Convergence of a Metropolized Integrator for Stochastic Differential Equations with Variable Diffusion Coefficient

TL;DR

This paper introduces an explicit Metropolized integrator for SDEs with variable diffusion coefficients that preserves equilibrium density and demonstrates weak convergence with order 1/2, supported by numerical experiments.

Contribution

The paper develops a new explicit numerical method for SDEs with variable diffusion coefficients that uses Metropolis-Hastings rejection to preserve equilibrium density.

Findings

Method is weakly convergent with order 1/2 for smooth coefficients.

Numerical experiments confirm convergence even for discontinuous coefficients.

The approach effectively preserves equilibrium density in simulations.

Abstract

We present an explicit method for simulating stochastic differential equations (SDEs) that have variable diffusion coefficients and satisfy the detailed balance condition with respect to a known equilibrium density. In Tupper and Yang (2012), we proposed a framework for such systems in which, instead of a diffusion coefficient and a drift coefficient, a modeller specifies a diffusion coefficient and a equilibrium density, and then assumes detailed balance with respect to this equilibrium density. We proposed a numerical method for such systems that works directly with the diffusion coefficient and equilibrium density, rather than the drift coefficient, and uses a Metropolis-Hastings rejection process to preserve the equilibrium density exactly. Here we show that the method is weakly convergent with order 1/2 for such systems with smooth coefficients. We perform numerical experiments demonstrating the convergence of the method for systems not covered by our theorem, including systems with discontinuous diffusion coefficients and equilibrium densities.