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). References: 6
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