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

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.


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
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