mean square convergent numerical methods for stochastic differential equations

Random differential equations are defined as differential equations involving random elements. In recent years, increasing interest in the numerical solution of random differential equations has led to the progressive development of several stochastic num, (A)Randomness may exist in the initial value, in the differential operator, in the non homogeneous function or among them.The solution of (A) is a stochastic process which is characterized by its probability density function or by obtaining some of its statistical moments, e.g. mean and covariance.Some numerical techniques were developed to solve the random differential equation (A), among of which is Euler and Runge- Kutta methods which are used in this thesis. This thesis is organized as follows:In chapter one, we give an introduction for the subject of thesis concerning review on probability theory , some properties of the expectation of random variables, stochastic processes, the classification and characteristics of stochastic processes, winneIn chapter three, we discuss Euler method for the deterministic and random ordinary equation and the Runge Kutta of the second order for the deterministic and random ordinary differential equations.In chapter four some selected linear and nonlinear problems are solved with proving the convergence of the obtained approximation of the solutions in mean square sense. Finally, conclusions and references are listed.