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Simulation and approximationof Lévy-driven stochastic differential equations

Published online by Cambridge University Press:  05 January 2012

Nicolas Fournier
Nicolas Fournier*
Affiliation:
LAMA UMR 8050, Faculté de Sciences et Technologies, Université Paris Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France; nicolas.fournier@univ-paris12.fr
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Abstract

We consider the approximate Euler scheme for Lévy-drivenstochastic differential equations.We study the rate of convergence in law of the paths.We show that when approximating the small jumps by Gaussianvariables, the convergence is much faster than when simplyneglecting them.For example, when the Lévy measure of the driving processbehaves like |z|−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the small jumps, see[S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of ordernα/(2−α), which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Lévy-driven S.D.E.by a Brownian S.D.E. when the Lévy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlós-Major-Tsunády [J. Komlós, P. Major and G. Tusnády, An approximation of partial sums of independentrvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].

Keywords

Lévy processesstochastic differential equationsMonte-Carlo methodssimulationWasserstein distance
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 233 - 248
Copyright
© EDP Sciences, SMAI, 2011

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