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Zbl 0791.60071
Kawazu, Kiyoshi; Tanaka, Hiroshi
On the maximum of a diffusion process in a drifted Brownian environment. (English)
[A] Azéma, J. (ed.) et al., Séminaire de probabilités XXVII. Berlin: Springer-Verlag. Lect. Notes Math. 1557, 78-85 (1993). ISBN 3-540-57282-1/pbk

The authors investigate the problem: how fast does $P\{\max\sb{t \ge 0} X(t) \ge x\}$ decay as $x \to \infty$, where $X(t)$ is a diffusion process with generator ${1 \over 2} \exp \{W(x)+cx\} {d \over dx} (\exp \{-W(x)-cx\} {d \over dx})$. It turns out that the answer depends on $c$ and varies according as $c>1$, $c=1$, $0<c<1$. This problem is a diffusion analog of {\it V. T. Afanas'ev's} problem [Theory Probab. Appl. 35, No. 2, 205-215 (1990); translation from Teor. Veroyatn. Primen. 35, No. 2, 209-219 (1990; Zbl 0714.60054)].
[N.M.Zinchenko (Kiev)]

MSC 2000:: *60J65 Brownian motion
60J60 Diffusion processes

Keywords: Brownian motion; Cameron-Martin-Maruyama-Girsanov formula; exponential functional

Citations: Zbl 0725.60074; Zbl 0714.60054

Cited in: Zbl 1044.60068

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