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(XT) by Eg(X̄n T), where (Xt, 0 ≤ t ≤ T) is the solution of a stochastic differential equation governed by a Levy process (Zt), (X̄n t) is defined by the Euler discretization scheme with step T/n. With appropriate assumptions on g(·), we show that the error Eg(XT) - Eg(X̄n T) 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.