Higher order numerical methods for SDEs without globally monotone coefficients

TL;DR

This paper introduces new higher order numerical schemes for stochastic differential equations with non-globally monotone coefficients, establishing their convergence rates and exponential integrability properties through a unified framework.

Contribution

It develops a novel family of stopped increment-tamed schemes for SDEs with non-globally monotone coefficients, providing rigorous convergence analysis and exponential integrability results.

Findings

Proposed schemes achieve optimal strong convergence rates.

Schemes possess exponential integrability properties.

Numerical experiments confirm theoretical results.

Abstract

In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and order 1.5 type for such SDEs. By formulating a novel unified framework, the proposed methods are shown to possess the exponential integrability properties, which are crucial to recovering convergence rates in the non-global monotone setting. Armed with such exponential integrability properties and by the arguments of perturbation estimates, we successfully identify the optimal strong convergence rates of the aforementioned methods in the non-global monotone setting. Numerical experiments are finally presented to corroborate the theoretical results.