# The Euler Scheme for Lévy driven Stochastic Differential Equations

1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : In relation with Monte-Carlo methods to solve some integro-differential equations, we study the approximation problem of $\ee g(X_T)$ by $\ee g(\bar X_T^n)$, where $(X_t,0\leq t\leq T)$ is the solution of a stochastic differential equation governed by a Lévy process $(Z_t)$, $(\bar X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$. With appropriate assumptions we show that the error $\ee g(X_T)-\ee g(\bar X_T^n)$ can be expanded in powers of $\frac1n$ if the Lévy measure of $Z$ has finite moments of order high enough. Otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the Lévy measure. The simulation of the increments of $(Z_t)$ is also discussed.
Keywords :
Type de document :
Rapport
[Research Report] RR-2621, INRIA. 1995, pp.31
Domaine :

https://hal.inria.fr/inria-00074065
Contributeur : Rapport de Recherche Inria <>
Soumis le : mercredi 24 mai 2006 - 14:25:01
Dernière modification le : samedi 27 janvier 2018 - 01:30:56
Document(s) archivé(s) le : jeudi 24 mars 2011 - 14:06:18

### Identifiants

• HAL Id : inria-00074065, version 1

### Citation

Philip Protter, Denis Talay. The Euler Scheme for Lévy driven Stochastic Differential Equations. [Research Report] RR-2621, INRIA. 1995, pp.31. 〈inria-00074065〉

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