Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

TL;DR

This paper establishes the strong mean-square convergence rate of 1/2 for the BDF2-Maruyama and backward Euler schemes applied to SDEs with global monotonicity, including superlinear growth, supported by numerical experiments.

Contribution

It is the first to prove a strong convergence rate for multi-step methods on SDEs with superlinear coefficients under global monotonicity.

Findings

Convergence rate of 1/2 for BDF2-Maruyama and backward Euler methods.

Numerical validation using a financial volatility model.

BDF2-Maruyama outperforms Euler methods in stiff or low-noise scenarios.

Abstract

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be $\frac{1}{2}$ for the two-step BDF-Maruyama scheme and for the backward Euler-Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the $\tfrac32$-volatility model from finance, which verify our results in practice and indicate that the BDF2-Maruyama method offers advantages over Euler-type methods if the stochastic differential equation is stiff or driven by a noise with small intensity.