Multilevel path simulation for weak approximation schemes

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 Lévy driven stochastic differential equations, and show that, given a weak convergence of order $\alpha \ge 1/2,$ the complexity of the corresponding "weak" MLMC estimate is of order ${\epsilon }^{-2}{\log }^2(\epsilon ).$ The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.