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

Abstract

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