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An Euler-Poisson Scheme for Lévy driven SDEs

We describe an Euler scheme to approximate solutions of Lévy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (2012). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate $\mathcal{O}(n^{-1/2})$. The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE; classic examples such as stable processes, subclasses of spectrally one sided Lévy processes and new families such as meromorphic Lévy processes (cf. Kuznetsov et al. (2011)) are some examples for which the implementation of our algorithm is straightforward.