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.

In this paper we propose the asymptotic error distributions of the Euler scheme for a stochastic differential equation driven by Itô semimartingales. Jacod (2004) studied this problem for stochastic differential equations driven by pure jump Lévy processes and obtained quite sharp results. We extend his results to a more general pure jump Itô semimartingale.

We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain the Lp-Hölder continuity of the solution of the FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on the Lp-Hölder estimate, we prove the convergence of the scheme when the number of time steps n goes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

Variance reduction has always been a central issue in Monte Carlo experiments. Population Monte Carlo can be used to this effect, in that a mixture of importance functions, called a D-kernel, can be iterativelyoptimized to achieve the minimum asymptotic variance for a function of interest among all possible mixtures.The implementation of this iterative scheme is illustrated for the computation of the price of a Europeanoption in the Cox-Ingersoll-Ross model. A Central Limit theorem as well as moderate deviations are established for the D-kernel Population Monte Carlo methodology.