The Euler scheme for Levy driven stochastic differential equations

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of Eg(X-T) by Eg((X) over bar(T)(n)), where (X-t, 0 less than or equal to t less than or equal to T) is the solution of a stochastic differential equation governed by a Levy process (Z(t)), ((X) over bar(t)(n)) is defined by the Euler discretization scheme with step T/n. With appropriate assumptions on g(.), we show that the error Eg(X-T) - Eg((X) over bar(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 is slower and its speed depends on the behavior of the tails of the Levy measure.