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.