Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes

Abstract

In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by Lévy processes. The underlying numerical schemes are based on jump-adapted Euler schemes. We prove stable convergence of an idealised scheme. Further, we deduce limit theorems for certain classes of functionals depending on the whole trajectory of the process. In particular, we allow dependence on marginals, integral averages and the supremum of the process. The idealised scheme is related to two practically implementable schemes and corresponding central limit theorems are given. In all cases, we obtain errors of order $N^{-1/2}(\operatorname{log}N)^{1/2}$ in the computational time $N$ which is the same order as obtained in the classical set-up analysed by Giles [Oper. Res. 56 (2008) 607–617]. Finally, we use the central limit theorems to optimise the parameters of the multilevel scheme.