An Euler–Poisson scheme for Lévy driven stochastic differential equations

Abstract

Abstract We describe an Euler scheme to approximate solutions of Lévy driven stochastic differential equations (SDEs) where the grid points are given by the arrival times of a Poisson process and thus are random. This result extends the previous work of Ferreiro-Castilla et al. (2014). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and prove that the mean-square error converges with rate 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. Classical examples, such as stable processes, subclasses of spectrally one-sided Lévy processes, and new families, such as meromorphic Lévy processes (Kuznetsov et al. (2012), are examples for which our algorithm provides an interesting alternative to existing methods, due to its straightforward implementation and its robustness with respect to the jump structure of the driving Lévy process.