ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF A RANDOM BOUNDARY-VALUE PROBLEM CONTAINING SMALL WHITE-NOISE

This paper considers a nonlinear random differential equation [formula presented] 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{lunate} is small, then x(t) = x0(t) + ε{lunate}x1(t) + O(ε{lunate}2), where x0(t) is the solution with ε{lunate} = 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. © 1993 Academic Press. Inc. All rights reserved.