Multiplicative functional for reflected Brownian motion via deterministic ODE

TL;DR

This paper establishes a convergence result for semi-discrete approximations of a multiplicative functional associated with reflected Brownian motion, using deterministic ODE analysis and excursion theory.

Contribution

It introduces a novel approach to approximate the multiplicative functional for reflected Brownian motion through semi-discrete schemes and deterministic analysis.

Findings

Proves convergence of semi-discrete approximations to the multiplicative functional.

Connects the functional to the Lyapunov exponent of the stochastic flow.

Utilizes deterministic ODE methods and excursion theory in the proof.

Abstract

We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.