Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm

TL;DR

This paper introduces a new explicit numerical scheme for stochastic differential equations with non-Lipschitz drift, using skew-symmetric distributions, which converges weakly and efficiently to the true diffusion and its equilibrium.

Contribution

It proposes a novel skew-symmetric sampling scheme that does not require Lipschitz conditions and demonstrates convergence and ergodic properties for long-time simulation.

Findings

Scheme converges weakly as step-size decreases

Converges to equilibrium at a geometric rate

Bias between scheme and true diffusion distributions quantified

Abstract

We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of skewness determined by the drift and volatility of the underlying process. We show that as the step-size decreases the scheme converges weakly to the diffusion of interest. We then consider the problem of simulating from the limiting distribution of an ergodic diffusion process using the numerical scheme with a fixed step-size. We establish conditions under which the numerical scheme converges to equilibrium at a geometric rate, and quantify the bias between the equilibrium distributions of the scheme and of the true diffusion process. Notably, our results do not require a global Lipschitz assumption on the drift, in contrast to those required for the Euler--Maruyama scheme for long-time simulation at fixed step-sizes. Our weak convergence result relies on an extension of the theory of Milstein \& Tretyakov to stochastic differential equations with non-Lipschitz drift, which could also be of independent interest. We support our theoretical results with numerical simulations.