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

TL;DR

This paper develops a new stability and convergence analysis framework for explicit and implicit Euler-type schemes applied to stochastic differential equations with super-linear growth, including practical methods like split-step backward Euler and a novel projected Euler-Maruyama.

Contribution

It introduces a generalized stability concept for stochastic schemes that does not require higher moment bounds and proves optimal convergence rates for specific schemes.

Findings

The split-step backward Euler method achieves optimal strong convergence rate.

The projected Euler-Maruyama method is a new explicit scheme with proven convergence.

Numerical experiments confirm theoretical convergence rates.

Abstract

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.