The Euler scheme for Lévy driven stochastic differential equations

Abstract

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of double-struck E sign g(XT) by double-struck E sign g(X̄nT), where (Xt, 0 ≤ t ≤ T) is the solution of a stochastic differential equation governed by a Lévy process (Zt), (X̄nt) is defined by the Euler discretization scheme with step T/n. With appropriate assumptions on g(·), we show that the error double-struck E sign g(XT) - double-struck E sign g(X̄nT) 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.