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Teor. Veroyatnost. i Primenen., 1999, Volume 44, Issue 2, Pages 466–472 (Mi tvp784)  

This article is cited in 8 scientific papers (total in 8 papers)

Short Communications

Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy's theorem

A. S. Chernya, A. N. Shiryaevb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The theorem proved by P. Lévy states that $(\sup B-B, \sup B)\stackrel{\mathrm{law}}{=}(|B|,L(B))$. Here, $B$ is a standard linear Brownian motion and $L(B)$ is its local time in zero. In this paper, we present an extension of P. Lévy's theorem to the case of a Brownian motion with a (random) drift as well as to the case of conditionally Gaussian martingales. We also give a simple proof of the equality $2\sup B^{\lambda}-B^{\lambda}\stackrel{\mathrm{law}}{=}|B^{\lambda}|+L(B^{\lambda})$, where $B^{\lambda}$ is the Brownian motion with a drift ${\lambda}\in\mathbb{R}$.

Keywords: P. Lévy's theorem, local time, Brownian motion with a drift, conditionally Gaussian martingales, Skorokhod's lemma.

DOI: https://doi.org/10.4213/tvp784

Full text: PDF file (371 kB)

English version:
Theory of Probability and its Applications, 2000, 44:2, 412–418

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Received: 25.01.1999

Citation: A. S. Cherny, A. N. Shiryaev, “Some distributional properties of a Brownian motion with a drift and an extension of P. Lévy's theorem”, Teor. Veroyatnost. i Primenen., 44:2 (1999), 466–472; Theory Probab. Appl., 44:2 (2000), 412–418

Citation in format AMSBIB
\by A.~S.~Cherny, A.~N.~Shiryaev
\paper Some distributional properties of a Brownian motion with a drift and an extension of P.~L\'evy's theorem
\jour Teor. Veroyatnost. i Primenen.
\yr 1999
\vol 44
\issue 2
\pages 466--472
\jour Theory Probab. Appl.
\yr 2000
\vol 44
\issue 2
\pages 412--418

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    This publication is cited in the following articles:
    1. Carr P., Geman H., Madan D.B., Yor M., “Stochastic volatility for Levy processes”, Math. Finance, 13:3 (2003), 345–382  crossref  mathscinet  zmath  isi  scopus
    2. Carmona P., Petit F., Yor M., “A trivariate law for certain processes related to perturbed Brownian motions”, Ann. Inst. H. Poincaré Probab. Statist., 40:6 (2004), 737–758  crossref  mathscinet  zmath  isi  scopus
    3. Roynette B., Vallois P., Yor M., “Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I”, Studia Sci. Math. Hungar., 43:2 (2006), 171–246  crossref  mathscinet  zmath  isi  elib  scopus
    4. A. N. Shiryaev, “O martingalnykh metodakh v zadachakh o peresechenii granits brounovskim dvizheniem”, Sovr. probl. matem., 8, MIAN, M., 2007, 3–78  mathnet  crossref  zmath
    5. Najnudel J., “Pénalisations de l'araignée brownienne”, Ann. Inst. Fourier (Grenoble), 57:4 (2007), 1063–1093  crossref  mathscinet  zmath  isi  scopus
    6. Carr P., Geman H., Madan D.B., Yor M., “Self-decomposability and option pricing”, Math. Finance, 17:1 (2007), 31–57  crossref  mathscinet  zmath  isi  scopus
    7. S. S. Sinel'nikov, “On the joint distribution of $(\sup X-X,\sup X)$ for a Lévy process $X$”, Russian Math. Surveys, 65:6 (2010), 1189–1191  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Kardaras C., “On the Stochastic Behaviour of Optional Processes Up To Random Times”, Ann. Appl. Probab., 25:2 (2015), 429–464  crossref  mathscinet  zmath  isi  scopus
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