Optimal Estimates in Problems of Extrapolation, Filtration, and Interpolation of Functionals of Random Processes with Va
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OPTIMAL ESTIMATES IN PROBLEMS OF EXTRAPOLATION, FILTRATION, AND INTERPOLATION OF FUNCTIONALS OF RANDOM PROCESSES WITH VALUES IN A HILBERT SPACE A. D. Shatashvili,1† I. Sh. Didmanidze,1‡ T. A. Fomina,1† and A. A. Fomin-Shatashvili1†
UDC 519.21
Abstract. General formulas are obtained for efficient calculation of optimal estimates for functionals of random processes with values in a Hilbert space. In a special case where the process under study is a solution of a nonlinear evolutionary differential equation with a small nonlinearity, the optimal estimates are expanded in powers of a small parameter and the expansion coefficients are given in the form of algorithms and are calculated explicitly in terms of known quantities of the differential equation. Keywords: algorithm, evolutionary differential equation, Radon–Nikodym density, extended stochastic integral, equivalence of measures, functional.
Problems of extrapolation and filtration of random processes in the analysis are most often estimated not completely but partially. For example, suppose that a random process is considered as a vector from a finite-dimensional space and some part of its components is only observed. Naturally, the problem of finding optimal estimates (both in extrapolation and in filtration problems) of the components of the considered random process that were not observed or of their functions and functionals is formulated. These problems are the most important ones in the theory of optimal estimates of random processes and the present study is devoted to them. Let {W, Ã, P } be a fixed probability space, H be a separable real Hilbert space with scalar product ( x, y) and norm || x || , x, y Î H , and à be s-algebra of measurable subsets of space H . Below, L2 = L2 {[0; a ], H } denotes the space of functions defined on the interval [0; a ] with values in H and square integrable with the norm H . In the Hilbert space L2 , let us introduce scalar product ( f , g ) L and norm || f || L , f , g Î L2 , as a
( f , g ) L = ò ( f ( t ), g ( t )) dt , 0
a
|| f || 2L = ò || f ( t ) || 2 dt , 0
where f , g ÎL2 , f ( t ), g ( t ) Î H . 1
Batumi Shota Rustaveli State University, Batumi, Georgia, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2018, pp. 79–92. Original article submitted October 26, 2017. 754
1060-0396/18/5405-0754 ©2018 Springer Science+Business Media, LLC
Let B ( t, s) be an operator function acting in space H for each t, s Î[0 ; a ]. Denote by | B ( t, s)| the norm of operator function in space H . It is generally known that operator function B ( t, s) , as a kernel, generates the integral operator B in space L2 : a
( Bj ) t = ò B ( t, s) j ( s) ds, j ÎL2 . 0
In space L2 , denote the norm of operator B by | B | L : aa
| B | 2L = ò ò | B ( t, s) | 2 dtds < ¥ .
(1)
00
The operator norm defined by formula (1) is called the Hilbert–Schmidt norm, and operator B that has such norm is called a Hilbert–Schmidt operator. Let us consider random process x ( t )
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