Construction of a Mean Square Error Adaptive Euler--Maruyama Method with Applications in Multilevel Monte Carlo

TL;DR

This paper develops an adaptive Euler--Maruyama method based on mean square error expansion, enhancing multilevel Monte Carlo efficiency for low-regularity stochastic differential equations with significant performance improvements.

Contribution

It introduces a novel a posteriori adaptive time stepping scheme for Euler--Maruyama, integrated into MLMC, optimized for low-regularity SDE problems.

Findings

Adaptive MLMC outperforms uniform MLMC by orders of magnitude.

Error bounds are maintained with high probability at near-optimal cost.

Method achieves O(TOL^{-2}log(TOL)^4) complexity for low-regularity problems.

Abstract

A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping Euler--Maruyama method for numerical solutions of SDE, and the resulting method is incorporated into a multilevel Monte Carlo (MLMC) method for weak approximations of SDE. This gives an efficient MSE adaptive MLMC method for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC method is shown to outperform the uniform time stepping MLMC method by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).