Distribution Estimates for Functionals of the Two-Parameter Wiener Process

Abstract

Bounds on absorption probabilities for Banach space-valued Brownian motion are obtained as expectations of estimates for the conditional probability given the endpoint of the path. The results are applied to the problem of computing the tail distributions of the supremum, $S$, of the two-parameter Wiener process and the supremum, $S'$, of its tied-down version. It is shown that for $\lambda \geqq 0$, $$P\{S' \geqq \lambda\} \geqq (2\lambda^2 + 1)\exp\lbrack -2\lambda^2 \rbrack$$ and $$P\{S \geqq \lambda\} \geqq 4 \int^\infty_\lambda sN(-s)ds$$ where $N(s)$ denotes the standard normal distribution. A corollary is that $P(S \geqq \lambda) \approx 4N(-\lambda)$ as $\lambda \rightarrow + \infty$.