Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon-Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations. Keywords: evolution differential equations, evolution family of bounded operators, Radon-Nikodym density, equivalence of probability measures, generating operator, Hilbert-Schmidt operator.Applied scientific and engineering problems are known to involve differential equations with random terms or coefficients that describe the behavior of systems in random media. The solutions of such equations obviously generate probability measures in infinite-dimensional spaces.One of the major problems in the theory of random processes is to establish the sufficient conditions for the equivalence of the above-mentioned probability measures with respect to some standard well-studied measures and to explicitly determine the corresponding Radon-Nikodym densities, naturally, in terms of known quantities, and in our case, in terms of the coefficients of the equations under study, their characteristics or abstract transformations. The existence of such densities allows an efficient solution of applied problems with the use of integration over known standard distributions (for example, Gaussian, Wiener, etc.) for which algorithms are available.The publications [1-42] address the absolute continuity and equivalence of probability measures for different classes of nonlinear transformations and nonlinear differential equations with Gaussian perturbation, which were applied to calculate optimal estimates in extrapolation and filtration problems for the solution of the differential equations [7,8,12,[34][35][36][37][38][39].The present paper can be considered as a continuation of the studies in this field for nonlinear evolution differential equations in a Hilbert space H and of the studies initiated in [44] for random fields being the solutions of Dirichlet and Neumann boundary-value problems for nonlinear elliptic differential equations in a finite-dimensional Euclidean space.In contrast to the previous studies, we will consider, for the first time, nonlinear differential equations with unbounded linear operators, which are a family of generating operators for the evolution family of bounded operators. We will use the concept of extended stochastic integral and the results from [9] to establish the sufficient conditions for the equivalence of the probability measures under study and to calculate explicitly the corresponding Radon-Nikodym densities.Denote by { } W, , Ã P a fixed probability space and by H a real separable Hilbert space with scalar product ( ,...

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

A general formula for effective calculations of optimal estimates for functionals of solutions of a certain nonlinear evolution differential equations in a Hilbert space H are received. In particular, when the actual process is the solution of some nonlinear evolution differential equation with a small nonlinearity, the obtained optimal estimates expend by powers of small parameter and the coefficients of this expansion are expressed by algorithms and are calculated in the obvious form through known parameters of this differential equation.2000 Mathematics Subject Classification. 60G35.In some problems of extrapolation and filtration the random processes are estimated mostly not completely, their influence as some arguments of functional dependence, or as some kind of function or functional of these random processes in some objects is particular, especially when the observations over these processes are not complete. For instance, suppose that the actual random process is considered as a vector from a finite dimensional space and only some part of its components is observed. Naturally, the problem of optimal estimation of non-observed components of random processes or their functions or functionals are considered. These problems are the most important problems in the theory of optimal estimates of random processes and this paper is devoted to these problems. Let ¹ ; }; P º be a fixed probability space, H a separable real Hilbert space with scalar product .x; y/ and norm kxk, x; y 2 H , let } H be a -algebra of measurable subsets of the space H . In the sequel, we denote by L 2 D L 2 ¹OE0; a; H º the space of all functions defined on the segment OE0; a with values in H and which are square integrable with respect to the norm of H . It is clear that the space L 2 is a Hilbert space. The scalar product in L 2 will be denoted by .f; g/ L and the norm by kf k L ; f; g 2 L 2 . They are introduced as follows:.f; g/ L D Z a 0 .f .t /; g.t // dt and kf k 2 L D Z a 0 kf k 2 dt;.1/ where f; g 2 L 2 ; f .t /; g.t/ 2 H; 0 Ä t a.Brought to you by |

519.21The paper considers the optimal extrapolation of a random process with values from a separable Hilbert space H. Formulas for random processes with bounded second moments are derived to efficiently calculate optimal (in the sense of minimum standard deviation) estimates in problems of random process extrapolation (prediction).Finding optimal estimates in problems of extrapolation and filtration of random processes is an important problem in the theory of random processes. It is of great significance in solving many topical applied problems of science and technology. Optimal extrapolation of a random process means predicting the future value of a random process in the best way from its past values.The optimal filtering problem is formulated as follows: let a random process be observed. It contains or combines a useful signal (first process) and random noise (second process). It is required to separate (filter) the noise from the signal and find the best (in a sense) approximation for the signal.The solution of the problem of linear extrapolation and filtration given earlier by A. N. Kolmogorov [10,11] and N. Wiener [1, 2] is optimal only for Gaussian processes, and is optimal for general processes only in the class of linear estimates and may appear not to be the best. Therefore, it is of theoretical and practical importance to develop methods for efficient solution of such problems. N. Wiener [1, 2], L. A. Zadeh [9], V. S. Mikhalevich [15-17], R. L. Stratonovich [23, 24], A. N. Shiryaev, R. Sh. Liptser, B. I. Grigelionis [12-14, 41-44, 6], et al. conducted studies in this field. However, the results they have obtained pertain mainly to different classes of Markov processes.The theory of linear extrapolation and filtering is completely developed, at least theoretically, by A. N. Kolmogorov, N. Wiener, A. M. Yaglom [1,2,10,11,[45][46][47][48] and many other authors. These results are presented in detail in the monograph by Yu. A. Rozanov [18]. As indicated above, linear extrapolation and filtering results in best estimates in the case of Gaussian processes. A. D. Shatashvili [33][34][35][36][37] showed for some class of random processes that if processes slightly differ from Gaussian ones, the order of deviation of the best estimates from linear ones is equal to the order of deviation of the process under study from the Gaussian one.It is natural that the necessity of considering all finite-dimensional distributions of processes under study substantially complicates the generalization of problems of linear extrapolation and filtering of non-Gaussian random processes. Therefore, problems of optimal estimation can efficiently be solved only if the information on all finite-dimensional distributions of the random process is obtained in closed form.From an abstract standpoint, the optimal solution to problems of extrapolation and filtration of random processes is rather simple and can be expressed using integration in a function space with respect to a measure depending on a functional 434

No abstract