Some necessary and sufficient conditions of the equivalence of two Gaussian measures induced by solutions of differential equations in an Euclid and a Hilbert spaces

Fomina, T. A.; Shatashvili, A. D.

doi:10.1515/156939703771891879

Cited by 4 publications

(8 citation statements)

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“…0 , is a Hilbert-Schmidt operator acting in H. As has been already noted, it is shown in [3][4][5][6][7][8] that there is a relation between solutions y t ( ) and x t ( ) to differential equations (4) and (5), respectively, in the form of the following nonlinear mappings:…”

Section: The Nonlinear Function F T Y T ( ( )) Defined On [ ]mentioning

confidence: 99%

“…Such equations were investigated by one of the authors together with T. A. Fomina in [3][4][5][6][7][8] in which conditions of existence and uniqueness of solutions to such equations and equivalence of measures generated by these solutions under Gaussian perturbations are found and also, which is most important, conditions for the existence of their Radon-Nikodim densities and an explicit expression for the densities in terms of coefficients of equations being considered are obtained. Hence, the results of Theorem 1 from [1] can be efficiently applied to solutions of evolutionary differential equations of the form (1).…”

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confidence: 99%

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A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. II

Fomin-Shatashvili¹,

Shatashvili

2008

Cybern Syst Anal

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519.21The investigation pursued in the previous article is continued. Using a general algorithm of calculating the optimum prediction for a random process, an optimum extrapolation estimate is found in explicit form for the decision of a nonlinear evolutionary differential equation in a Hilbert space with unbounded linear operators. If a differential equation contains a small nonlinearity, then such an estimate is developed as a series in powers of a small parameter.This article is the continuation of the authors' investigations originated in [1]. We use the results obtained in Theorem 1 in computing the optimal prediction for the solution of a nonlinear evolutionary differential equation under a Gaussian perturbation in the right side of the form dy t dt A t y t A t y t f t y t

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Section: The Nonlinear Function F T Y T ( ( )) Defined On [ ]mentioning

confidence: 99%

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confidence: 99%

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. II

Fomin-Shatashvili¹,

Shatashvili

2008

Cybern Syst Anal

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“…As was shown in [26][27][28][29][30], if some sufficient conditions are observed, the solutions y t ( ) and x t ( ) of Eqs. (3) and (4) can be related by the nonlinear mappings…”

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confidence: 90%

“…One of the authors of the present paper investigated systems (3) and (4). Together with T. A. Fomina [25][26][27][28][29][30], he obtained sufficient conditions for the equivalence of the measures m y and m x and computed the Radon-Nikodym densities in explicit form in terms of the coefficients of Eqs. (3) and (4) with the use of the extended stochastic integral introduced by A. V. Skorokhod [19].…”

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A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

Fomin-Shatashvili¹,

Shatashvili

2008

Cybern Syst Anal

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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

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Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part I

2011

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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 ( ,...

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Some necessary and sufficient conditions of the equivalence of two Gaussian measures induced by solutions of differential equations in an Euclid and a Hilbert spaces

Cited by 4 publications

References 5 publications

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. II

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. II

A method for efficient computation of optimal estimates in the extrapolation of solutions of nonlinear evolutionary differential equations in a Hilbert space. I

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space, disturbed by Gaussian processes. Part I

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