I have a system of two (one-dimensional, Ito) stochastic differential equations, one describing the evolution of $X_t$ and the other the evolution of $Y_t$.
The two SDEs are coupled through the noise, in the sense that the equation for $X_t$ does not depend on $Y_t$, and vice versa, but the Brownian motion driving the two SDEs is the same.
A toy example is the following:
$d X_t = - a X_t dt + \sigma d W_t \\ d Y_t = \mu Y_t dt + \eta Y_t d W_t $
Both SDEs can be solved analytically. In fact, it turns out that (for any time $t$) $X_t$ is normally distributed, and $Y_t$ is log-normally distributed.
The question is: what is their joint density?