First exit of Brownian motion from a one-sided moving boundary

TL;DR

This paper revisits Uchiyama's 1980 result on Brownian motion's exit probability from a moving boundary, providing a simplified proof for decreasing boundaries and linking the integral test to Bessel process behavior.

Contribution

It offers an elementary, concise proof for the decreasing boundary case and connects the integral test to Bessel process repulsion effects.

Findings

Elementary proof for decreasing boundary case

Integral test linked to Bessel process repulsion

Potential for generalization to other processes like FBM

Abstract

We revisit a result of Uchiyama (1980): given that a certain integral test is satisfied, the rate of the probability that Brownian motion remains below the moving boundary $f$ is asymptotically the same as for the constant boundary. The integral test for $f$ is also necessary in some sense. After Uchiyama's result, a number of different proofs appeared simplifying the original arguments, which strongly rely on some known identities for Brownian motion. In particular, Novikov (1996) gives an elementary proof in the case of an increasing boundary. Here, we provide an elementary, half-page proof for the case of a decreasing boundary. Further, we identify that the integral test is related to a repulsion effect of the three-dimensional Bessel process. Our proof gives some hope to be generalized to other processes such as FBM.