Let $X_T,Y_T$ be the terminal values of two price processes following Continuous Gaussian Motion (I.E.) let us assume no jumps. Further assume the correct forwards/futures price is given by $F^X_{t,T} = E_t[X_T],F^Y_{t,T} = E_t[Y_T]$.
We know then the following equation is True:
$cov_t(X_T,Y_T) = E_t[X_TY_T] - F_{t,T}^XF_{t,T}^Y$
I read in a recent text that it is possible to further decompose this conditional co-variance as:
[EDIT: I misread the original text] $cov_t(X_T,Y_T) = E_t[cov(X_T,Y_T|F_{exp}^X,F_{exp}^Y)]+cov_t(E[X_T|F^X_{exp}],E[Y_T|F^Y_{exp}])$
Where "exp" refers to the expiration/settlement time of the forward. Although the author does not state it explicitly, I presume $t>\text{"exp"}$
My goal is to try to determine if the claim the 2nd equation makes holds true?