I have a stochastic process $X_t$, and I have a function $a(x | t)$ that reflects my beliefs about the value of $X_t$ ($a$ is a density function in its first parameter). I am studying the properties of the stochastic process $Y_t = \int_0^t X_s ds$. I am thinking of using the following method to find a density function $b(y | t)$ for $Y_t$:
Let $M_n$ be a function that returns the $n^{th}$ moment of a random variable. By Fubini's Theorem, $\int_0^t M_n(X_s) ds = M_n(\int_0^t X_s ds) = M_n(Y_t)$. Since this gives me a function that spits out all moments of $Y_t$, and since a random variable is uniquely determined by its moments, this is enough information to find $Y_t$.
My questions: (1) Have I applied Fubini's Theorem correctly? I'm having trouble formalizing the proof of that first equality, but I intuitively feel that it's true. (2) Are there any other obvious flaws with this method?
Thank you.