} (2010); X
in \textit{Theory Stoch. Process.} \textbf{14}:2
(2008). ArXiv:math/0510575. MR:2253307.
Zbl:1121.60004.},
ISSN = {0081-6906},
}
B. Roynette, P. Vallois, and M. Yor :
“Some penalisations of the Wiener measure Jpn. J. Math. (3)
1 : 1
(April 2006 ),
pp. 263–290 .
MR
2261065 Zbl
1160.60315 article
Abstract
People
BibTeX
A number of limit laws, which are obtained from various penalisations of the Wiener measure on \( C(\mathbb{R}_+,\mathbb{R}^d) \) , are shown to exist, and are described thoroughly, with the help of path decompositions.
@article {key2261065m,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {Some penalisations of the {W}iener measure},
JOURNAL = {Jpn. J. Math. (3)},
FJOURNAL = {Japanese Journal of Mathematics. 3rd
Series},
VOLUME = {1},
NUMBER = {1},
MONTH = {April},
YEAR = {2006},
PAGES = {263--290},
DOI = {10.1007/s11537-006-0507-0},
NOTE = {MR:2261065. Zbl:1160.60315.},
ISSN = {0289-2316},
}
B. Roynette, P. Vallois, and M. Yor :
“Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement Brownien et à son maximum unilatère Penalizations and some extensions of Pitman’s theorem relative to Brownian motion and its one-sided maximum ],
pp. 305–336
in
In memoriam Paul-André Meyer: Séminaire de probabilités XXXIX
[In memoriam Paul-André Meyer: Thirty-ninth probability seminar ].
Edited by M. Émery and M. Yor .
Lecture Notes in Mathematics 1874 .
Springer (Berlin ),
2006 .
MR
2276902 Zbl
1124.60034 incollection
People
BibTeX
@incollection {key2276902m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {P\'enalisations et quelques extensions
du th\'eor\`eme de {P}itman, relatives
au mouvement {B}rownien et \`a son maximum
unilat\`ere [Penalizations and some
extensions of {P}itman's theorem relative
to {B}rownian motion and its one-sided
maximum]},
BOOKTITLE = {In memoriam {P}aul-{A}ndr\'e {M}eyer:
{S}\'eminaire de probabilit\'es {XXXIX}
[In memoriam {P}aul-{A}ndr\'e {M}eyer:
{T}hirty-ninth probability seminar]},
EDITOR = {\'Emery, M. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1874},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2006},
PAGES = {305--336},
DOI = {10.1007/978-3-540-35513-7_20},
NOTE = {MR:2276902. Zbl:1124.60034.},
ISSN = {0075-8434},
ISBN = {9783540309949},
}
J. Bertoin, T. Fujita, B. Roynette, and M. Yor :
“On a particular class of self-decomposable random variables: The durations of Bessel excursions straddling independent exponential times Probab. Math. Stat.
26 : 2
(2006 ),
pp. 315–366 .
MR
2325310 Zbl
1123.60063 article
Abstract
People
BibTeX
The distributional properties of the duration of a recurrent Bessel process straddling an independent exponential time are studied in detail. Althrough our study may be considered as a particular case of M. Winkel’s in [2005], the infinite divisibility structure of these Bessel durations is particularly rich and we develop algebraic properties for a family of random variables arising from the Lévy measures of these durations.
@article {key2325310m,
AUTHOR = {Bertoin, J. and Fujita, T. and Roynette,
B. and Yor, M.},
TITLE = {On a particular class of self-decomposable
random variables: {T}he durations of
{B}essel excursions straddling independent
exponential times},
JOURNAL = {Probab. Math. Stat.},
FJOURNAL = {Proability and Mathematical Statistics},
VOLUME = {26},
NUMBER = {2},
YEAR = {2006},
PAGES = {315--366},
URL = {https://hal.sorbonne-universite.fr/hal-00104879/},
NOTE = {MR:2325310. Zbl:1123.60063.},
ISSN = {0208-4147},
}
B. Roynette, P. Vallois, and M. Yor :
“Some extensions of Pitman and Ray–Knight theorems for penalized Brownian motions and their local times, IV ,”
Studia Sci. Math. Hung.
44 : 4
(2007 ),
pp. 469–516 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2361439 Zbl
1164.60355 article
Abstract
People
BibTeX
We show that Pitman’s theorem relating Brownian motion and the \( BES(3) \) process, as well as the Ray–Knight theorems for Brownian local times remain valid, mutatis mutandis, under the limiting laws of Brownian motion penalized by a function of its one-sided maximum.
@article {key2361439m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Some extensions of {P}itman and {R}ay--{K}night
theorems for penalized {B}rownian motions
and their local times, {IV}},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {44},
NUMBER = {4},
YEAR = {2007},
PAGES = {469--516},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
V in \textit{Studia Sci. Math. Hung.}
\textbf{45}:1 (2008); VI in \textit{ESAIM
Probab. Stat.} \textbf{13} (2009); VII
in \textit{Ann. Inst. Henri Poincar\'e,
Probab. Stat.} \textbf{45}:2 (2009);
VIII in \textit{J. Funct. Anal.} \textbf{255}:9
(2008); IX in \textit{ESAIM Probab.
Stat.} \textbf{14} (2010); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
MR:2361439. Zbl:1164.60355.},
ISSN = {0081-6906},
}
J. Najnudel, B. Roynette, and M. Yor :
“A remarkable \( \sigma \) -finite measure on \( \mathcal{C}(\mathbf{R}_+,\mathbf{R}) \) related to many Brownian penalisations C. R. Math. Acad. Sci. Paris
345 : 8
(2007 ),
pp. 459–466 .
MR
2367926 Zbl
1221.60003 article
Abstract
People
BibTeX
In this note, we study a \( \sigma \) -finite measure \( \mathcal{W} \) on the space \( \mathcal{C}(\mathbf{R}_+,\mathbf{R}) \) , strongly related to Wiener measure, and we construct a large class of Brownian martingales from \( \mathcal{W} \) . Some of these martingales appear naturally in the study of Brownian penalisations made by B. Roynette, P. Vallois and M. Yor.
@article {key2367926m,
AUTHOR = {Najnudel, Joseph and Roynette, Bernard
and Yor, Marc},
TITLE = {A remarkable \$\sigma\$-finite measure
on \$\mathcal{C}(\mathbf{R}_+,\mathbf{R})\$
related to many {B}rownian penalisations},
JOURNAL = {C. R. Math. Acad. Sci. Paris},
FJOURNAL = {Comptes Rendus Math\'ematique. Acad\'emie
des Sciences. Paris},
VOLUME = {345},
NUMBER = {8},
YEAR = {2007},
PAGES = {459--466},
DOI = {10.1016/j.crma.2007.09.015},
NOTE = {MR:2367926. Zbl:1221.60003.},
ISSN = {1631-073X},
}
B. Roynette, P. Vallois, and M. Yor :
“Penalizing a \( \mathrm{BES}(d) \) process \( (0\lt d\lt 2) \) with a function of its local time, V Studia Sci. Math. Hung.
45 : 1
(2008 ),
pp. 67–124 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4(2007)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2401169 Zbl
1164.60307 article
Abstract
People
BibTeX
We describe the limit laws, as \( t\to\infty \) , of a Bessel process \( (R_s \) , \( s\leq t) \) of dimension \( d\in (0,2) \) penalized by an integrable function of its local time \( L_t \) at 0, thus extending our previous work of this kind, relative to Brownian motion.
@article {key2401169m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Penalizing a \$\mathrm{BES}(d)\$ process
\$(0\lt d\lt 2)\$ with a function of its
local time, {V}},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {45},
NUMBER = {1},
YEAR = {2008},
PAGES = {67--124},
DOI = {10.1556/SScMath.2007.1042},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4(2007); VI in \textit{ESAIM
Probab. Stat.} \textbf{13} (2009); VII
in \textit{Ann. Inst. Henri Poincar\'e,
Probab. Stat.} \textbf{45}:2 (2009);
VIII in \textit{J. Funct. Anal.} \textbf{255}:9
(2008); IX in \textit{ESAIM Probab.
Stat.} \textbf{14} (2010); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
MR:2401169. Zbl:1164.60307.},
ISSN = {0081-6906},
}
C. Donati-Martin, B. Roynette, P. Vallois, and M. Yor :
“On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension \( d = 2(1-\alpha) \) , \( 0 \lt \alpha \lt 1 \) Studia Sci. Math. Hung.
45 : 2
(2008 ),
pp. 207–221 .
MR
2417969 Zbl
1212.60070 article
Abstract
People
BibTeX
The precise choice of the local time at 0 for a Bessel process with dimension \( d\in (0,2) \) plays some role in explicit computations or limiting results involving excursion theory for these processes. Starting from one specific choice, and deriving the main related formulae, it is shown how the various multiplicative constants corresponding to other choices made in the literature enter into these formulae.
@article {key2417969m,
AUTHOR = {Donati-Martin, Catherine and Roynette,
Bernard and Vallois, Pierre and Yor,
Marc},
TITLE = {On constants related to the choice of
the local time at 0, and the corresponding
{I}t\^o measure for {B}essel processes
with dimension \$d = 2(1-\alpha)\$, \$
0 \lt \alpha \lt 1\$},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica},
VOLUME = {45},
NUMBER = {2},
YEAR = {2008},
PAGES = {207--221},
DOI = {10.1556/SScMath.2007.1033},
NOTE = {MR:2417969. Zbl:1212.60070.},
ISSN = {0081-6906},
}
B. Roynette and M. Yor :
“Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII J. Funct. Anal.
255 : 9
(November 2008 ),
pp. 2606–2640 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2473270 Zbl
1180.60070 article
Abstract
People
BibTeX
We penalise Brownian motion by a function of its one-sided supremum considered up to the last zero before \( t \) , respectively first zero after \( t \) , of that Brownian motion. This study presents some analogy with penalisation by the longest length of Brownian excursions, up to time \( t \) .
@article {key2473270m,
AUTHOR = {Roynette, B. and Yor, M.},
TITLE = {Ten penalisation results of {B}rownian
motion involving its one-sided supremum
until first and last passage times,
{VIII}},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {255},
NUMBER = {9},
MONTH = {November},
YEAR = {2008},
PAGES = {2606--2640},
DOI = {10.1016/j.jfa.2008.04.024},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VI in \textit{ESAIM Probab. Stat.} \textbf{13}
(2009); VII in \textit{Ann. Inst. Henri
Poincar\'e, Probab. Stat.} \textbf{45}:2
(2009); IX in \textit{ESAIM Probab.
Stat.} \textbf{14} (2010); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
MR:2473270. Zbl:1180.60070.},
ISSN = {0022-1236},
}
L. F. James, B. Roynette, and M. Yor :
“Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples Probab. Surv.
5
(2008 ),
pp. 346–415 .
MR
2476736 Zbl
1189.60035 ArXiv
0708.3932 article
Abstract
People
BibTeX
In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (GGC) variables, their Wiener–Gamma representations, and relation with the Dirichlet processes.
To a GGC variable, one may associate a unique Thorin measure. Let \( G \) a positive r.v. and \( \Gamma_t(G) \) (resp. \( \Gamma_t(1/G) \) the Generalized Gamma Convolution with Thorin measure \( t \) -times the law of \( G \) (resp. the law of \( 1/G \) ). In Section 2, we compare the laws of \( \Gamma_t(G) \) and \( \Gamma_t(1/G) \) .
In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.
@article {key2476736m,
AUTHOR = {James, Lancelot F. and Roynette, Bernard
and Yor, Marc},
TITLE = {Generalized gamma convolutions, {D}irichlet
means, {T}horin measures, with explicit
examples},
JOURNAL = {Probab. Surv.},
FJOURNAL = {Probability Surveys},
VOLUME = {5},
YEAR = {2008},
PAGES = {346--415},
DOI = {10.1214/07-PS118},
NOTE = {ArXiv:0708.3932. MR:2476736. Zbl:1189.60035.},
ISSN = {1549-5787},
}
B. Roynette, P. Vallois, and M. Yor :
“Penalisations of Brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding, X Theory Stoch. Process.
14 : 2
(2008 ),
pp. 116–138 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)
MR
2479739 Zbl
1224.60076 article
Abstract
People
BibTeX
We develop a Brownian penalisation procedure related to weight processes \( (F_t) \) of the type: \( F_t := f(I_t,S_t) \) where \( f \) is a bounded function with compact support and \( S_t \) (resp. \( I_t \) ) is the one-sided maximum (resp. minimum) of the Brownian motion up to time \( t \) . Two main cases are treated: either \( F_t \) is the indicator function of \( \{I_t\geq\alpha \) , \( S_t\leq\beta\} \) or \( F_t \) is null when \( \{S_t - I_t \gt c\} \) for some \( c \gt 0 \) . Then we apply these results to some kind of asymptotic Skorokhod embedding problem.
@article {key2479739m,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {Penalisations of {B}rownian motion with
its maximum and minimum processes as
weak forms of {S}korokhod embedding,
{X}},
JOURNAL = {Theory Stoch. Process.},
FJOURNAL = {Theory of Stochastic Processes},
VOLUME = {14},
NUMBER = {2},
YEAR = {2008},
PAGES = {116--138},
URL = {https://hal.archives-ouvertes.fr/hal-00275179},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VI in \textit{ESAIM Probab. Stat.} \textbf{13}
(2009); VII in \textit{Ann. Inst. Henri
Poincar\'e, Probab. Stat.} \textbf{45}:2
(2009); VIII in \textit{J. Funct. Anal.}
\textbf{255}:9 (2008); IX in \textit{ESAIM
Probab. Stat.} \textbf{14} (2010). MR:2479739.
Zbl:1224.60076.},
ISSN = {0321-3900},
}
D. Madan, B. Roynette, and M. Yor :
“Option prices as probabilities Financ. Res. Lett.
5 : 2
(June 2008 ),
pp. 79–87 .
article
Abstract
People
BibTeX
Four distribution functions are associated with call and put prices seen as functions of their strike and maturity. The random variables associated with these distributions are identified when the process for moneyness defined as the stock price relative to the forward price is a positive local martingale with no positive jumps that tends to zero at infinity. Results on calls require moneyness to be a continuous martingale as well. It is shown that for puts the distributions in the strike are those for the remaining supremum while for calls, they relate to the remaining infimum. In maturity we see the distribution functions for the last passage times of moneyness to strike.
@article {key57450396,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Option prices as probabilities},
JOURNAL = {Financ. Res. Lett.},
FJOURNAL = {Finance Research Letters},
VOLUME = {5},
NUMBER = {2},
MONTH = {June},
YEAR = {2008},
PAGES = {79--87},
DOI = {10.1016/j.frl.2008.02.002},
ISSN = {1544-6123},
}
B. Roynette and M. Yor :
Existence and properties of pseudo-inverses for Bessel and related processes Preprint ,
December 2008 .
techreport
Abstract
People
BibTeX
@techreport {key50830307,
AUTHOR = {Roynette, B. and Yor, M.},
TITLE = {Existence and properties of pseudo-inverses
for {B}essel and related processes},
TYPE = {preprint},
MONTH = {December},
YEAR = {2008},
URL = {https://hal.archives-ouvertes.fr/hal-00344697/},
}
D. Madan, B. Roynette, and M. Yor :
“Unifying Black–Scholes type formulae which involve Brownian last passage times up to a finite horizon Asia-Pac. Financ. Mark.
15 : 2
(June 2008 ),
pp. 97–115 .
Zbl
1163.91414 article
Abstract
People
BibTeX
The authors recently discovered some interesting relations between the Black–Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a finite horizon. This is achieved in the present paper, where Yuri’s formula, as originally presented in Akahori et al. (On the pricing of options written on the last exit time, 2008), is also derived.
@article {key1163.91414z,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Unifying {B}lack--{S}choles type formulae
which involve {B}rownian last passage
times up to a finite horizon},
JOURNAL = {Asia-Pac. Financ. Mark.},
FJOURNAL = {Asia-Pacific Financial Markets},
VOLUME = {15},
NUMBER = {2},
MONTH = {June},
YEAR = {2008},
PAGES = {97--115},
DOI = {10.1007/s10690-008-9068-y},
NOTE = {Zbl:1163.91414.},
ISSN = {1387-2834},
}
B. Roynette and M. Yor :
Penalising Brownian paths Lecture Notes in Mathematics 1969 .
Springer (Berlin ),
2009 .
Dedicated to Frank Knight (1933–2007).
MR
2504013 Zbl
1190.60002 book
People
BibTeX
@book {key2504013m,
AUTHOR = {Roynette, Bernard and Yor, Marc},
TITLE = {Penalising {B}rownian paths},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1969},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2009},
PAGES = {xiv+275},
DOI = {10.1007/978-3-540-89699-9},
NOTE = {Dedicated to Frank Knight (1933--2007).
MR:2504013. Zbl:1190.60002.},
ISSN = {0075-8434},
ISBN = {9783540896982},
}
B. Roynette, P. Vallois, and M. Yor :
“Penalisations of multidimensional Brownian motion, VI ESAIM Probab. Stat.
13
(January 2009 ),
pp. 152–180 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2518544 Zbl
1189.60069 article
Abstract
People
BibTeX
As in preceding papers in which we studied the limits of penalized 1-dimensional Wiener measures with certain functionals \( \Gamma_t \) , we obtain here the existence of the limit, as \( t\to\infty \) , of \( d \) -dimensional Wiener measures penalized by a function of the maximum up to time \( t \) of the Brownian winding process (for \( d = 2 \) ), or in \( d\geq 2 \) dimensions for Brownian motion prevented to exit a cone before time \( t \) . Various extensions of these multidimensional penalisations are studied, and the limit laws are described. Throughout this paper, the skew-product decomposition of \( d \) -dimensional Brownian motion plays an important role.
@article {key2518544m,
AUTHOR = {Roynette, Bernard and Vallois, Pierre
and Yor, Marc},
TITLE = {Penalisations of multidimensional {B}rownian
motion, {VI}},
JOURNAL = {ESAIM Probab. Stat.},
FJOURNAL = {European Series in Applied and Industrial
Mathematics (ESAIM): Probability and
Statistics},
VOLUME = {13},
MONTH = {January},
YEAR = {2009},
PAGES = {152--180},
DOI = {10.1051/ps:2008003},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VII in \textit{Ann. Inst. Henri Poincar\'e,
Probab. Stat.} \textbf{45}:2 (2009);
VIII in \textit{J. Funct. Anal.} \textbf{255}:9
(2008); IX in \textit{ESAIM Probab.
Stat.} \textbf{14} (2010); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
MR:2518544. Zbl:1189.60069.},
ISSN = {1292-8100},
}
B. Roynette, P. Vallois, and M. Yor :
“Brownian penalisations related to excursion lengths, VII Ann. Inst. Henri Poincaré, Probab. Stat.
45 : 2
(2009 ),
pp. 421–452 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)J. Funct. Anal. 255 :9 (2008)ESAIM Probab. Stat. 14 (2010)Theory Stoch. Process. 14 :2 (2008)
MR
2521408 Zbl
1181.60046 article
Abstract
People
BibTeX
Limiting laws, as \( t\to\infty \) , for Brownian motion penalised by the longest length of excursions up to \( t \) , or up to the last zero before \( t \) , or again, up to the first zero after \( t \) , are shown to exist, and are characterized.
@article {key2521408m,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {Brownian penalisations related to excursion
lengths, {VII}},
JOURNAL = {Ann. Inst. Henri Poincar\'e, Probab.
Stat.},
FJOURNAL = {Annales de l'Institut Henri Poincar\'e.
Probabilit\'es et Statistiques},
VOLUME = {45},
NUMBER = {2},
YEAR = {2009},
PAGES = {421--452},
DOI = {10.1214/08-AIHP177},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VI in \textit{ESAIM Probab. Stat.} \textbf{13}
(2009); VIII in \textit{J. Funct. Anal.}
\textbf{255}:9 (2008); IX in \textit{ESAIM
Probab. Stat.} \textbf{14} (2010); X
in \textit{Theory Stoch. Process.} \textbf{14}:2
(2008). MR:2521408. Zbl:1181.60046.},
ISSN = {0246-0203},
}
J. Najnudel, B. Roynette, and M. Yor :
A global view of Brownian penalisations MSJ Memoirs 19 .
Mathematical Society of Japan (Tokyo ),
2009 .
MR
2528440 Zbl
1180.60004 ArXiv
0905.2220 book
People
BibTeX
@book {key2528440m,
AUTHOR = {Najnudel, J. and Roynette, B. and Yor,
M.},
TITLE = {A global view of {B}rownian penalisations},
SERIES = {MSJ Memoirs},
NUMBER = {19},
PUBLISHER = {Mathematical Society of Japan},
ADDRESS = {Tokyo},
YEAR = {2009},
PAGES = {xii+137},
DOI = {10.2969/msjmemoirs/019010000},
NOTE = {ArXiv:0905.2220. MR:2528440. Zbl:1180.60004.},
ISSN = {2189-1494},
ISBN = {9784931469525},
}
D. Madan, B. Roynette, and M. Yor :
“Put option prices as joint distribution functions in strike and maturity: The Black–Scholes case Int. J. Theor. Appl. Finance
12 : 8
(2009 ),
pp. 1075–1090 .
MR
2598932 Zbl
1183.91179 article
Abstract
People
BibTeX
For a large class of \( \mathbb{R}_+ \) valued, continuous local martingales \( (M_t \) , \( t\geq 0) \) , with \( M_0 = 1 \) and \( M_{\infty} = 0 \) , the put quantity:
\[ \Pi_M(K,t) = E((K-M_t)^+) \]
turns out to be the distribution function in both variables \( K \) and \( t \) , for \( K\leq 1 \) and \( t\geq 0 \) , of a probability \( \gamma_M \) on \( [0,1]{\times} [0, \infty) \) . In this paper, the first in a series of three, we discuss in detail the case where
\[ M_t = \mathcal{E}_t := \exp\bigl(B_t - \tfrac{t}{2}\bigr) ,\]
for \( (B_t \) , \( t\geq 0) \) a standard Brownian motion.
@article {key2598932m,
AUTHOR = {Madan, D. and Roynette, B. and Yor,
M.},
TITLE = {Put option prices as joint distribution
functions in strike and maturity: {T}he
{B}lack--{S}choles case},
JOURNAL = {Int. J. Theor. Appl. Finance},
FJOURNAL = {International Journal of Theoretical
and Applied Finance},
VOLUME = {12},
NUMBER = {8},
YEAR = {2009},
PAGES = {1075--1090},
DOI = {10.1142/S0219024909005580},
NOTE = {MR:2598932. Zbl:1183.91179.},
ISSN = {0219-0249},
}
B. Roynette, P. Vallois, and M. Yor :
“A family of generalized gamma convoluted variables Probab. Math. Statist.
29 : 2
(2009 ),
pp. 181–204 .
MR
2792539 Zbl
1197.60012 article
Abstract
People
BibTeX
This paper consists of three parts: in the first part, we describe a family of generalized gamma convoluted (abbreviated as GGC) variables. In the second part, we use this description to prove that several r.v.’s, related to the length of excursions away from 0 for a recurrent linear diffusion on \( \mathbb{R}_+ \) , are GGC. Finally, in the third part, we apply our results to the case of Bessel processes with dimension \( d = 2(1 - \alpha) \) (\( 0 \lt d \lt 2 \) , or \( 0 \lt \alpha \lt 1 \) ).
@article {key2792539m,
AUTHOR = {Roynette, B. and Vallois, P. and Yor,
M.},
TITLE = {A family of generalized gamma convoluted
variables},
JOURNAL = {Probab. Math. Statist.},
FJOURNAL = {Probability and Mathematical Statistics},
VOLUME = {29},
NUMBER = {2},
YEAR = {2009},
PAGES = {181--204},
URL = {http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=29.2&nrA=1&ppB=181&ppE=204},
NOTE = {MR:2792539. Zbl:1197.60012.},
ISSN = {0208-4147},
}
C. Profeta, B. Roynette, and M. Yor :
Option prices as probabilities: A new look at generalized Black–Scholes formulae Springer Finance .
Springer (Berlin ),
2010 .
MR
2582990 Zbl
1188.91004 book
Abstract
People
BibTeX
To the best of our knowledge this book discusses in a unique way last passage times. The Black–Scholes formula plays a central role in Mathematical Finance; it gives the right price at which buyer and seller can agree with, in the geometric Brownian framework, when strike \( K \) and maturity \( T \) are given. This yields an explicit well-known formula, obtained by Black and Scholes in 1973. The present volume gives another representation of this formula in terms of Brownian last passages times, which, to our knowledge, has never been made in this sense. The volume is devoted to various extensions and discussions of features and quantities stemming from the last passages times representation in the Brownian case such as: past-future martingales, last passage times up to a finite horizon, pseudo-inverses of processes.. They are developed in eight chapters, with complements, appendices and exercises.
@book {key2582990m,
AUTHOR = {Profeta, Christophe and Roynette, Bernard
and Yor, Marc},
TITLE = {Option prices as probabilities: {A}
new look at generalized {B}lack--{S}choles
formulae},
SERIES = {Springer Finance},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2010},
PAGES = {xxii+270},
DOI = {10.1007/978-3-642-10395-7},
NOTE = {MR:2582990. Zbl:1188.91004.},
ISSN = {1616-0533},
ISBN = {9783642103940},
}
B. Roynette and M. Yor :
“Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX ESAIM Probab. Stat.
14
(March 2010 ),
pp. 65–92 .
Parts I–X have very different titles. I was published in Studia Sci. Math. Hung. 46 :2 (2003)Studia Sci. Math. Hung. 43 :3 (2006)Period. Math. Hung. 50 :1–2 (2005)Studia Sci. Math. Hung. 44 :4 (2007)Studia Sci. Math. Hung. 45 :1 (2008)ESAIM Probab. Stat. 13 (2009)Ann. Inst. Henri Poincaré, Probab. Stat. 45 :2 (2009)J. Funct. Anal. 255 :9 (2008)Theory Stoch. Process. 14 :2 (2008)
MR
2654548 Zbl
1219.60036 article
Abstract
People
BibTeX
We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional:
\[ \Bigl(A_t^- := \int_0^t 1_{\{X_s\lt 0\}}\,ds; \,t\geq 0 \Bigr) .\]
On the other hand, we describe Feynman–Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung. 43 (2006) 171–246]).
@article {key2654548m,
AUTHOR = {Roynette, Bernard and Yor, Marc},
TITLE = {Local limit theorems for {B}rownian
additive functionals and penalisation
of {B}rownian paths, {IX}},
JOURNAL = {ESAIM Probab. Stat.},
FJOURNAL = {European Series in Applied and Industrial
Mathematics (ESAIM): Probability and
Statistics},
VOLUME = {14},
MONTH = {March},
YEAR = {2010},
PAGES = {65--92},
DOI = {10.1051/ps:2008028},
NOTE = {Parts I--X have very different titles.
I was published in \textit{Studia Sci.
Math. Hung.} \textbf{46}:2 (2003); II
in \textit{Studia Sci. Math. Hung.}
\textbf{43}:3 (2006); III in \textit{Period.
Math. Hung.} \textbf{50}:1--2 (2005);
IV in \textit{Studia Sci. Math. Hung.}
\textbf{44}:4 (2007); V in \textit{Studia
Sci. Math. Hung.} \textbf{45}:1 (2008);
VI in \textit{ESAIM Probab. Stat.} \textbf{13}
(2009); VII in \textit{Ann. Inst. Henri
Poincar\'e, Probab. Stat.} \textbf{45}:2
(2009); VIII in \textit{J. Funct. Anal.}
\textbf{255}:9 (2008); X in \textit{Theory
Stoch. Process.} \textbf{14}:2 (2008).
MR:2654548. Zbl:1219.60036.},
ISSN = {1292-8100},
}
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},
}
F. Hirsch, C. Profeta, B. Roynette, and M. Yor :
“Constructing self-similar martingales via two Skorokhod embeddings 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},
}
M. Jeanblanc and F. Hirsch :
“Marc Yor 1949–2014 ,”
Matapli
103
(2014 ),
pp. 51–55 .
With remembrances of Yor by Bernard Roynette.
MR
3235376 Zbl
1365.01056 article
People
BibTeX
@article {key3235376m,
AUTHOR = {Jeanblanc, Monique and Hirsch, Francis},
TITLE = {Marc {Y}or 1949--2014},
JOURNAL = {Matapli},
FJOURNAL = {Matapli},
VOLUME = {103},
YEAR = {2014},
PAGES = {51--55},
NOTE = {With remembrances of Yor by Bernard
Roynette. MR:3235376. Zbl:1365.01056.},
ISSN = {0762-5707},
}
F. Hirsch and B. Roynette :
“Marc Yor et les peacocks ”
[Marc Yor and peacocks ],
pp. 111–120
in
Marc Yor: La passion du mouvement brownien
[Marc Yor: The passion of Brownian motion ].
Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi .
Société Mathématique de France (Paris ),
2015 .
Gazette des Mathématiciens and Matapli special issue.
incollection
People
BibTeX
@incollection {key73960677,
AUTHOR = {Hirsch, Francis and Roynette, Bernard},
TITLE = {Marc {Y}or et les peacocks [Marc {Y}or
and peacocks]},
BOOKTITLE = {Marc {Y}or: {L}a passion du mouvement
brownien [Marc {Y}or: {T}he passion
of {B}rownian motion]},
EDITOR = {Bertoin, Jean and Jeanblanc, Monique
and Le Gall, Jean-Fran\c{c}ois and Shi,
Zhan},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {2015},
PAGES = {111--120},
NOTE = {Gazette des Math\'ematiciens and Matapli
special issue.},
ISBN = {9782856298015},
}
F. Hirsch, B. Roynette, and M. Yor :
“Kellerer’s theorem revisited 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},
}
B. Roynette :
“Le travail de Marc sur les pénalisations Marc’s work on penalizations ],
pp. 103–110
in
Marc Yor: La passion du mouvement brownien
[Marc Yor: The passion of Brownian motion ].
Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi .
Société Mathématique de France (Paris ),
2015 .
Gazette des Mathématiciens and Matapli special issue.
incollection
Abstract
People
BibTeX
Marc travailla sur la théorie de la pénalisation une petite dizaine d’années, jusqu’en 2009 environ. Il publia sur ce thème avec ses co-auteurs environ une quinzaine d’articles, monographie et livre. Il n’est donc pas question de détailler ici l’ensemble de ses résultats mais seulement d’en esquisser les idées principales.
@incollection {key49201586,
AUTHOR = {Roynette, Bernard},
TITLE = {Le travail de {M}arc sur les p\'enalisations
[Marc's work on penalizations]},
BOOKTITLE = {Marc {Y}or: {L}a passion du mouvement
brownien [Marc {Y}or: {T}he passion
of {B}rownian motion]},
EDITOR = {Bertoin, Jean and Jeanblanc, Monique
and Le Gall, Jean-Fran\c{c}ois and Shi,
Zhan},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {2015},
PAGES = {103--110},
URL = {https://hal.archives-ouvertes.fr/hal-01285925},
NOTE = {Gazette des Math\'ematiciens and Matapli
special issue.},
ISBN = {9782856298015},
}
