Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes

TL;DR

This paper establishes mean-square convergence of order 1 for Milstein-type schemes applied to SDEs with super-linear growth, using stochastic C-stability and B-consistency analysis, supported by numerical experiments.

Contribution

It introduces a new projected Milstein scheme and extends stochastic C-stability and B-consistency analysis to these schemes for SDEs with super-linear coefficients.

Findings

Both schemes are mean-square convergent of order 1.

The analysis applies to equations with super-linear growth.

Numerical experiments confirm theoretical results.

Abstract

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.