Asymptotic Approximation of the Solution of a Random Boundary Value Problem Containing Small White Noise

Abstract

This paper considers a nonlinear random differential equation [formula] where α(ω) is F1-measurable and w is an Rm-valued Wiener process. By introducing a weak problem, the shooting method can be used to prove the uniqueness of the Rn-valued Ft-measurable solution x(t) in the meaning of large probability. If the parameter ϵ is small, then x(t) = x0(t) + ϵx1(t) + O(ϵ2), where x0(t) is the solution with ϵ = 0 and x1(t) satisfies a linear random boundary value problem. For simplicity the discussion is given in the scalar case, but extensions to higher dimensions are readily apparent.