Worst case complexity of multivariate Feynman–Kac path integration

Abstract

We study the multivariate Feynman–Kac path integration problem. This problem was studied in Plaskota et al. (J. Comp. Phys. 164 (2000) 335) for the univariate case. We describe an algorithm based on uniform approximation, instead of the L 2-approximation used in Plaskota et al. (2000). Similarly to Plaskota et al. (2000), our algorithm requires extensive precomputing. We also present bounds on the complexity of our problem. The lower bound is provided by the complexity of a certain integration problem, and the upper bound by the complexity of the uniform approximation problem. The algorithm presented in this paper is almost optimal for the classes of functions for which uniform approximation and integration have roughly the same complexities.