Multilevel Monte Carlo for stochastic differential equations with small noise

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.