Assume that $X_t$ and $Y_t$ are two $\mathbb{R}^d$ valued solutions to the following linear SDE (for some initial condition) $$ dX_t = A\circ X_t dt + \sigma dB_t $$
$$ dY_t = B\circ Y_t dt + \sigma dB'_t $$
where $B_t, B'_t$ are standard Brownian motions on $\mathbb{R}^d$ and $\sigma$ is diagonal but allowed to have $0$ entries in the diagonal. There is a paper by Duncan titled "Mutual Information for Stochastic Differential Equations" which says that the mutual information between $X_s$ and $Y_s$ up to $s \leq t$ should be (he does it in a more general form where coefficients of both SDEs can depend on the both variables $X_t$ and $Y_t$) $$ I(X,Y) = E(\int_0^t [(AX_s - \hat{AX}_s)^2 - (AY_s - \hat{AY}_s)^2] ds), $$ where for instance $$ \hat{AX}_s = E[AX_s |_{\mathcal{F}_{X_s}}]. $$ Here $\mathcal{F}_{X_s}$ is the filtration adapted to $X_s$ and since $AX_s$ is Martingales and linear image of Martingales is Martingales we get $ \hat{AX}_s = AX_s$. So $I(X_t,Y_t)=0$. But this is definitely not true for all solutions of linear SDEs that is there are those mutual information is non-zero. For instance if one takes $Y_t = X_{t +\tau}$ for some fixed $\tau$ the solutions are correlated to each other so I would not expect the mutual information to be $0$. So I do not quite understand what I am doing wrong here. One difference of the paper from the setting above is he assume only standard Brownian motion with no non-degeneracy. But even in that case it seems wrong that mutual information is always $0$ for SDEs as above. So
1- Where am I doing a mistake in using the result of Duncan on linear SDEs?
2- Assume you have two Langevin equations for the variables $X_t$,$Y_t$ with linear force terms (which looks like the equations above). Since the solution to the SDE is known is it possible to compute the probability densities $p_X, p_Y$ and the joint probability density $p_{XY}$ starting from the solutions for $X_t$ and $Y_t$. The Fokker-Plank equations for $p_X$ and $p_Y$ still look complicated so I guess that does not give one explicit solutions. I am asking this because if one knows the explicit forms of the densities above one can also compute $I(X_t,Y_t)$ directly rather then using Duncan's theorem.