How to calculate the PSD of a stochastic process

Question

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô integrals (and maybe not).

In any of these cases, how can we compute the power spectral density of the process?

For instance, for the Ornstein-Uhlenbeck process, the steps would be

1. Find autocorrelation function. This is easily done with the Itô isometry and the properties of the Wiener process.
2. Compute Fourier transform

And that's it, because the Wiener-Khinchin theorem assures us that the process autocorrelation and the PSD are a Fourier transform pair as long as the process is wide sense stationary (and the Ornstein-Uhlenbeck process satisfies this). However, The Ornstein-Uhlenbeck process is too nice in reality.

What can be done to obtain the PSD in a more general, non stationary case? What can be then said about the autocorrelation function relation to the PSD? Could you provide any references with theory/worked examples?

For instance, I would really appreciate your help in the not-so-nice (but still nice) second order harmonic oscillator with additive noise term $$ \ddot{x} + \gamma\dot{x} + \Omega^2x = \sigma \frac{\mathrm{d}W_t}{\mathrm{d}t} $$ (forgive my abuse of notation). I have seen physicists doing awful things with this stochastic process, and I'm in need of some (mathematical) cleanliness and mental peace.

Answer 1

You ask for the spectral analysis of a nonstationary stochastic process. Because the autocorrelation function $C(s,t)$ now depends on the two times $s$ and $t$ separately, and not only on their difference, the power spectral density $P(\omega,\omega')$ will depend on two frequencies, and not just on a single frequency. Alternatively, one can define a power spectrum $P_t(\omega)$ that depends on one frequency and one time variable, and has the interpretation of a time dependent power spectral density. This makes sense if the nonstationary characteristics change sufficiently slowly in time (in a way that can be made precise).

This approach was developed by M.B. Priestley, formally in Evolutionary Spectra and Non-Stationary Processes, with applications in Power spectral analysis of non-stationary random processes.