Z. Shi and M. Yor :
“Integrability and lower limits of the local time of iterated Brownian motion ,”
Studia Sci. Math. Hung.
33 : 1–3
(1997 ),
pp. 279–298 .
Dedicated to Endre Csáki on his sixtieth birthday.
MR
1454115 Zbl
0909.60068 article
Abstract
People
BibTeX
We study the extraordinarily large and small values of the local time of iterated Brownian motion. It is known that the local time has exponential moments for deterministic times. We prove that, taken at appropriate random times, the local time has finite \( \alpha \) -th moment if and only if \( \alpha \lt 2/3 \) . We also investigate the almost sure lower asymptotics of both the local time at a fixed level and the maximum local time. The critical rate functions for these two processes are obtained, which improves previous results of Csáki et al. [1996]. Our approach essentially relies on Ray–Knight theorems and the general theory of stochastic calculus, notably some refined martingale inequalities.
@article {key1454115m,
AUTHOR = {Shi, Zh. and Yor, M.},
TITLE = {Integrability and lower limits of the
local time of iterated {B}rownian motion},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {33},
NUMBER = {1--3},
YEAR = {1997},
PAGES = {279--298},
NOTE = {Dedicated to Endre Cs\'aki on his sixtieth
birthday . MR:1454115. Zbl:0909.60068.},
ISSN = {0081-6906},
}
M. Yor :
“Generalized meanders as limits of weighted Bessel processes, and an elementary proof of Spitzer’s asymptotic result on Brownian windings ,”
Studia Sci. Math. Hung.
33 : 1–3
(1997 ),
pp. 339–343 .
Dedicated to Professor E. Csáki on his sixtieth birthday.
MR
1454119 Zbl
0909.60070 article
People
BibTeX
@article {key1454119m,
AUTHOR = {Yor, M.},
TITLE = {Generalized meanders as limits of weighted
{B}essel processes, and an elementary
proof of {S}pitzer's asymptotic result
on {B}rownian windings},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {33},
NUMBER = {1--3},
YEAR = {1997},
PAGES = {339--343},
NOTE = {Dedicated to Professor E. Cs\'aki on
his sixtieth birthday . MR:1454119.
Zbl:0909.60070.},
ISSN = {0081-6906},
}
C. Donati-Martin, H. Matsumoto, and M. Yor :
“On striking identities about the exponential functionals of the Brownian bridge and Brownian motion Period. Math. Hung.
41 : 1–2
(November 2000 ),
pp. 103–119 .
Dedicated to Professor Endre Csáki on the occasion of his 65th birthday.
MR
1812799 Zbl
1062.60080 article
Abstract
People
BibTeX
The negative moment of order 1, resp. of order \( 1/2 \) , for the integral on \( (0,1) \) of the exponential of times the Brownian bridge, resp. the Brownian motion, does not depend on \( \alpha \) . We give a simple explanation and a reinforcement of this property in the case of the Brownian bridge. We then discuss how different the case of the Brownian motion is.
@article {key1812799m,
AUTHOR = {Donati-Martin, Catherine and Matsumoto,
Hiroyuki and Yor, Marc},
TITLE = {On striking identities about the exponential
functionals of the {B}rownian bridge
and {B}rownian motion},
JOURNAL = {Period. Math. Hung.},
FJOURNAL = {Periodica Mathematica Hungarica. Journal
of the J\'anos Bolyai Mathematical Society},
VOLUME = {41},
NUMBER = {1--2},
MONTH = {November},
YEAR = {2000},
PAGES = {103--119},
DOI = {10.1023/A:1010308203346},
NOTE = {Dedicated to Professor Endre Cs\'aki
on the occasion of his 65th birthday.
MR:1812799. Zbl:1062.60080.},
ISSN = {0031-5303},
}
G. Pap and M. Yor :
“The accuracy of Cauchy approximation for the windings of planar Brownian motion Period. Math. Hung.
41 : 1–2
(November 2000 ),
pp. 213–226 .
Dedicated to Professor Endre Csáki on the occasion of his 65th birthday.
MR
1812807 Zbl
1074.60507 article
People
BibTeX
@article {key1812807m,
AUTHOR = {Pap, Gyula and Yor, Marc},
TITLE = {The accuracy of {C}auchy approximation
for the windings of planar {B}rownian
motion},
JOURNAL = {Period. Math. Hung.},
FJOURNAL = {Periodica Mathematica Hungarica. Journal
of the J\'anos Bolyai Mathematical Society},
VOLUME = {41},
NUMBER = {1--2},
MONTH = {November},
YEAR = {2000},
PAGES = {213--226},
DOI = {10.1023/A:1010380808797},
NOTE = {Dedicated to Professor Endre Cs\'aki
on the occasion of his 65th birthday.
MR:1812807. Zbl:1074.60507.},
ISSN = {0031-5303},
}
E. Csáki, Z. Shi, and M. Yor :
“Fractional Brownian motions as ‘higher-order’ fractional derivatives of Brownian local times ,”
pp. 365–387
in
Limit theorems in probability and statistics: Fourth Hungarian colloquium on limit theorems in probability and statistics
(Balatonlelle, Hungary, 28 June–2 July 1999 ),
vol. 1 .
Edited by I. Berkes, E. Csáki, and M. Csörgő .
János Bolyai Mathematical Society (Budapest ),
2002 .
Dedicated to Pál Révész on the occasion of his 65th birthday.
MR
1979974 Zbl
1030.60073 incollection
Abstract
People
BibTeX
Fractional derivatives \( \mathcal{D}^{\gamma} \) of Brownian local times are well defined for all \( \gamma \lt 3/2 \) . We show that, in the weak convergence sense, these fractional derivatives admit themselves derivatives which feature all fractional Brownian motions. Strong approximation results are also developed as counterparts of limit theorems for Brownian additive functionals which feature the fractional derivatives of Brownian local times.
@incollection {key1979974m,
AUTHOR = {Cs\'aki, E. and Shi, Z. and Yor, M.},
TITLE = {Fractional {B}rownian motions as ``higher-order''
fractional derivatives of {B}rownian
local times},
BOOKTITLE = {Limit theorems in probability and statistics:
{F}ourth {H}ungarian colloquium on limit
theorems in probability and statistics},
EDITOR = {Berkes, I. and Cs\'aki, E. and Cs\"org\H{o},
M.},
VOLUME = {1},
PUBLISHER = {J\'anos Bolyai Mathematical Society},
ADDRESS = {Budapest},
YEAR = {2002},
PAGES = {365--387},
NOTE = {(Balatonlelle, Hungary, 28 June--2 July
1999). Dedicated to P\'al R\'ev\'esz
on the occasion of his 65th birthday.
MR:1979974. Zbl:1030.60073.},
ISBN = {9639453013},
}
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},
}
P. Salminen and M. Yor :
“On hitting times of affine boundaries by reflecting Brownian motion and Bessel processes Period. Math. Hung.
62 : 1
(March 2011 ),
pp. 75–101 .
Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays.
MR
2772384 Zbl
1274.60251 ArXiv
1012.2038 article
Abstract
People
BibTeX
Firstly, we compute the distribution function for the hitting time of a linear time-dependent boundary \( t\mapsto a + bt \) , \( a\geq 0 \) , \( b\in\mathbb{R} \) , by a reflecting Brownian motion. The main tool hereby is Doob’s formula which gives the probability that Brownian motion started inside a wedge does not hit this wedge. Other key ingredients are the time inversion property of Brownian motion and the time reversal property of diffusion bridges. Secondly, this methodology can also be applied for the three-dimensional Bessel process. Thirdly, we consider Bessel bridges from 0 to 0 with dimension parameter \( \delta \gt 0 \) and show that the probability that such a Bessel bridge crosses an affine boundary is equal to the probability that this Bessel bridge stays below some fixed value.
@article {key2772384m,
AUTHOR = {Salminen, Paavo and Yor, Marc},
TITLE = {On hitting times of affine boundaries
by reflecting {B}rownian motion and
{B}essel processes},
JOURNAL = {Period. Math. Hung.},
FJOURNAL = {Periodica Mathematica Hungarica},
VOLUME = {62},
NUMBER = {1},
MONTH = {March},
YEAR = {2011},
PAGES = {75--101},
DOI = {10.1007/s10998-011-5075-2},
NOTE = {Dedicated to Endre Cs\'aki and P\'al
R\'ev\'esz on the occasion of their
75th birthdays. ArXiv:1012.2038. MR:2772384.
Zbl:1274.60251.},
ISSN = {0031-5303},
}
