Asymptotic error distributions for the Euler method for stochastic differential equations. (English) Zbl 0937.60060

Jacod, Jean; Protter, Philip

Ann. Probab. 26, No. 1, 267-307 (1998).

Summary: 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 behaviour 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 Skorokhod sense, and the limit is given an explicit form.

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30	Numerical solutions to stochastic differential and integral equations
60G44	Martingales with continuous parameter
60F17	Functional limit theorems; invariance principles

Keywords:

stochastic differential equations; Euler scheme; error distributions; Lévy processes; numerical approximation

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