Nonlinear interpolation of components of diffusion Markov processes

A diffusion Markov process defined by the Ito equations (3) is considered. For the a posteriori probability densities $\pi_{\alpha\beta}(t,\tau)$, $\pi_\alpha(t,\tau)$, $0\le t\le\tau\le T$ defined in (2), differential equations in $\tau$ are deduced (see (21) and (13)). In §2 for the coefficients (31), it is shown that $\pi_\alpha(t,\tau)$ and $\pi_{\alpha\beta}(t,\tau)$ are Gaussian densities in $\alpha$ with parameters defined by (37), (38) and (65), (66).