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 $n$th order filter. We carry out a perturbation analysis based on a small parameter $\varepsilon$ 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 $\varepsilon$, 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.