Reverse time differentiation and smoothing formulae for a finite state Markov process. (English) Zbl 0595.60045

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 \[ \hat F_{s,t}(x)={\mathbb{E}}[F_{s,t}(x)| w_ v-w_ u,s\leq u\leq v\leq t]. \]
Reviewer: O.Enchev


60G35 Signal detection and filtering (aspects of stochastic processes)
93E11 Filtering in stochastic control theory
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93C10 Nonlinear systems in control theory
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