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April 2003 The first exit time of a Brownian motion from an unbounded convex domain
Wenbo V. Li
Ann. Probab. 31(2): 1078-1096 (April 2003). DOI: 10.1214/aop/1048516546

Abstract

Consider the first exit time $\tau_D$ of a $(d+1)$-dimensional Brownian motion from an unbounded open domain $D=\set{ (x, y) \in \R^{d+1} \dvtx y > f(x), x \in \R^d }$ starting at\vspace{0.5pt} $(x_0, f(x_0)+1)\in \R^{d+1}$ for some $x_0 \in \R^d$, where the function $f(x)$ on $\R^d$ is convex and $f(x) \to \infty$ as the Euclidean norm $|x| \to \infty$. Very general estimates for the asymptotics of $\log \pr{\tau_D >t}$ are given by using Gaussian techniques. In particular, for $f(x)=\exp\{ |x|^p\}$, $p >0$, \[ \lim_{t\to\infty} t^{-1} (\log t)^{2/p} \log \pr{\tau_D>t}=-j_\nu^2/2, \] where $\nu=(d-2)/2$ and $j_\nu$ is the smallest positive zero of the Bessel function $J_{\nu}$.

Citation

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Wenbo V. Li. "The first exit time of a Brownian motion from an unbounded convex domain." Ann. Probab. 31 (2) 1078 - 1096, April 2003. https://doi.org/10.1214/aop/1048516546

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1030.60032
MathSciNet: MR1964959
Digital Object Identifier: 10.1214/aop/1048516546

Subjects:
Primary: 60G40
Secondary: 60J65

Keywords: asymptotic tail distribution , Bessel process , Brownian motion , exit probabilities , Slepian's inequality

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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