Some basic questions about Stochastic Calculus I have a transition function for a Markov process $X_t$.  I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$.  Some questions about this:


Is this the same as the transition function for the stochastic process $Y'_t := \int_0^t 1 \,dX_s$?  If not, what is the difference in real-world meaning between $Y_t$ and $Y'_t$?
In normal calculus, one typically integrates by invoking the Fundamental Theorem and taking antiderivatives.  Is there a Fundamental Theorem analog in stochastic calculus that might be useful here?
My best lead so far is to use the Feynman-Kac formula on $X_t$ to find the characteristic function of the density function of $Y_t$, then invert it.  The transition function for $X_t$ is very long and ugly, so this would be hugely complicated to execute.  I've got Mathematica as my integration calculator, so I can handle some amount of ugliness, but this strategy was too complicated for Mathematica to stomach, and it froze up.  Is there a known simpler method?


Thanks - advice on any of these three questions is appreciated.
For (1), since $dX_s \equiv X_{s+ds}-X_s$, $Y'_t = \int_0^t dX_s = X_t - X_0$.
For (1), since $dX_s \equiv X_{s+ds}-X_s$, $Y'_t = \int_0^t dX_s = X_t - X_0$.