Simulation of Reflected Brownian motion on two dimensional wedges

TL;DR

This paper derives explicit density formulas and develops recursive algorithms for simulating reflected and stopped Brownian motion within two-dimensional wedges, addressing computational challenges and complexity bounds.

Contribution

It introduces explicit density formulas using Bessel functions and recursive algorithms for simulating reflected Brownian motion in wedges, extending the reflection principle to two dimensions.

Findings

Explicit density formulas involving Bessel functions.

Recursive algorithms for simulation of reflected Brownian motion.

Analysis of algorithm complexity and computational considerations.

Abstract

We study a correlated Brownian motion in two dimensions, which is reflected, stopped or killed in a wedge represented as the intersection of two half spaces. First, we provide explicit density formulas, hinted by the method of images. These explicit expressions rely on infinite oscillating sums of Bessel functions and may demand computationally costly procedures. We propose suitable recursive algorithms for the simulation of the laws of reflected and stopped Brownian motion which are based on generalizations of the reflection principle in two dimensions. We study and give bounds for the complexity of the proposed algorithms.