Multilevel Monte Carlo for stochastic differential equations with small noise

TL;DR

This paper analyzes the efficiency of multilevel Monte Carlo methods combined with Euler-Maruyama discretization for estimating expectations of solutions to small noise stochastic differential equations, demonstrating optimal asymptotic complexity.

Contribution

It introduces a detailed variance analysis for coupled paths in multilevel Monte Carlo, showing its efficiency in small noise regimes and matching idealized complexity.

Findings

Multilevel Monte Carlo with Euler-Maruyama is often most efficient in small noise SDEs.

The method achieves asymptotic complexity comparable to exact sampling at constant cost.

Simulations confirm theoretical asymptotic efficiency results.

Abstract

We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the commonly occurring small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. Under the assumptions we make on the underlying model, the multilevel method combined with Euler-Maruyama is often found to be the most efficient option. Moreover, under a wide range of scalings the multilevel method is found to give the same asymptotic complexity that would arise in the idealized case where we have access to exact samples of the required distribution at a cost of $O(1)$ per sample. A key step in our analysis is to analyze the variance between two coupled paths directly, as opposed to their $L^2$ distance. Careful simulations are provided to illustrate the asymptotic results.