A multilevel Monte Carlo algorithm for stochastic differential equations driven by countably dimensional Wiener process and Poisson random measure

Abstract

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density $n\in N$ and truncation dimension parameter $M\in N$, is of the order $n^{-1/2}+\delta \left(M\right)$ such that $\delta (\cdot )$ is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported.