Accuracy Improvement of Stochastic Linear Multi-Step Methods for Solving Stochastic Ordinary Differential Equations

The main objectives of this thesis may be oriented toward three directions. The first objective is a study, in details, the basic theory of stochastic calculus and study the linear multistep methods for solving stochastic differential equations and prove some results related to this topic, as well as, studying the Itô-Taylor series expansion and its applications. The second objective is a study the two steps Maruyama method and also the solution of stochastic ordinary differential equations using implicit methods which are treated by using the methods for solving nonlinear algebraic equations resulting from the used implicit method, such as Newton- Raphson method and predictor corrector method, also proposing a new approach for solving stochastic ordinary differential equations using variable step size method have been proposed. The third objective is to introduce the higher-order Richardson extrapolation method and variable order method for solving stochastic
ordinary differential equations, which has the utility of improving the accuracy of the obtained results.