Time-frequency analysis of stochastic differential equations

Most of the stochastic processes used to model physical processes are nonstationary, and yet most of the theoretical results on stochastic processes are related to the stationary case. We consider a nonstationary random process defined as the solution of a stochastic differential equation. We first transform the stochastic equation to the Wigner spectrum domain, where we obtain a deterministic differential equation. Then, by applying the Laplace transform, we obtain the exact solution of the deterministic equation. Finally, we rewrite the general solution in a form which clarifies the structure of the nonstationary stochastic process, and which highlights the connection to the classical results obtained by Fourier analysis.