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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.

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  • $\begingroup$ 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". $\endgroup$
    – Roger V.
    Oct 21, 2020 at 11:10
  • $\begingroup$ Consider to spell out acronyms. $\endgroup$
    – Qmechanic
    Oct 21, 2020 at 11:21

1 Answer 1

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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.

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  • $\begingroup$ 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) $\endgroup$
    – ernst
    Oct 21, 2020 at 12:05
  • $\begingroup$ I have seen recently something very similar treated with Monte Carlo (but their diffusion part is more complex): ki.tu-berlin.de//fileadmin/fg135/publikationen/… $\endgroup$
    – Roger V.
    Oct 21, 2020 at 12:17
  • $\begingroup$ it seems useful. I was actually trying to go in the opposite direction: from discrete time update equations that in the infinitesimal spacing limit converge to the mentioned SDE and then obtain the stationary distributions with the Kolmogorov equations $\endgroup$
    – ernst
    Oct 21, 2020 at 13:04

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