The Annals of Applied Probability

Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction

Steffen Dereich

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Abstract

We introduce and analyze multilevel Monte Carlo algorithms for the computation of $\mathbb {E}f(Y)$, where Y=(Yt)t∈[0, 1] is the solution of a multidimensional Lévy-driven stochastic differential equation and f is a real-valued function on the path space. The algorithm relies on approximations obtained by simulating large jumps of the Lévy process individually and applying a Gaussian approximation for the small jump part. Upper bounds are provided for the worst case error over the class of all measurable real functions f that are Lipschitz continuous with respect to the supremum norm. These upper bounds are easily tractable once one knows the behavior of the Lévy measure around zero.

In particular, one can derive upper bounds from the Blumenthal–Getoor index of the Lévy process. In the case where the Blumenthal–Getoor index is larger than one, this approach is superior to algorithms that do not apply a Gaussian approximation. If the Lévy process does not incorporate a Wiener process or if the Blumenthal–Getoor index β is larger than 4∕3, then the upper bound is of order τ−(4−β)∕(6β) when the runtime τ tends to infinity. Whereas in the case, where β is in [1, 4∕3] and the Lévy process has a Gaussian component, we obtain bounds of order τβ∕(6β−4). In particular, the error is at most of order τ−1∕6.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 1 (2011), 283-311.

Dates
First available in Project Euclid: 17 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1292598035

Digital Object Identifier
doi:10.1214/10-AAP695

Mathematical Reviews number (MathSciNet)
MR2759203

Zentralblatt MATH identifier
1220.60040

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 60H05: Stochastic integrals 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes

Keywords
Multilevel Monte Carlo Komlós–Major–Tusnády coupling weak approximation numerical integration Lévy-driven stochastic differential equation

Citation

Dereich, Steffen. Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Probab. 21 (2011), no. 1, 283--311. doi:10.1214/10-AAP695. https://projecteuclid.org/euclid.aoap/1292598035


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