An application of orthogonal and biorthogonal systems to filtering problems. (English) Zbl 0923.60043

Daletskij, Yuri; Salehi, Habib; Skorokhod, Anatolij

Theory Probab. Math. Stat. 56, 175-182 (1998) and Teor. Jmovirn. Mat. Stat. 56, 169-176 (1997).

The general filtering problem is to calculate the conditional expectation of the form \(E(\Phi(\xi)/\eta),\) where \(\xi\) is a random variable in a measurable space \((X,\Xi),\) \(\eta\) is a random variable in a measurable space \((Y,\Psi),\) and \(\Phi :X\to R\) is a measurable function. The expansion for the conditional expectation is proposed. Orthogonal systems in \(X\) and \(Y\) are used for this expansion. A special biorthogonal system in \(Y\) is constructed for the representation of the conditional characteristic function of the random variable \(\xi\) if the space \(X\) is linear.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60G35	Signal detection and filtering (aspects of stochastic processes)
60G57	Random measures
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)

Keywords:

orthogonal system; biorthogonal system; filtering; conditional expectation; admissible shift; conditional characteristic function