R. Douady, M. Yor, and A. N. Shiryaev :
“On the probability characteristics of ‘drop’ variables in standard Brownian motion Teor. Veroyatnost. i Primenen.
44 : 1
(1999 ),
pp. 3–13 .
An English translation was published in Theory Probab. Appl. 44 :1 (1999)
MR
1751185 article
People
BibTeX
@article {key1751185m,
AUTHOR = {Douady, R. and Yor, M. and Shiryaev,
A. N.},
TITLE = {On the probability characteristics of
``drop'' variables in standard {B}rownian
motion},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Teoriya Veroyatnoste\u\i\ i ee Primeneniya.
Rossi\u\i skaya Akademiya Nauk},
VOLUME = {44},
NUMBER = {1},
YEAR = {1999},
PAGES = {3--13},
DOI = {10.4213/tvp594},
NOTE = {An English translation was published
in \textit{Theory Probab. Appl.} \textbf{44}:1
(1999). MR:1751185.},
ISSN = {0040-361X},
}
R. Douady, A. N. Shiryaev, and M. Yor :
“On probability characteristics of ‘downfalls’ in a standard Brownian motion Theory Probab. Appl.
44 : 1
(1999 ),
pp. 29–38 .
Original Russian version was published in Teor. Veroyatnost. i Primenen. 44 :1 (1999)
Zbl
0959.60073 article
Abstract
People
BibTeX
For a Brownian motion \( B=(B_t)_{t\leq 1} \) with \( B_0=0 \) , \( \textbf{\textrm{E}}B_t=0 \) , \( \textbf{\textrm{E}}B_t^2=t \) problems of probability distributions and their characteristics are considered for the variables
\begin{align*} & \mathbb{D} =\sup_{0\leq t\leq t^{\prime}\leq 1}(B_t-B_{t^{\prime}}),\\ & \mathbb{D}_1 =B_\sigma-\inf_{\sigma\leq t^{\prime}\leq 1}B_{t^{\prime}}, \\ & \mathbb{D}_2 =\sup_{0\leq t\leq\sigma^{\prime}}B_{t}-B_{\sigma^{\prime}}, \end{align*}
where \( \sigma \) and \( \sigma^{\prime} \) are times (non-Markov) of the absolute maximum and absolute minimum of the Brownian motion on \( [0,1] \) , i.e.,
\[ B_{\sigma}=\sup_{0\leq t\leq 1}B_t, \quad B_{\sigma^{\prime}}=\inf_{0\leq t^{\prime}\leq 1}B_{t^{\prime}} .\]
@article {key0959.60073z,
AUTHOR = {Douady, R. and Shiryaev, A. N. and Yor,
M.},
TITLE = {On probability characteristics of ``downfalls''
in a standard {B}rownian motion},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {44},
NUMBER = {1},
YEAR = {1999},
PAGES = {29--38},
DOI = {10.1137/S0040585X97977306},
NOTE = {Original Russian version was published
in \textit{Teor. Veroyatnost. i Primenen.}
\textbf{44}:1 (1999). Zbl:0959.60073.},
ISSN = {0040-585X},
}
A. S. Cherny, A. N. Shiryaev, and M. Yor :
“Limit behaviour of the ‘horizontal-vertical’ random walk and some extensions of the Donsker–Prokhorov invariance principle Teor. Veroyatnost. i Primenen.
47 : 3
(2002 ),
pp. 498–517 .
An English translation was published in Theory Probab. Appl. 47 :3 (2002)
MR
1975425 article
Abstract
People
BibTeX
We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane \( \{y\gt x\} \) and in the vertical direction on the half-plane \( \{y\leq x\} \) . The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.
@article {key1975425m,
AUTHOR = {Cherny, A. S. and Shiryaev, A. N. and
Yor, M.},
TITLE = {Limit behaviour of the ``horizontal-vertical''
random walk and some extensions of the
{D}onsker--{P}rokhorov invariance principle},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Teoriya Veroyatnoste\u{\i} i e\"e Primeneniya},
VOLUME = {47},
NUMBER = {3},
YEAR = {2002},
PAGES = {498--517},
DOI = {10.4213/tvp3689},
NOTE = {An English translation was published
in \textit{Theory Probab. Appl.} \textbf{47}:3
(2002). MR:1975425.},
ISSN = {0040-361X},
}
A. S. Cherny, A. N. Shiryaev, and M. Yor :
“Limit behaviour of the ‘horizontal-vertical’ random walk and some extensions of the Donsker–Prokhorov invariance principle Theory Probab. Appl.
47 : 3
(2002 ),
pp. 377–394 .
English translation of Russian original published in Teor. Veroyatnost. i Primenen. 47 :3 (2002)
Zbl
1034.60076 article
Abstract
People
BibTeX
We consider a two-dimensional random walk that moves in the horizontal direction on the half-plane \( \{y\gt x\} \) and in the vertical direction on the half-plane \( \{y\leq x\} \) . The limit behavior (as the time interval between two steps and the size of each step tend to zero) of this “horizontal-vertical” random walk is investigated. In order to solve this problem, we prove an extension of the Donsker–Prokhorov invariance principle. The extension states that the discrete-time stochastic integrals with respect to the appropriately renormalized one-dimensional random walk converge in distribution to the corresponding stochastic integral with respect to a Brownian motion. This extension enables us to construct a discrete-time approximation of the local time of a Brownian motion. We also provide discrete-time approximations of skew Brownian motions.
@article {key1034.60076z,
AUTHOR = {Cherny, A. S. and Shiryaev, A. N. and
Yor, M.},
TITLE = {Limit behaviour of the ``horizontal-vertical''
random walk and some extensions of the
{D}onsker--{P}rokhorov invariance principle},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {47},
NUMBER = {3},
YEAR = {2002},
PAGES = {377--394},
DOI = {10.1137/S0040585X97979834},
NOTE = {English translation of Russian original
published in \textit{Teor. Veroyatnost.
i Primenen.} \textbf{47}:3 (2002). Zbl:1034.60076.},
ISSN = {0040-585X},
}
A. N. Shiryaev and M. Yor :
“On stochastic integral representations of functionals of Brownian motion, I Teor. Veroyatnost. i Primenen.
48 : 2
(2003 ),
pp. 375–385 .
An English translation was published in Theory Probab. Appl. 48 :2 (2003)
MR
2015458 article
Abstract
People
BibTeX
For functionals \( S=S(\omega) \) of the Brownian motion \( B \) , we propose a method for finding stochastic integral representations based on the Itô formula for the stochastic integral associated with \( B \) . As an illustration of the method, we consider functionals of the “maximal” type: \( S_T \) , \( S_{T_{-a}} \) , \( S_{g_T} \) , and \( S_{\theta_T} \) , where
\[ S_T=\max_{t\leq T}B_t, \qquad S_{T_{-a}}=\max_{t\leq T_{-a}}B_t \]
with \( T_{-a}=\inf\{t \gt 0\mid B_t=-a\} \) , \( a\gt 0 \) , and
\[ S_{g_T}=\max_{t\leq g_T} B_t, \qquad S_{\theta_T}=\max_{t\leq \theta_T}B_t ,\]
\( g_T \) and \( \theta_T \) are non-Markov times: \( g_T \) is the time of the last zero of Brownian motion on \( [0,T] \) and \( \theta_T \) is a time when the Brownian motion achieves its maximal value on \( [0,T] \) .
@article {key2015458m,
AUTHOR = {Shiryaev, A. N. and Yor, M.},
TITLE = {On stochastic integral representations
of functionals of {B}rownian motion,
{I}},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Teoriya Veroyatnoste\u{\i} i e\"e Primeneniya},
VOLUME = {48},
NUMBER = {2},
YEAR = {2003},
PAGES = {375--385},
DOI = {10.4213/tvp290},
NOTE = {An English translation was published
in \textit{Theory Probab. Appl.} \textbf{48}:2
(2003). MR:2015458.},
ISSN = {0040-361X},
}
A. N. Shiryaev and M. Yor :
“On the problem of stochastic integral representations of functionals of the Brownian motion, I Theory Probab. Appl.
48 : 2
(2003 ),
pp. 304–313 .
English translation of Russian original published in Teor. Veroyatnost. i Primenen. 48 :2 (2003)
Zbl
1057.60057 article
People
BibTeX
@article {key1057.60057z,
AUTHOR = {Shiryaev, A. N. and Yor, M.},
TITLE = {On the problem of stochastic integral
representations of functionals of the
{B}rownian motion, {I}},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {48},
NUMBER = {2},
YEAR = {2003},
PAGES = {304--313},
DOI = {10.1137/S0040585X97980427},
NOTE = {English translation of Russian original
published in \textit{Teor. Veroyatnost.
i Primenen.} \textbf{48}:2 (2003). Zbl:1057.60057.},
ISSN = {0040-585X},
}
J. Obłój and M. Yor :
“On local martingale and its supremum: Harmonic functions and beyond 517–533
in
From stochastic calculus to mathematical finance: The Shiryaev Festschrift
(Meatbief, France, 9–15 January 2005 ).
Edited by Y. Kabanov, R. Lipster, and J. Stoyanov .
Springer (Berlin ),
2006 .
MR
2234288 Zbl
1120.60045 ArXiv
math/0412196 incollection
Abstract
People
BibTeX
We discuss certain facts involving a continuous local martingale \( N \) and its supremum \( N \) . A complete characterization of \( (N,\overline{N}) \) -harmonic functions is given. This yields an important family of martingales, the usefulness of which is demonstrated, by means of examples involving the Skorohod embedding problem, bounds on the law of the supremum, or the local time at 0, of a martingale with a fixed terminal distribution, or yet in some Brownian penalization problems. In particular we obtain new bounds on the law of the local time at 0, which involve the excess wealth order.
@incollection {key2234288m,
AUTHOR = {Ob\l \'oj, Jan and Yor, Marc},
TITLE = {On local martingale and its supremum:
{H}armonic functions and beyond},
BOOKTITLE = {From stochastic calculus to mathematical
finance: {T}he {S}hiryaev {F}estschrift},
EDITOR = {Kabanov, Yuri and Lipster, Robert and
Stoyanov, Jordan},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2006},
PAGES = {517--533},
DOI = {10.1007/978-3-540-30788-4_25},
NOTE = {(Meatbief, France, 9--15 January 2005).
ArXiv:math/0412196. MR:2234288. Zbl:1120.60045.},
ISBN = {9783642068034},
}
S. È. Graversen, A. N. Shiryaev, and M. Yor :
“On stochastic integral representations of functionals of Brownian motion, II Teor. Veroyatn. Primen.
51 : 1
(2006 ),
pp. 64–77 .
An English translation was published in Theory Probab. Appl. 51 :1 (2007)
MR
2324166 article
Abstract
People
BibTeX
In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl. , 48 (2004), pp. 304–313], a method of obtaining stochastic integral representations of functionals \( S(\omega) \) of Brownian motion \( B = (B_t)_{t\geq 0} \) was stated. Functionals \( \max_{t\leq T}B_t \) and \( \max_{t\leq T_{-a}}B_t \) where
\[ T_{-a} = \inf\{t\mid B_t=-a\}, \quad a \gt 0 ,\]
were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional \( \max_{t\leq g_T}B_t \) , where (non-Markov time)
\[ g_T=\sup\{0\leq t\leq T\mid B_t=0\} \]
are given.
@article {key2324166m,
AUTHOR = {Graversen, S. \`E. and Shiryaev, A.
N. and Yor, M.},
TITLE = {On stochastic integral representations
of functionals of {B}rownian motion,
{II}},
JOURNAL = {Teor. Veroyatn. Primen.},
FJOURNAL = {Teoriya Veroyatnoste\u\i iee Primeneniya.
Rossi\u\i skaya Akademiya Nauk},
VOLUME = {51},
NUMBER = {1},
YEAR = {2006},
PAGES = {64--77},
DOI = {10.4213/tvp146},
NOTE = {An English translation was published
in \textit{Theory Probab. Appl.} \textbf{51}:1
(2007). MR:2324166.},
ISSN = {0040-361X},
}
S. È. Graversen, A. N. Shiryaev, and M. Yor :
“On stochastic integral representations of functionals of Brownian motion, II Theory Probab. Appl.
51 : 1
(2007 ),
pp. 65–77 .
English translation of Russian original published in Teor. Veroyatn. Primen. 51 :1 (2007)
Zbl
1116.60022 article
Abstract
People
BibTeX
In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl. , 48 (2004), pp. 304–313], a method of obtaining stochastic integral representations of functionals \( S(\omega) \) of Brownian motion \( B = (B_t)_{t\geq 0} \) was stated. Functionals \( \max_{t\leq T}B_t \) and \( \max_{t\leq T_{-a}}B_t \) where
\[ T_{-a} = \inf\{t\mid B_t=-a\}, \quad a \gt 0 ,\]
were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional \( \max_{t\leq g_T}B_t \) , where (non-Markov time)
\[ g_T=\sup\{0\leq t\leq T\mid B_t=0\} \]
are given.
@article {key1116.60022z,
AUTHOR = {Graversen, S. \`E. and Shiryaev, A.
N. and Yor, M.},
TITLE = {On stochastic integral representations
of functionals of {B}rownian motion,
{II}},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {51},
NUMBER = {1},
YEAR = {2007},
PAGES = {65--77},
DOI = {10.1137/S0040585X97982190},
NOTE = {English translation of Russian original
published in \textit{Teor. Veroyatn.
Primen.} \textbf{51}:1 (2007). Zbl:1116.60022.},
ISSN = {0040-585X},
}
J. Bertoin and M. Yor :
“Retrieving information from subordination 97–106
in
Prokhorov and contemporary probability theory .
Edited by A. N. Shiryaev, S. R. S. Varadhan, and E. L. Presman .
Springer Proceedings in Mathematics & Statistics 33 .
Springer (Heidelberg ),
2013 .
MR
3070468 Zbl
1284.60084 ArXiv
1005.3187 incollection
Abstract
People
BibTeX
We recall some instances of the recovery problem of a signal process hidden in an observation process. Our main focus is then to show that if \( (X_s \) , \( s\geq 0) \) is a right-continuous process,
\[ Y_t = \int_0^t X_s \,ds \]
its integral process and \( \tau =(\tau_u \) , \( u\geq 0) \) a subordinator, then the time-changed process \( (Y_{\tau_u} \) , \( u\geq 0) \) allows to retrieve the information about \( (X_{\tau_v} \) , \( v\geq 0) \) when \( \tau \) is stable, but not when \( \tau \) is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.
@incollection {key3070468m,
AUTHOR = {Bertoin, Jean and Yor, Marc},
TITLE = {Retrieving information from subordination},
BOOKTITLE = {Prokhorov and contemporary probability
theory},
EDITOR = {Shiryaev, Albert N. and Varadhan, S.
R. S. and Presman, Ernst L.},
SERIES = {Springer Proceedings in Mathematics
\& Statistics},
NUMBER = {33},
PUBLISHER = {Springer},
ADDRESS = {Heidelberg},
YEAR = {2013},
PAGES = {97--106},
DOI = {10.1007/978-3-642-33549-5_5},
NOTE = {ArXiv:1005.3187. MR:3070468. Zbl:1284.60084.},
ISSN = {2194-1009},
ISBN = {9783642335488},
}
M. Jeanblanc-Picqué and A. N. Shiryaev :
“In memory of Marc Yor Teor. Veroyatnost. i Primenen.
59 : 1
(2015 ),
pp. 205–206 .
English translation of Russian original published in Theory Probab. Appl. 59 :1 (2005)
article
People
BibTeX
@article {key22208149,
AUTHOR = {Jeanblanc-Picqu\'e, M. and Shiryaev,
A. N.},
TITLE = {In memory of {M}arc {Y}or},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Teoriya Veroyatnoste\u{\i} i e\"e Primeneniya},
VOLUME = {59},
NUMBER = {1},
YEAR = {2015},
PAGES = {205--206},
DOI = {10.4213/tvp4561},
NOTE = {English translation of Russian original
published in \textit{Theory Probab.
Appl.} \textbf{59}:1 (2005).},
ISSN = {0040-361X},
}
M. Jeanblanc and A. Shiryaev :
“In memory of Marc Yor Theory Probab. Appl.
59 : 1
(2015 ),
pp. 180 .
English translation of Russian original published in Teor. Veroyatnost. i Primenen. 59 :1 (2005)
MR
3416074 Zbl
1314.01019 article
Abstract
People
BibTeX
Obituary of Marc Yor, who passed away on January 9, 2014. Professor Yor was a distinguished probabilist and exceptional mathematician. Throughout his career he made many valuable contributions to the fields of probability theory and stochastic processes.
@article {key3416074m,
AUTHOR = {Jeanblanc, M. and Shiryaev, A.},
TITLE = {In memory of {M}arc {Y}or},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {59},
NUMBER = {1},
YEAR = {2015},
PAGES = {180},
DOI = {10.1137/S0040585X97987028},
NOTE = {English translation of Russian original
published in \textit{Teor. Veroyatnost.
i Primenen.} \textbf{59}:1 (2005). MR:3416074.
Zbl:1314.01019.},
ISSN = {0040-585X},
}
