Boundary Crossings of Brownian Motion

Abstract

Let $B$ be a standard Brownian motion and let $b_\gamma$ be a piecewise linear continuous boundary function. In this paper we obtain an exact asymptotic expansion of $P\{ B(t) < b_\gamma(t), \forall t\in [0,1]\} $ provided that the boundary function satisfies $\lim_{\gamma \to \infty} b_\gamma(t^*)= -\infty$ for some $t^*\in (0,1]$.