A stochastic oscillator with time-dependent damping

We study stochastic forced oscillations of a mass-spring system with time-dependent, stochastic damping. The main purpose is to analyze the effect of the time-dependent damping. The oscillations are governed by the second-order stochastic differential equation ẍ + (a0 + α0ηWt)dotx + θ2x = T0ηWt, where x denotes the motion, Wt white noise, and a0, α0, η, θ2, T0 are constants. This equation represents a simplified model of slow-drift motions of moored offshore structures of large volume. The equation is transformed into a stochastic Volterra equation, which is solved by means of stochastic calculus and the Wick product. The solution is used to deduce probabilistic properties of the motions. It is shown that the mean value and the variance of the motion is determined by the time-averaged damping, the mass-spring characteristics and the exciting force, for large value of the time.