The topic of this thesis is the study of approximation schemes of jump processes whose driving noise is a Levy process.

In the first part of our work we study properties of the driving noise. We present a novel approximation method for the density of a Levy process. The scheme makes use of a continuous time Markov chain defined through a careful analysis of the generator. We identify the rate of convergence and carry out a detailed analysis of the error. We also analyse the case of multidimensional Levy processes in the form of subordinate Brownian motion. We provide a weak scheme to approximate the density that does not rely on discretising the Levy measure and results in better convergence rates.

The second part of the thesis concerns the analysis of schemes for BSDEs driven by Brownian motion and a Poisson random measure. Such equations appear naturally in hedging problems, stochastic control and they provide a natural probabilistic approach to the solution of certain semi linear PIDEs. While the numerical approximation of the continuous case has been studied in the literature, there has been relatively little progress in the study of such equations with a discontinuous driver. We present a weak Monte Carlo scheme in this setting based on Picard iterations. We discuss its convergence and provide a numerical illustration.