Approximation theorems for Lévy-driven Marcus (canonical) stochastic differential equations

In this thesis, we consider the problem of the numerical approximation of the Marcus (canonical) stochastic differential equations (SDEs) driven by a Brownian motion and an independent the pure jump Lévy process. The numerical scheme used in this thesis is the non-linear discrete time approximation based on the Wong–Zakai approximation scheme. The main results of this thesis are presented in two parts. In the first part, we prove the uniform strong approximation theorem for solutions of the Marcus SDEs. This result is an extension of the approximation results known for Stratonovich SDEs driven by a Brownian motion. We also estimate the convergence rate of strong approximations. The approximation scheme requires the explicit knowledge of the increments of the pure jump Lévy process. In the second part, we apply the method suggested by Asmussen and Rosiński, and approximate the increments of the pure jump Lévy process by a sum of Gaussian and a compound Poisson random variables that can be simulated explicitly. Hence, we examine the weak and strong convergence of the modified Wong–Zakai approximations and also determine the convergence rates. We illustrate our results by a numerical example.