Multi-level Monte Carlo methods with the Truncated Euler-Maruyama Scheme for Stochastic Differential Equations

TL;DR

This paper combines the truncated Euler-Maruyama scheme with multi-level Monte Carlo methods to efficiently approximate expectations of solutions to stochastic differential equations, providing theoretical convergence and cost analysis.

Contribution

It introduces a novel integration of the truncated EM method with MLMC for SDEs, with proven convergence rates and cost efficiency under specific coefficient conditions.

Findings

Proves convergence rate of the combined method.

Analyzes computational cost and efficiency.

Numerical examples validate theoretical results.

Abstract

In this paper, the truncated Euler-Maruyama (EM) method is employed together with the Multi-level Monte Carlo (MLMC) method to approximate the expectations of functions of solutions to stochastic differential equations (SDEs). The convergence rate and the computational cost of the approximations using the truncated EM method with the MLMC method are proved when the coefficients of SDEs fulfill the local Lipschitz and Khasminskii-type conditions. Numerical examples are given to demonstrate the theoretical results.