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January 1998 Asymptotic error distributions for the Euler method for stochastic differential equations
Jean Jacod, Philip Protter
Ann. Probab. 26(1): 267-307 (January 1998). DOI: 10.1214/aop/1022855419

Abstract

We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equations the rate is $1/ \sqrt{n}$; we provide a necessary and sufficient condition for this rate to be $1/ \sqrt{n}$ when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law.

The rate can also differ from $1/ \sqrt{n}$: this is the case for instance if the driving process is deterministic, or if it is a Lévy process without a Brownian component. It is again $1/ \sqrt{n}$ when the driving process is Lévy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finite-dimensional sense only, while the discretized normalized error processes converge in law in the Skorohod sense, and the limit is given an explicit form.

Citation

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Jean Jacod. Philip Protter. "Asymptotic error distributions for the Euler method for stochastic differential equations." Ann. Probab. 26 (1) 267 - 307, January 1998. https://doi.org/10.1214/aop/1022855419

Information

Published: January 1998
First available in Project Euclid: 31 May 2002

zbMATH: 0937.60060
MathSciNet: MR1617049
Digital Object Identifier: 10.1214/aop/1022855419

Subjects:
Primary: 60H10, 65U05
Secondary: 60F17, 60G44

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.26 • No. 1 • January 1998
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