The Euler scheme for stochastic differential equations: error analysis with Malliavin calculus

We study the approximation problem of Ef(XT) by Ef(XTn), where (Xt) is the solution of a stochastic differential equation, (Xtn) is defined by the Euler discretization scheme with step Tn, and f is a given function. For smooth f's, Talay and Tubaro had shown that the error Ef(XT) − Ef(XTn) can be expanded in powers of Tn, 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 Hörmander condition for the infinitesimal generator of (Xt)): this is obtained with Malliavin's calculus. We also get an estimate on the difference between the density of the law of XT and the density of the law of XTn.