An optimal Monte Carlo algorithm for multivariate Feynman–Kac path integrals

Abstract

We study a Monte Carlo algorithm for approximating multivariate Feynman–Kac path integrals parameterized by initial value and potential d-variate functions. This problem suffers from the curse of dimensionality in the worst case deterministic setting. That is why we study the randomized setting in which the functions are sampled at randomized points and the algorithm uses a finite number of such samples. We achieve optimality of the randomized algorithm due to variance reduction obtained by Smolyak’s algorithm for approximating tensor product functions. The optimal convergence depends on the smoothness of initial value and potential functions. When the initial value and potential functions are r times continuously differentiable we obtain the optimal convergence of order m−1∕2−r∕d for m function samples. Thus, if r∕d is negligible we do not gain much over the commonly used classical Monte Carlo algorithm whose convergence is of order m−1∕2. Hence, the classical Monte Carlo algorithm turns out to be almost optimal if r∕d is small. On the other hand, we can significantly improve the classical Monte Carlo convergence if r∕d is not negligible.