# SDE with drift multiplied by telegraph like random process

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.

• I think you could get more feedback in the Cross Validated or even Finance forum, since in those fields people often have stronger background in stochastic processes than physicists. Although it will be probably also more "mathy". Commented Oct 21, 2020 at 11:10
• Consider to spell out acronyms. Commented Oct 21, 2020 at 11:21

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.
• Thank you. It is indeed correct that the process $S_t$ is not a telegraph process, I was looking to the closest analog process to give an intuition. I guess I'll have to reformulate the thing considering maybe asymmetric transition probabilities for 0->1 and 1->0. The actual formulation seems to corresponds to something like $\nu\rightarrow\infty$. In any case I'll look into the references to check whether the problem has already been solved ( the process will be a combination of telegraph processes and diffusions) Commented Oct 21, 2020 at 12:05