Let $(X_t,Y_t)$ be a pair of stochastic processes such that $$ \begin{aligned} dX_t =& A_t X_t dt + C_t dW_t,\\ dY_t = & H_t X_t dt + K_tdB_t \end{aligned} $$ for some non-random matrix-valued functions $A,C,H,K$ of appropriate dimension satisfying the usual conditions of the Kalman-Bucy filter. It's clear that $X_t$ follows a (multidimensional) Ornstein-Uhlembeck process so is distributed according to this wiki post. However, what is the distribution of $Y_t$? It's not Gaussian... is it?