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January 1997 The Euler scheme for Lévy driven stochastic differential equations
Philip Protter, Denis Talay
Ann. Probab. 25(1): 393-423 (January 1997). DOI: 10.1214/aop/1024404293

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.

Citation

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Philip Protter. Denis Talay. "The Euler scheme for Lévy driven stochastic differential equations." Ann. Probab. 25 (1) 393 - 423, January 1997. https://doi.org/10.1214/aop/1024404293

Information

Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0876.60030
MathSciNet: MR1428514
Digital Object Identifier: 10.1214/aop/1024404293

Subjects:
Primary: 60H10, 65U05
Secondary: 60E07, 60J30, 65C5, 65R20

Rights: Copyright © 1997 Institute of Mathematical Statistics

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Vol.25 • No. 1 • January 1997
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