The law of the Euler scheme for stochastic differential equations .1. Convergence rate of the distribution function

We study the approximation problem of IE f(X(T)) by IE f(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 Tin, and f is a given function. For smooth f's, Talay and Tubaro have shown that the error IE f(X(T)) - f(X(T)(n)) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when f is only supposed measurable and bounded, under an additional nondegeneracy condition of Hormander type for the infinitesimal generator of (X(t)) : to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law of X(T)(n) and compare it to the density of the law of X(T).