THE EULER SCHEME FOR STOCHASTIC DIFFERENTIAL-EQUATIONS - ERROR ANALYSIS WITH MALLIAVIN CALCULUS

We study the approximation problem of Ef(X(T)) by Ef(X(T)(n)), where (X(t)) is the solution of a stochastic differential equation, (X(t)(n)) is defined by the Euler discretization scheme with step T/n, and f is a given function. For smooth f's, Talay and Tubaro had shown that the error Ef(X(T)) - Ef(X(T)(n)) can be expanded in powers of T/n, which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we present our following recent result: the expansion exists also when f is only supposed measurable and bounded, under a nondegeneracy condition (essentially, the Hormander condition for the infinitesimal generator of (X(t))): this is obtained with Malliavin's calculus. We also get an estimate on the difference between the density of the law of X(T) and the density of the law of X(T)(n).