Improving Brownian approximations for boundary crossing problems

TL;DR

This paper enhances Brownian motion approximations for boundary crossing problems of random walks by deriving correction terms, leading to more accurate predictions of crossing times and locations.

Contribution

It introduces correction terms that improve the accuracy of boundary crossing approximations based on Brownian motion for random walks.

Findings

Derived correction terms improve approximation accuracy

Enhanced predictions of crossing times and locations

Applicable to nonlinear boundary crossing problems

Abstract

Donsker's theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear boundary. In this paper, correction terms are derived to improve the accuracy of these approximations.