The walk on moving spheres: a new tool for simulating Brownian motion's exit time from a domain

TL;DR

This paper presents a novel simulation method for Brownian motion exit times using Bessel process connections, avoiding complex series inversion and splitting schemes, resulting in a fast and accurate numerical approach.

Contribution

Introduces a new method leveraging Bessel process hitting times to efficiently simulate Brownian motion exit times without complex series inversion.

Findings

Method outperforms existing techniques in speed and accuracy.

Numerical experiments validate the effectiveness of the approach.

Avoids splitting time schemes and complex series inversion.

Abstract

In this paper we introduce a new method for the simulation of the exit time and position of a $\delta$-dimensional Brownian motion from a domain. The main interest of our method is that it avoids splitting time schemes as well as inversion of complicated series. The idea is to use the connexion between the $\delta$-dimensional Bessel process and the $\delta$-dimensional Brownian motion thanks to an explicit Bessel hitting time distribution associated with a particular curved boundary. This allows to build a fast and accurate numerical scheme for approximating the hitting time. Numerical comparisons with existing methods are performed.