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July 2004 The Euler scheme for Lévy driven stochastic differential equations: limit theorems
Jean Jacod
Ann. Probab. 32(3): 1830-1872 (July 2004). DOI: 10.1214/009117904000000667

Abstract

We study the Euler scheme for a stochastic differential equation driven by a Lévy process Y. More precisely, we look at the asymptotic behavior of the normalized error process un(XnX), where X is the true solution and Xn is its Euler approximation with stepsize 1/n, and un is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (un) is a rate, which, in addition, is sharp when the limiting process (or processes) is not trivial.

We suppose that Y has no Gaussian part (otherwise a rate is known to be $u_{n}=\sqrt {n}$ ). Then rates are given in terms of the concentration of the Lévy measure of Y around 0 and, further, we prove the convergence of the sequence un(XnX) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Lévy processes whose Lévy measure behave like a stable Lévy measure near the origin. For example, when Y is a symmetric stable process with index α(0,2), a sharp rate is un=(n/logn)1/α; when Y is stable but not symmetric, the rate is again un=(n/logn)1/α when α>1, but it becomes un=n/(logn)2 if α=1 and un=n if α<1.

Citation

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Jean Jacod. "The Euler scheme for Lévy driven stochastic differential equations: limit theorems." Ann. Probab. 32 (3) 1830 - 1872, July 2004. https://doi.org/10.1214/009117904000000667

Information

Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1054.65008
MathSciNet: MR2073179
Digital Object Identifier: 10.1214/009117904000000667

Subjects:
Primary: 60J75 , 65C30
Secondary: 60F17 , 60J30

Keywords: Euler scheme , Lévy process , rate of convergence

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • July 2004
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