The Annals of Probability

The Euler scheme for Lévy driven stochastic differential equations

Philip Protter and Denis Talay

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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

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

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 65U05
Secondary: 65C5 60J30 60E07: Infinitely divisible distributions; stable distributions 65R20: Integral equations

Stochastic differenctial equations Lévy processes Euler method Monte Carlo methods simulation


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.

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