The First Exit Time of a Brownian Motion from an Unbounded Convex Domain1 By
نویسنده
چکیده
Consider the first exit time τ D of a (d + 1)-dimensional Brownian motion from an unbounded open domain D = {(x, y) ∈ R d+1 : y > f (x), x ∈ R d } starting at (x 0 , f (x 0) + 1) ∈ R d+1 for some x 0 ∈ R d , where the function f (x) on R d is convex and f (x) → ∞ as the Euclidean norm |x| → ∞. Very general estimates for the asymptotics of log P(τ D > t) are given by using Gaussian techniques. In particular, for f (x) = exp{|x| p }, p > 0, lim t →∞ t −1 (log t) 2/p log P(τ D > t) = −j 2 ν /2, where ν = (d − 2)/2 and j ν is the smallest positive zero of the Bessel function J ν. 1. Introduction. Let B(t) = (B 1 (t),. .. , B d (t)) ∈ R d , t ≥ 0, be a standard d-dimensional Brownian motion, where B i (t), 1 ≤ i ≤ d, are independent Brown-ian motions starting at 0. Consider the first exit time τ D of a (d + 1)-dimensional Brownian motion from the unbounded open domain
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تاریخ انتشار 2003