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We consider the problem of numerically estimating expectations of solutionsto stochastic differential equations driven by Brownian motions in the commonlyoccurring small noise regime. We consider (i) standard Monte Carlo methodscombined with numerical discretization algorithms tailored to the small noisesetting, and (ii) a multilevel Monte Carlo method combined with a standardEuler-Maruyama implementation. Under the assumptions we make on the underlyingmodel, the multilevel method combined with Euler-Maruyama is often found to bethe most efficient option. Moreover, under a wide range of scalings themultilevel method is found to give the same asymptotic complexity that wouldarise in the idealized case where we have access to exact samples of therequired distribution at a cost of $O(1)$ per sample. A key step in ouranalysis is to analyze the variance between two coupled paths directly, asopposed to their $L^2$ distance. Careful simulations are provided to illustratethe asymptotic results.

Multilevel Monte Carlo for stochastic differential equations with small noise