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By Silviu Guiasu, M. Malitza

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1 Introduction This chapter first considers a discrete-time, finite-state Markov chain which is observed through a real- or vector-valued function whose values are corrupted by noise. For simplicity, we assume Gaussian noise. The main tool is again a version in discrete time of Girsanov’s theorem. d. random variables. Working under P we obtain unnormalized, recursive estimators and smoothers for the state of the Markov chain, given the observations. Furthermore, recursive estimators and smoothers are derived for the number of jumps of the chain from one state to another, for the occupation time in any state, and for processes related to the observation process.

6) Then E V˜k+1 | Fk , Yk+1 = E [ Xk+1 | Fk , Yk+1 ] − A˜ XkYk+1 = A˜ XkYk+1 − A˜ XkYk+1 = 0. In summary then, we have the following. 8). The entries of A˜ satisfy 46 2 Discrete States and Discrete Observations N ∑ α jir = 1, α jir ≥ 0. 8) j=1 V˜k is a martingale increment satisfying E V˜k+1 | Fk , Yk+1 = 0. Next, we derive filters and smoothers for various processes. d. random variables, uniformly distributed over the set of standard unit vectors { f1 , . . , fM } of RM . 3. Using Bayes’ Theorem we see that P [ Xk+1 = e j | Fk , Yk+1 ] = E = = E Xk+1 , e j | Fk , Yk+1 Xk+1 , e j Λk+1 | Fk , Yk+1 E [ Λk+1 | Gk , Yk+1 ] Λk+1 E Xk+1 , e j | Fk , Yk+1 Λk+1 = P [ Xk+1 = e j | Fk , Yk+1 ] = P [ Xk+1 = e j | Xk ,Yk+1 ] .

So E [Vk+1 | Gk , wk+1 ] = E [Vk+1 | Fk ] = 0. Consequently, E [Vk+1 | Yk+1 ] = E [E [Vk+1 | Gk , wk+1 ] Yk+1 ] = 0 and Xˆk+1 = E [ Xk+1 | Yk+1 ] = E [ AXk +Vk+1 | Yk+1 ] = AE [ Xk | Yk+1 ] . 1) the result follows. 2) is that it is not linear in Xˆk .

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