Strong Convergence for Euler-Maruyama and Milstein Schemes with Asymptotic Method

TL;DR

This paper proves strong convergence of accelerated Euler-Maruyama and Milstein schemes for perturbed stochastic differential equations, with applications to multi-level Monte Carlo methods and numerical validation on the SABR model.

Contribution

It introduces accelerated schemes with proven strong convergence for perturbed SDEs, extending existing methods and demonstrating their effectiveness through numerical experiments.

Findings

Accelerated Euler-Maruyama scheme achieves strong convergence.

Milstein scheme with acceleration also converges strongly.

Numerical experiments confirm efficiency on SABR model.

Abstract

Motivated by weak convergence results in the paper of Takahashi and Yoshida (2005), we show strong convergence for an accelerated Euler-Maruyama scheme applied to perturbed stochastic differential equations. The Milstein scheme with the same acceleration is also discussed as an extended result. The theoretical results can be applied to analyzing the multi-level Monte Carlo method originally developed by M.B. Giles. Several numerical experiments for the SABR stochastic volatility model are presented in order to confirm the efficiency of the schemes.