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

TL;DR

This paper develops a multilevel Monte Carlo algorithm for weak approximation of SDEs driven by infinite-dimensional Wiener processes and Poisson measures, providing complexity bounds and numerical validation.

Contribution

It introduces a multilevel Monte Carlo method tailored for infinite-dimensional SDEs with Poisson jumps, establishing complexity bounds and demonstrating efficiency improvements.

Findings

Complexity bounds depend on parameters for dimension truncation and grid density.

Multilevel Monte Carlo outperforms standard Monte Carlo in computational efficiency.

Numerical experiments confirm theoretical complexity estimates and implementation feasibility.

Abstract

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