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(X̄Tⁿ), where (Xₜ, 0 ≤ t ≤ T) is the solution of a stochastic differential equation governed by a Lévy process (Zₜ), (X̄ₜⁿ) 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(X̄Tⁿ) 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.