Adapted Numerical Methods for Stochastic Differential Equations

Abstract

[EN]The use of deterministic differential equations is among the most common ways to describe how a system changes mathematically. However, in practice, it may be impossible to account for all the factors that influence the system, or some of these factors may be random in nature. In this case, it is natural to modify the deterministic model to obtain one given by a stochastic differential equation. In recent years, there has been a growing interest in the stochastic generalization of differential equations, stochastic differential equations (SDEs). One reason for this is the wide-ranging applications of these equations, which are used, e.g., in Finance to model financial derivatives, in Biology to model the spread of diseases, in Physics to describe particle motion, and in Computer Science for generative modeling. Another reason for the increased popularity of stochastic differential equations is advances in computational power combined with the development of numerical schemes, which are often required to solve these equations numerically. In this thesis, we will focus on three aspects of SDEs: their numerical simulation, their stability, and the approximation of associated operators.