Mean Square Numerical Methods for Initial Value Random Differential Equations

Randomness may exist in the initial value or in the differential operator or both. In [1,2], the authors discussed the general order conditions and a global convergence proof is given for stochastic Runge-Kutta methods applied to stochastic ordinary differential equations (SODEs) of Stratonovich type. In [3,4], the authors discussed the random Euler method and the conditions for the mean square convergence of this problem. In [5], the authors considered a very simple adaptive algorithm based on controlling only the drift component of a time step. Platen, E. [6] discussed discrete time strong and weak approximation methods that are suitable for different applications. Other numerical methods are discussed in [7-12]. In this paper the random Euler and random RungeKutta of the second order methods are used to obtain an approximate solution for Equation (1.1). This paper is organized as follows. In Section 2, some important preliminaries are discussed. In Section 3, the existence and uniqueness of the solution of random differential initial value problem is discussed and the convergence of random Euler and random Runge-Kutta of the second order methods is discussed. In Section 4, the statistical properties for the exact and numerical solutions are studied. Section 5 presents the solution of some numerical examples of first order random differential equations using random Euler and random Runge-Kutta of the second order methods showing the convergence of the numerical solutions to the exact ones (if possible). The general conclusions are presented in the end section.