Stability of Ordinary Differential Equations with Colored Noise Forcing

We present a perturbation method for determining the moment stability of linear ordinary differential equations with parametric forcing by colored noise. In particular, the forcing arises from passing white noise through an nth order filter. We carry out a perturbation analysis based on a small parameter ε that gives the amplitude of the forcing. Our perturbation analysis is based on a ladder operator approach to the vector Ornstein–Uhlenbeck process. We can carry out our perturbation expansion to any order in ε, for a large class linear filters, and for quite arbitrary linear systems. As an example we apply our results to the stochastically forced Mathieu equation.