Stochastic Equations of Nonlinear Filtering of Markovian Jump Processes

Let $(\theta_t,\eta_t)$ be a Markov process where $\theta_t$ is a non-observable component which is a Markovian jump process, and $\eta_t$ is the observable component satisfying the equation
$$d\eta_t=A(\theta_t,\eta_t,t)dt+B(\eta_t,t)dW_t,\,\eta_0=0$$
. This paper derives stochastic equations which the a posteriori probabilities $\pi_t(\mathfrak A)=\mathbf P\{\theta_t\in\mathfrak A/\eta(\tau),\,\tau\leq t\}$ satisfy [see Eq. (4)] and which are sufficient statistics in various problems in nonlinear filtering, extrapolation, in optimal control problems, pattern recognition, etc.