The General Filtering Problem and the Stochastic Equation of the Optimal Filter (Part I)
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 13)
Before discussing the filtering problem, we prove a number of results in preparation for the martingale approach to the stochastic differential equation of the optimal filter which will be derived in the later sections of this chapter. Let us recall that (Ω,A,P) is a complete probability space and (ℱ t ) (t ∈ R + ) is an increasing family of sub σ-fields of A, and that it will be assumed that all P-null sets belong to ℱ0. The following processes are given on Ω: (S t ) called the signal or system process; (Z t ), the observation process; and (B t ), the noise process. All three are related by the model
$$Z_t = S_t + B_t $$
KeywordsMarkov Process Stochastic Differential Equation Innovation Process Wiener Process Stochastic Equation
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