The First Exit Time Of The Maximum Value Of Multiple Brownian Motions From An Unbounded Convex Domain

Based on the large deviation theory and the small deviation theory, this paper con-cerns the first exit probability of the maximum value of multiple Brownian motions from an unbounded convex domain. Throughout the paper, consider Bi(t), i=1,2,...n be n in-dependent d-dimensional Brownian motions, W be a standard one-dimensional Brownian motion starting at 0, independent of{Bi(t),t≥0}, and h(x) be a reversible nondecreasing lower semi-continuous convex function on [0,∞) with h(0) finite. The exit probability is P(h(maxl≤i≤{‖Bi(s)‖})≤W(s)+h(0)+1,0≤s≤t). We develop former researches. Very useful upper and lower asymptotic estimates of logP(·) are given by using Gaussian technique and Slepian inequality.This dissertation is organized as follows:In chapter 1, we introduce the background of the research and relative definitions and properties.In chapter 2, we introduce the theoretical basis of this paper and important theories and lemmas.Chapter 3 is the core of this dissertation. We obtain the upper and lower asymptotic estimates of the first exit probability of the maximum value of multiple Brownian motions from an unbounded convex domain.In the end, we come to the conclusion and expectation.