On first exit times and their means for Brownian bridges

TL;DR

This paper analyzes the behavior of the mean first exit times for Brownian and Bessel bridges from small intervals, revealing an $O(h^2)$ behavior and a surprising connection to the Kolmogorov distribution, with applications to option pricing schemes.

Contribution

It establishes the asymptotic behavior of mean first exit times for Brownian and Bessel bridges and links these to the Kolmogorov distribution, aiding in financial option valuation analysis.

Findings

Mean first exit time from $(-h,h)$ behaves as $O(h^2)$ as $h \to 0$ for Brownian and Bessel bridges.

The mean exit time has a representation involving the Kolmogorov distribution.

Application to estimate convergence of binomial tree schemes for European options.

Abstract

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.