Suppose we are given a stochastic process $X: [0, T] \to \mathbb{R}^D$ where $(X(0), X(T)) \in \mathbb{R}^{2D}$ is normally distributed with mean $\mu$ and covariance $\Sigma$. Suppose we are also given that conditionally on $(X(0), X(T))$, $X$ evolves according to the SDE: $$dX_t = f(X_t, t; X_T) dt + L dW_t$$ where $X_T \equiv X(T)$, $W_t$ is a standard $D$-dimensional Brownian motion, and $L \in \mathbb{R}^{D\times D}$.
My question is whether we can say the unconditional process (integrating out the dependence on $(X(0), X(T))$) evolves according to some SDE as well, and if so, what the drift term of this SDE would be. It would be great, for instance, if the drift of the SDE describing the unconditional process was: $$\tilde{f}(x, t) = \mathbb{E}[f(x, t; X_T)]$$
A proof or references would be appreciated. Feel free to assume smoothness or nice conditions on $f$. Please let me know if anything is unclear or if my terminology is off, as I am new to this area.