I have been studying the fascinating subject of stochastic processes and have constructed various equations I can think of and then I try to look up literature for that process.
One process I have thought of is this:
$$dS = f(s,t)dt + \alpha N(t)\,dt$$
Here $f(s,t)$ is a deterministic function and $N(t)$ is a simple random walk consisting of step functions representing the count of all increment/decrement events upto $t$. Is this a standard class of processes that I can read up on?
The closest thing I have been able to find are jump-diffusion-drift models but these are not quite what I have, as my process involves the time integration of the stochastic process $N(t)$ which is in continuous time but has discrete state space.