The Annals of Probability

Asymptotic error distributions for the Euler method for stochastic differential equations

Jean Jacod and Philip Protter

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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.

Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 267-307.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855419

Digital Object Identifier
doi:10.1214/aop/1022855419

Mathematical Reviews number (MathSciNet)
MR1617049

Zentralblatt MATH identifier
0937.60060

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 65U05
Secondary: 60G44: Martingales with continuous parameter 60F17: Functional limit theorems; invariance principles

Keywords
Stochastic differential equations Euler scheme error distributions Lévy processes numerical approximation

Citation

Jacod, Jean; Protter, Philip. Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 (1998), no. 1, 267--307. doi:10.1214/aop/1022855419. https://projecteuclid.org/euclid.aop/1022855419


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