Numerical Solutions of Some Stochastic Ordinary Differential Equations Using Runge-Kutta Methods

The main objectives of this thesis may be oriented towards three directions. The first objective is to study in details the theory of stochastic calculus and stochastic ordinary differential equations using the Itô and
Stratonovich formulae, as well as, studying the Taylor series expansion and its applications. The second objective is to study and prove some theoretical results for solving stochastic differential equations analytically, in case of linear and nonlinear equations, and proved of some results related to the subject. The third objective is a study of the numerical solution of stochastic differential equations using one of the most well known methods, which are the Runge-Kutta methods. The derivation of the numerical methods (seem to be new to the best of our knowledge) based on a modified approach considering the Taylor series expansion as two parts, the first one is called the deterministic part and the other is the stochastic part.