Understanding average Brownian exit time

Abstract

Let W denote Brownian motion starting from the origin. The idea of this paper is give a computation of the expected exit time Eτ [ a, b] from an interval [ a, b], where a<0< b, without the aid of Wald's Identity. Instead, the Strong Markov Property and other fundamental properties of Brownian motion are used directly to show that Eτ [ a, b] is linear in both a and b, and then a limiting result about Brownian motion is used to compute the constant of linearity. As a part of the proof of the linearity of the expected exit time, we compute the distribution of W τ [ a, b] .