Consider the diffusion process $$ d X = \mu(X, t) dt + \sigma(X, t) dY. $$ When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general case where $Y$ is an independent and stationary process such as the Poisson process or a weighted sum of a Brownian motion and a Poisson process. How can we derive the associated Fokker-Planck equation? Thanks!
See, for example, Generalized Fokker-Planck equation: Derivation and exact solutions
We derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Lévy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises.