The law of the Euler scheme for stochastic differential equations

Abstract

We study the approximation problem ofE f(X T ) byE f(X ), where (X t ) is the solution of a stochastic differential equation, (X ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X T ) −f(X ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf 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 ofX and compare it to the density of the law ofX T .