The Euler scheme for Lévy 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 (XTⁿ), where (Xₜ, 0 t T) is the solution of a stochastic differential equation governed by a Lévy process (Zₜ), (Xₜⁿ) is defined by the Euler discretization scheme with step T/n. With appropriate assumptions on g (), we show that the error of Eg (XT) - Eg (XTⁿ) can be expanded in powers of 1/n if the Lévy 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 Lévy measure.