Multilevel Monte Carlo for L\'evy-driven SDEs: Central limit theorems for adaptive Euler schemes

TL;DR

This paper develops multilevel Monte Carlo methods for Le9vy-driven SDEs using jump-adapted Euler schemes, proving convergence and limit theorems, and optimizing scheme parameters for efficient expectation computation.

Contribution

It introduces stable convergence results and central limit theorems for multilevel Monte Carlo schemes applied to Le9vy-driven SDEs with jump-adapted Euler methods, including practical implementation insights.

Findings

Errors of order N^{-1/2}(0)^{1/2} achieved

Limit theorems established for functionals of the process

Parameter optimization for multilevel schemes provided

Abstract

In this article, we consider multilevel Monte Carlo for the numerical computation of expectations for stochastic differential equations driven by L\'{e}vy 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 $N^{-1/2}(\log N)^{1/2}$ 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.