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This article is cited in 9 scientific papers (total in 9 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.
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
Linking options:
https://www.mathnet.ru/eng/tvp784https://doi.org/10.4213/tvp784 https://www.mathnet.ru/eng/tvp/v44/i2/p466
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