Multilevel path simulation for weak approximation schemes

TL;DR

This paper explores the use of multilevel Monte Carlo methods for weak approximation schemes, demonstrating that simple couplings can achieve significant complexity reductions, exemplified with Euler schemes for Lévy-driven SDEs.

Contribution

It introduces a novel approach to applying MLMC to weak schemes using simple couplings, achieving complexity gains similar to strong convergence scenarios.

Findings

Complexity of order ε^{-2} log^2(ε) for weak MLMC estimates

Effective coupling strategy for consecutive discretization levels

Numerical examples confirm theoretical complexity improvements

Abstract

In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler scheme for L\'evy driven stochastic differential equations, and show that, given a weak convergence of order $\alpha\geq 1/2,$ the complexity of the corresponding "weak" MLMC estimate is of order $\varepsilon^{-2}\log ^{2}(\varepsilon).$ The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.