Numerical solution of systems of random differential equations with Gaussian statistics (Q761037)

This paper presents a numerical method for approximating the mean and covariances of the solution of the random system \(X'(t)=F(t,X(t),A(t)),\) \(X(0)=X_ 0\) where A(t) is a known Gaussian vector process and \(X_ 0\) is a known Gaussian vector. The method involves expanding the components of the approximate \(X(t_{i+1})\) generated by a (deterministic) one-step method in quadratic random Taylor series of the components of \(X(t_ i)\). The Taylor series are then averaged to approximate the means of the components of \(X(t_{i+1})\) and multiplied by each other and averaged to approximate the covariances of the components of \(X(t_{i+1})\). A theorem is given to establish convergence when X(t) is Gaussian and F is quadratic in X(t) and A(t). Three examples with numerical results are given to demonstrate the effectiveness of the method.