The Annals of Applied Probability

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

Steffen Dereich and Sangmeng Li

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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.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 136-185.

Dates
Received: March 2014
Revised: November 2014
First available in Project Euclid: 5 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1452003236

Digital Object Identifier
doi:10.1214/14-AAP1087

Mathematical Reviews number (MathSciNet)
MR3449315

Zentralblatt MATH identifier
1338.65004

Subjects
Primary: 65C05: Monte Carlo methods
Secondary: 60G51: Processes with independent increments; Lévy processes 60F05: Central limit and other weak theorems

Keywords
Multilevel Monte Carlo central limit theorem Lévy-driven stochastic differential equation Euler scheme jump-adapted scheme stable convergence

Citation

Dereich, Steffen; Li, Sangmeng. Multilevel Monte Carlo for Lévy-driven SDEs: Central limit theorems for adaptive Euler schemes. Ann. Appl. Probab. 26 (2016), no. 1, 136--185. doi:10.1214/14-AAP1087. https://projecteuclid.org/euclid.aoap/1452003236


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