# 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?

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The LHS cannot be a function of $t$. Do you mean $Y(t_2)$? Likewise, $g(X,t)$ makes no sense. Do you mean $g(X(t),t)$? – Did Dec 8 '12 at 13:32
thanks for the advice, I corrected it. – the_candyman Dec 8 '12 at 14:34
Still g(X,s) is mysterious. – Did Dec 8 '12 at 14:46
@did: Well, implicitely, $g(X,s) = g(X(s), s)$. I think that it is clear from the context and I don't understand the downvote – the_candyman Dec 9 '12 at 11:06
Your identity is finally correct, this is nice. (In case your last comment implies that I downvoted your question, please note that: (1.) you have no way of knowing this, (2.) I did not downvote, (3.) I do resent very much having had to say that I did not.) – Did Dec 9 '12 at 14:50