  The connection between forward backward doubly stochastic differential
equations and the optimal filtering problem is established without using the
Zakai's equation. The solutions of forward backward doubly stochastic
differential equations are expressed in terms of conditional law of a partially
observed Markov diffusion process. It then follows that the adjoint
time-inverse forward backward doubly stochastic differential equations governs
the evolution of the unnormalized filtering density in the optimal filtering
problem.
