I'm facing with a problem of notation and I hope stack could help me!
Let $X(t)$ be a time-continuous stochastic process, with pdf $p_X(x, t)$. Let $g(x, t)$ be a generic function. Now, consider the following:
$ Y(t) = \int_{0}^{t} g(X(s),s)dX(s)$ where $Y(t)$ is itself a time-continuous stochastic process. In which way I must interpret/deal with this integral?
I mean, how do I perform the integration, since integration domain is over time while I only have $dX(t)$?
I feel like I'm missing something, and most likely I must perform a "change of variable" by using the pdf $p_X(x, t)$.
Can someone bring me some light?