Is this a valid method for time-integrating a stochastic process? 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.
 A: Here is an illustration of why knowing $a(x|t)$ is not enough to estimate $b(y|t)$.  I'm using discrete time processes $X_n$ and $Y_n = \sum_{k=1}^n X_k$ to simplify notation, but the same idea applies in the continuous setting.
Consider the following two processes:


*

*$X_n^{(1)} = \xi$ where $P(\xi=-1) = P(\xi=+1) = 1/2$, i.e. a random decision is made at the start, and after this the process has constant paths.

*$X_n^{(2)} \in \{-1, +1\}$ i.i.d., with the same distribution as $\xi$, i.e.
$P\bigl(X_n^{(2)} = -1\bigr) = P\bigl(X_n^{(2)} = +1\bigr) = 1/2$.  Here, each of the values is picked randomly, independent of the others.
Both processes are Markov chains, and they have the same marginals:
$$P\bigl(X_n^{(k)} = -1\bigr) = P\bigl(X_n^{(k)} = +1\bigr) = 1/2$$
for all $n$ and $k \in \{1,2\}$.  Thus your "beliefs about the value of $X_n$" are likely to be the same for both processes, leading to $a^{(1)}(x|t) = a^{(2)}(x|t)$.
The processes have very different $Y_n$:  For process $X_n^{(1)}$, all the added values are the same, so we have $Y^{(1)}_n = n \xi$.  The process $Y^{(1)}_n$ moves with constant speed 1, either up or down.  In constrast, the terms in the sum for $Y^{(2)}_n$ can cancel.  $Y^{(2)}_n$ is a random walk, and the magnitude of $Y^{(2)}_n$ only grows like $\sqrt{n}$.
Thus we have a situation where two processes have the same $a$ but very different $b$.  This shows that knowing $a$ is not enough to find $b$, even if you manage to work around the problem with the application of Fubini's theorem.  You will need some information about the trajectories of $X$ rather than just the marginals for fixed times.
