I have a bunch of covariance expressions for two correlated RVs like $\mathrm{Cov}(X',Y')=E[X'Y']$, where $X'=X-E[X]$, $E[X]$ is the mean of $X$ and $X'$ is the fluctuation. I do not have any given densities associated with these RVs, I only know the nature of correlation either positive or negative. Now I want to derive the dynamical equations for covariance terms: i.e., I want to derive $\frac{\partial E[X'Y']}{\partial t}$. How will I do that?
I thought to use the Fokker-Planck equation, and for a try, I made assumptions as follows:
$X'$, $Y'$ have linear dynamics, i.e., $\frac{dX'}{dt}=k_X$ and $\frac{dY'}{dt}=k_Y$; where $k_Y$ and $k_X$ are some constants.
The diffusion coefficients $D_A$ and $D_B$ are constant. The FP equation then becomes
$$\frac{\partial P}{\partial t}=-k_X\frac{\partial(X'P)}{\partial X'}-k_Y\frac{\partial(Y'P)}{\partial Y'}+D_X\frac{\partial^2(P)}{\partial X'^2}+D_Y\frac{\partial^2(P)}{\partial Y'^2}$$ We are interested in $\frac{dE[X'Y']}{dt}$ and we know $E[X'Y']=\mathrm{Cov}(X',Y')$. Expressing $E[X'Y']$ in terms of joint probability:
$$E[X'Y']=\int X'Y'~P(X',Y',t)~dX'dY'$$ Then differentiating
$$\frac{d}{dt}E[X'Y']=\frac{d}{dt}\int X'Y'~P(X',Y',t)~dX'dY'$$
Then plugging the FP equation in this integral above.
$$\frac{d}{dt}E[X'Y']=\int X'Y'~[-k_X\frac{\partial(X'P)}{\partial X'}-k_Y\frac{\partial(Y'P)}{\partial Y'}+D_X\frac{\partial^2(P)}{\partial X'^2}+D_Y\frac{\partial^2(P)}{\partial Y'^2}]~dX'dY'$$
(assuming $\frac{d}{dt}\int A'B'PdA'dB'=\int A'B'\frac{d}{dt}P dA'dB'$ I'm not sure if it is actually allowed)
The issue is that I do not have a particular form of $P(X',Y',t)$, I only know the qualitative correlation between $X'$ and $Y'$. I want to ask whether this is the right approach? Or any other clues would be helpful.