Reverse time differentiation and smoothing formulae for a finite state Markov process (Q1077814)

First a Markov process \(X_{s,t}(x)\), \(0\leq s\leq t,\) \(x\in S\) \(=\) the state space of X (assumed to be finite), is appropriately defined, with its definition involving another Poisson point process. Then \(X_{s,t}(x)\) is interpreted as a signal and the following observation process model is considered: \[ dy_ t=h(X_{s,t}(x),t)dt+\alpha (t)dw_ t \] (h is bounded, \(w\amalg X\) and \(\alpha\) is measurable with bounded inverse). The central role of Bayes' rule for filtering and smoothing is played by the quantity \(F_{s,t}(x)=f(X_{s,t}(x))\Lambda_{s,t}(x),\) where \[ \Lambda_{s,t}(x)=\exp \{\int^{t}_{s}h(X_{s,u}(x),u)\alpha^{- 1}(u)dy_ u-\int^{t}_{s}h(X_{s,u}(x),u)^ 2\alpha^{-2}(u)du\}. \] The main result is the derivation of the reverse-time stochastic differential equation for \[ \hat F_{s,t}(x)={\mathbb{E}}[F_{s,t}(x)| w_ v-w_ u,s\leq u\leq v\leq t]. \]