The law of the Euler scheme for stochastic differential equations

In relation with Monte-Carlo methods to solve parabolic Partial Differential Equations or some integro-differential equations, we study two approximation problems. The first problem is the approximation of Eg(X(T)) by Eg(X(T)(n)), where (X(t)) is the solution of a stochastic differential equation governed by a Levy process (Z(t)), (X(t)(n)) is defined by the Euler discretion scheme with step T/n. With appropriate assumptions on g(.), the error Eg(X(T)) - Eg(X(T)(n)) can be expanded in powers of 1/n if the Levy measure of Z has finite moments of order high enough. Otherwise the rate of convergence depends on the behavior of the tails of the Levy measure. The second problem concerns the case where Z is simply a Brownian motion. 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): the difference between the densities can also be expanded in terms of 1/n.