Stabilized Numerical Methods for Stochastic Differential Equations driven by Diffusion and Jump-Diffusion Processes

Stochastic models that account for sudden, unforeseeable events play a crucial role in many different fields such as finance, economics, biology, chemistry, physics and so on. That kind of stochastic problems can be modeled by stochastic differential equations driven by jump-diffusion processes. In addition, there are situations, where a stochastic model is based on stochastic differential equations with multiple scales. Such stochastic problems are called stiff and lead for classical explicit integrators such as the Euler-Maruyama method to time stepsize restrictions due to stability issues. This opens the door for stabilized explicit numerical methods to efficiently tackle such situations. In this thesis we introduce first a stabilized multilevel Monte Carlo method for stiff stochastic differential equations. Using S-ROCK methods we show that this approach is very efficient for stochastic problems with multiple scales, but also for nonstiff problems with a significant noise part. Further, we present an improved version of the stabilized multilevel Monte Carlo method by considering S-ROCK methods with a higher weak order of convergence. Then we extend the S-ROCK methods to jump-diffusion processes. We study in detail the strong order of convergence of the newly introduced methods and we discuss the corresponding mean square stability domains. In the next part we present the multilevel Monte Carlo method for jump-diffusion processes. We state and prove a theorem that indicates the computational cost required to achieve a certain mean square accuracy. In the numerical section we compare the multilevel Monte Carlo approach to two variance reduction techniques, the antithetic and the control variates. We also show how the S-ROCK method for jump-diffusion processes, introduced in this thesis, can be used to create a stabilized multilevel Monte Carlo method for jump-diffusions that handles stiffness and considers the inclusion of jumps at the same time. Finally, we propose in this thesis a variable time stepping algorithm that uses S-ROCK methods to approximate weak solutions of stiff stochastic differential equations. A rigorous analytical study is carried out to derive a computable leading term of the time discretization error and an adaptive algorithm is suggested that adapts the time grid and adjusts the number of stages of the S-ROCK method simultaneously.