An Asymptotic Result for Non Crossing Probabilities of Brownian Motion with Trend

Kurzfassung/Abstract

Let $B$ be a Brownian motion with paths in $C([0,1])$ and covariance kernel $K(s,t)=\min\{s,t\}$ and let $h$ be a nonnegative continuous function. The authors improve the previously known results on an asymptotic expansion $(\gamma \to \infty)$ for a non crossing probability $$P\{\forall t \in [0,1]:B(t)<u(t)-\gamma h(t)\},$$ where $u(t)$ is a real-valued measurable function.