The First Time of a Brownian Motion from an Unbounded Convex Domain
Date
2002
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Department of Mathematical Sciences
Abstract
Consider the first exit time, ˝D of a (d + 1)-dimensional Brownian motion from an
unbounded open domain D =
(x; y) 2 R
d+1 : y > f(x); x 2 R
d
starting at (x0; f(x0) + 1) 2 R
d+1
for some x0 2 R
d , where the function f(x) on R
d is convex and f(x) ! 1 as the Euclidean norm
j x j ! 1 . Very general estimates for the asymptotics of log P (˝D > t) are given by using Gaussian
techniques. In particular, for f(x) = exp fj x j
p
g , p > 0,
lim
t !1
t
Description
Keywords
Brownian Motion, Bessel process, asymptotic tail distribution, exit probabilities, Slepian's inequality