On conditional distributions of diffusion processes

For a two-component diffusion process $(x,y)$ on the Euclidean space $R^n$ ($n\geqslant2$), we consider the question of the existence of the density $\pi_{t,s}$ of the distribution $P(x_t\in\nobreak\cdot\,|\,y_\tau,\ \tau\leqslant s)$, $s\leqslant t$, with respect to Lebesgue measure, and we study its analytic properties. We also consider the question of the existence and uniqueness of the solution of the equation for $\pi_{t,t}$ (the filtering equation).
Bibliography: 18 titles.