SDE with drift multiplied by telegraph like random process

Question

Let the stochastic process $\{X_t\}$ be defined by the following SDE (Ito's convention for discretization)

$dX_t=\frac{1}{p}S_tg(X_t)dt+\sqrt{2}dW_t$

where $W_t$ is a standard Wiener process, $g: \mathrm{R}\rightarrow \mathrm{R}$ is a $C^{\infty}$ function and $S_t$ is an other (white) stochastic process described by

\begin{align} &p(S_t=1)=p,\quad p(S_t=0)=1-p\\ & S_t\perp S_\tau, \quad \forall t\neq\tau \end{align} where $p\in(0,1]$

Informally you could think of $S_t$ as a telegraph process https://en.wikipedia.org/wiki/Telegraph_process

My questions are the following:

1. Does it make sense to define a process like $S_t$?

2. If 1) has positive answer, can I use the infinitesimal generator formalism (https://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes)) to derive, for example the Fokker Planck equations for the process $X_t$?

If you have link to similar problems solved in any physics area it would be awesome. Forgive me for being formally imprecise, I am not trained in pure math but I do use SDEs simulation schemes for engineering/physics problems.

Answer 1

I am not an authority in the subject, so the following remarks are based on my limited experience:

1. Telegraph process is usually thought of as a switching process, which changes its value between zero and one. To define such a process one needs to introduce a probability of a switching event at time $t$, given that the previous event happened at $t'$. E.g., $$ P(t|t') = e^{-\nu (t-t')}, $$ in which case the statistics of the switching events is a Poisson process. In other words, I think that your definition of the process is insufficient.
2. Telegraph process is a Markov process. However it is a jump rather than a diffusion process, and as such it is described by a somewhat more general version of Kolmogorov (=Fokker-Planck) equations, called sometimes Kolmogorov-Feller equations. These are more complex than the FPE, but for the simple versions of a telegraph process some exact results can be derived.

My own knowledge about telegraph process and the associated results traces back to the course on Functional methods for stochastic processes for physicists, based on the book by Klyatskin (in Russian). A google search showed me that there are several of his books published in English with somewhat different titles, and I am not sure which one corresponds to the one that I studied from: see here and here.