Multilevel Monte Carlo for stochastic differential equations with small noise

We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the 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. The multilevel method combined with Euler-Maruyama is found to be the most efficient option under the assumptions we make on the underlying model. Further, under a wide range of scalings the multilevel method is found to be optimal in the sense that it has the same asymptotic computational complexity that arises from Monte Carlo with direct sampling from the exact distribution --- something that is typically impossible to do. The variance between two coupled paths, as opposed to the L2 distance, is directly analyzed in order to provide sharp estimates in the multilevel setting.