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 $\ee g(X_T)$ by $\ee g(\bar X_T^n)$, where $(X_t,0\leq t\leq T)$ is the solution of a stochastic differential equation governed by a Levy process $(Z_t)$, $(\bar X^n_t)$ is defined by the Euler discretization scheme with step $\fracTn$. With appropriate assumptions we show that the error $\ee g(X_T)-\ee g(\bar X_T^n)$ can be expanded in powers of $\frac1n$ 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. The simulation of the increments of $(Z_t)$ is also discussed.

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