Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align} dY_t &= - Y_t\ X_t\ dW^Y_t \\ dX_t &= - \kappa X_t dt + \sigma dW^X_t \end{align}
where the brownians $W_t^Y$ and $W_t^X$ are uncorrelated. Many thanks in advance !
I am not sure I can give a reference for this particular question, but it can be done easily enough.
If the two Brownian motions are independent you can find the probability distribution for $Y_t$ conditioned on a path of $X_t$. You can write $$ \log Y_t = \log Y_0 - \int_0^t X_t dW^Y_t - \frac12 \int_0^t X_t^2\,dt. $$ The point is that $\int_0^t X_t^2\,dt = Z_t$ is just a random variable and, conditioned on a particular path of $X_t$, the first integral is an Ito integral of a deterministic function, so $$ \log Y_t = \log Y_0 - N(0, Z_t) - \frac12 Z_t.$$ So the distribution of $\log Y_t$ conditioned on the value of $Z_t=\int_0^t X_t^2\,dt$ is a normal distribution, as above, and the distributions of $X_t$ and $Z_t$ you can calculate independently of this question.
The particular $X_t$ that you wrote down is an Ornstein-Uhlenbeck process. See Wikipedia, for example, which also gives the pdfs for $X_t$, $X_t^2$, etc., from which you can find the distribution of $Z_t$.