Stochastic process integration - Notation 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?

This is a stochastic integral. In order to grasp the notion, I think it is a good idea to follow a whole course on that subject, maybe even a broader course on stochastic processes in continuous time and martingales. 
I recommend Kuo's stochastic integration textbook for an introduction specifically targetting such integrals. There's also Øksendal's Stochastic differential equations textbook. The first chapter of Kuo's book has a very nice pedagogical explanation of the heuristics of that integral.

As an extra bonus, I also noticed that Mörters and Peres' book on Brownian motion is available online on Mörters webpage at this link.
