The Law of the Euler Scheme for Stochastic Differential Equations: I. Convergence Rate of the Distribution Function

We study the approximation problem of IEf(XT ) by IEf(Xn T ), where (Xt) is the solution of a stochastic differential equation, (Xn t ) is defined by the Euler discretization scheme with step Tn , and f is a given function. For smooth f ’s, Talay and Tubaro have shown that the error IEf(XT )−f(Xn T ) 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 Hörmander type for the infinitesimal generator of (Xt): 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 Xn T and compare it to the density of the law of XT . AMS(MOS) classification: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.