## The Annals of Probability

### The Euler scheme for Lévy driven stochastic differential equations

#### Abstract

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.

#### Article information

Source
Ann. Probab. Volume 25, Number 1 (1997), 393-423.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404293

Digital Object Identifier
doi:10.1214/aop/1024404293

Mathematical Reviews number (MathSciNet)
MR1428514

Zentralblatt MATH identifier
0876.60030

#### Citation

Protter, Philip; Talay, Denis. The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25 (1997), no. 1, 393--423. doi:10.1214/aop/1024404293. https://projecteuclid.org/euclid.aop/1024404293

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• WEST LAFAYETTE, INDIANA 47907-1395 B.P. 93 E-MAIL: protter@math.purdue.edu 06902 SOPHIA-ANTIPOLIS FRANCE