Asymptotic error distributions for the Euler method for stochastic differential equations

Jacod, Jean ; Protter, Philip

Ann. Probab., Tome 26 (1998) no. 1, p. 267-307 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1998-01-14
Classification:  Stochastic differential equations,  Euler scheme,  error distributions,  Lévy processes,  numerical approximation,  60H10,  65U05,  60G44,  60F17