Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

TL;DR

This paper develops a new theoretical framework to analyze and establish moment bounds and strong convergence for numerical schemes approximating SDEs with non-globally Lipschitz coefficients, which are common in various scientific fields.

Contribution

It introduces a rare events-based approach to prove moment bounds and convergence for explicit and implicit Euler schemes for challenging SDEs with superlinear growth.

Findings

Established moment bounds for new explicit schemes

Proved strong convergence of the proposed methods

Demonstrated applicability to SDEs in finance, physics, biology, and chemistry

Abstract

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.