The Euler scheme for Lévy driven stochastic differential equations

Protter, Philip ; Talay, Denis

Ann. Probab., Tome 25 (1997) no. 4, p. 393-423 / Harvested from Project Euclid

Résumé

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of $\mathbb{E}g(X_T)$ by $\mathbb{E}g(\overline{X}_T^n)$, where $(X_t, 0 \leq t \leq T)$ is the solution of a stochastic differential equation governed by a Lévy process $(Z_t), (\overline{X}_t^n)$ is defined by the Euler discretization scheme with step $T/n$. With appropriate assumptions on $g(\cdot)$, we show that the error of $\mathbb{E}g(X_T) - \mathbb{E}g(\overline{X}_T^n)$ can be expanded in powers of $1/n$ 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.

Détail

Publié le : 1997-01-14
Classification:  Stochastic differenctial equations,  Lévy processes,  Euler method,  Monte Carlo methods,  simulation,  60H10,  65U05,  65C5,  60J30,  60E07,  65R20