ASYMPTOTIC ERROR DISTRIBUTIONS FOR THE EULER METHOD FOR STOCHASTIC DIFFERENTIAL-EQUATIONS

We are interested in the rate of convergence of the Euler scheme appro ximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymp totic behavior of the associated normalized error. It is well known th at for Ito's equations the rate is 1/root n;we provide a necessary and sufficient condition for this rate to be 1 root n when the driving se mimartingale is a continuous martingale, or a continuous semimartingal e under a mild additional assumption; we also prove that in these case s the normalized error processes converge in law. The rate can also di ffer fi om 1 root n: this is the case for instance if the driving proc ess is deterministic, or if it is a Levy process without a Brownian co mponent. It is again 1/root n when the driving process is Levy with a nonvanishing Brownian component, but then the normalized error process es converge in law in the finite-dimensional sense only, while the dis cretized normalized error processes converge in law in the Skorohod se nse, and the limit is given an explicit form.