A lower bound for boundary crossing probabilities of Brownian bridge/motion with trend

Kurzfassung/Abstract

We show a lower bound for the boundary crossing probability ${\bold P}\{\exists z\in[0,1]$: $h(z)+B_0(z)>u(z)\}$ with $B_0$ a Brownian bridge, $h$ a trend function and $u$ a boundary function. By that we get also a lower bound for the boundary crossing probability ${\bold P}\{\exists z\in[0,1]:h(z)+B_0(z) <u(z)\}$. It turns out that the bound improves the asymptotic result given by the authors, {\it F. Miller} and {\it J Hüsler} [Methodol. Comput. Appl. Probab. 5, 271--287 (2003; Zbl 1048.60025)] when considering a trend function $\gamma h$, $\gamma\to\infty$. The usual tools to obtain boundary crossing probabilities are finite-dimensional approximations of the above probability. However, we use the Cameron-Martin-Girsanov formula to obtain a bound of the above probability.