The Law of the Euler Scheme for Stochastic Differential Equations: II. Convergence Rate of the Density

Abstract

In the first part of this work~\cite{Bally-Talay-94-1} we have studied the approximation problem of $\ee f(X_T)$ by $\ee f(X_T^n)$, where $(X_t)$ is the solution of a stochastic differential equation, $(X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$, and $f(\cdot)$ is a given function, only supposed measurable and bounded. We have proven that the discretization error can be expanded in terms of powers of $\frac1n$ under a nondegeneracy condition of Hormander type for the infinitesimal generator of $(X_t)$. In this second part, we consider the density of the law of a small perturbation of $X_T^n$ and we compare it to the density of the law of $X_T$: we prove that the difference between the densities can also be expanded in terms of $\frac1n$. \noindent{\bf AMS(MOS) classification}: 60H07, 60H10, 60J60, 65C05, 65C20, 65B05.