Optimal filtering, interpolation and extrapolation of Markov processes with a countable number of states

Abstract

The present chapter will be concerned with a pair of random processes $\left( {\theta ,\,\xi } \right)\, = \,\left( {{\theta _t},\,{\xi _t}} \right),\,0 \leqslant \,t\, \leqslant \,T$ where the unobservable component 9 is a Markov process with a finite or countable number of states, and the observable process permits the stochastic differential $d{\xi _t}\, = \,{A_t}\left( {{\theta _t},\,\xi } \right)dt\, + \,{B_t}\left( \xi \right)d{W_t},$ where W _t is a Wiener process.