On a method for an effective calculation of optimal estimates in problems of a filtration of random processes for certain nonlinear evolution differential equations in a Hilbert space. Part 1

Abstract

In the first part of this paper it is received the closed general expression which can be considered as algorithm for obvious calculation of an optimum estimation in problems of a filtration of random processes accepting the values in abstract Hilbert spaces. Using a condition of existence of density of Radon-Nikodym's measure generated by random process concerning some Gaussian measure (the optimum filter) it is shown that calculation of conditional mathematical expectation is reduced to calculation of an unconditional population mean on the base of obvious expressions.In this paper, we continue the studies, initiated in [1, 2], with a focus on optimal filtering problems. Let us give a general setting of the problem considered here. Consider a random process in a Hilbert space H, simultaneously containing two processes, an unobservable process, which is the one we are interested in and is called a "useful signal", and the second process, called the "random noise", superposed with the "useful signal" distorting or completely contained in it. The optimal filtering problem consists in the following: using observable values of the process under consideration of any of its components, to separate the "random noise" and the "useful signal", and to construct, in some sense, the best approximation for the unobserved process, i.e., the "useful signal", possibly with time anticipation.The paper consists of two parts. In the first part, we will consider a general setting of the problem in an abstract Hilbert space H and obtain general formulas and algorithms for an explicit calculation of optimal estimates in filtration problems.In the second part, the obtained algorithms will be used for finding, in an explicit form, optimal estimates in the filtration problems for systems of nonlinear evolution differential equations perturbed with Gaussian random processes in a Hilbert space H. In the case where these equations contain a small nonlinearity, the obtained estimates are expanded in powers of the small parameter at the nonlinear term; the principal term Brought to you by | HEC Bibliotheque Maryriam ET J. Authenticated Download Date | 6/8/15 5:30 AM

On a method for an effective calculation of optimal estimates in problems of filtration of random processes for certain nonlinear evolution differential equations in Hilbert space. Part II

“…Denote by kxk i and .x; y/ i , x; y 2 H i , the norm and the scalar product correspondingly in a Hilbert space H i , i D 1; 2. As we have already mentioned in the first part of this paper (see [2]), the general formulation of the problem of optimal filtration of random process looks like the following problem: let in a certain Hilbert space H a certain random process z.t/ 2 H be considered containing simultaneously two random processes, for instance y 1 .t/ and y 2 .t/, i.e. z.t/ D z.y 1 .t /; y 2 .t//.…”

“…We will use Theorem 1 and the results obtained in the first part of this paper (see [2]) for the solution of this problem. The main conditions of these results are the condition of the equivalency of measures y 1 y 2 and x 1 x 2 , y 1 y 2…”

“…/ in order to use the results of Theorem 2 and the algorithms obtained in the first part of our paper (see [2]). Then we solve the problem of the optimal filtration in the obvious form using its estimate for the solution z.t/ of the nonlinear evolutionary differential equation (1).…”

“…in the form of formula (35) and conditions for its existence we can apply it for the problem of optimal filtration putting it in the general formula of the algorithm of the optimal estimate obtaining in the first part of this paper (see [2]). For this reason we should identify a little the expression for the density (35) and write it in a more convenient form for the application of it for the calculation of an optimal estimate in the filtration problem.…”

“…In particular, we will consider these problems for the functionals of random processes-solutions of some evolutionary differential equations in the Hilbert space H , and also their expansion by the power of small parameter˛at nonlinear summand (see [2]). The obtained Brought to you by | Universitaet Giessen Authenticated Download Date | 5/31/15 2:46 PM results will give the completed picture of our investigations in the problem of optimal estimates of random processes-solutions of nonlinear evolutionary differential equations and they generalize significantly all early known results in this direction and expend the region of their application.…”