I have two random processes $X(t)$ and $Y(t)$, both are stationary processes, I know the correlation functions $R_{X}(\tau)$ and $R_{Y}(\tau)$. Further assume that $Y(t)$ and $Y(t+\tau)$ to be jointly Normal, same for $X(t)$ and $X(t+\tau)$, and assume that all have zero mean. We also know that $X(t)$ and $Y(t)$ are jointly Gaussian with known correlation coefficient $\rho$.
I am interested in knowing the correlation between $X(t)$ and $Y(t+\tau)$, is it possible to infer that from the above information? Are there some typical assumptions in this area that I can use to simplify arrive for reasonable solution.
Assuming first order regression for each of the processes would lead to an intuitive solution: $$E\{X(t)Y(t+\tau)\} = R_{Y}(\tau) \times \rho $$ One might argue another intuitive solution (with symmetric correlation function) $$E\{X(t)Y(t+\tau)\} = R_{X}(\tau) \times \rho $$
Thanks.
Your own intuitive argument shows that the information you have is not enough for finding that cross correlation.
Consider the case one with $Y(t)=\rho X(t)+Z(t)$ with an independent zero mean process $Z(t)$ and the case two with $X(t)=\rho Y(t)+Z(t)$. In both situations you can satisfy your assumptions but you have two different cross correlations.
You can also assume that $Y(t)=X(t+\delta)$ and compute the cross correlation and get another expression which cannot be obtained from your information.