Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Abstract

In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo. We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the bias and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance.