Terminal filtering of stochastic processes observed on a finite time interval
Authors: Sokolov S.V. , Chub E.G. , Manin A.A. 
Published: November 2021

Annotation. Currently, the problem of evaluating stochastic processes observed under noisy conditions on a finite time interval is solved only for datasets in the form of time series using a limited number of statistical variational or spectral analysis methods, as well as various modifications of regression methods. In this case, parametric criteria are used that depend on individual parameters of the distribution density of the observed process, and not on the density itself, which significantly limits the possibilities of increasing the estimation accuracy. To solve the problem of high-precision estimation of stochastic processes on a finite time interval of their observation, an approach is proposed, firstly, providing optimal estimation according to the criterion depending on the posterior distribution density - the most informative characteristic of the observed process, and secondly, taking into account the dynamic structure of the process and the finiteness of the interval observation. A numerical example is considered to illustrate the effectiveness of the developed approach. Relevance. Currently, the problem of evaluating stochastic processes observed under noisy conditions on a finite time interval (terminal filtering problem) is solved only for datasets in the form of time series using a limited number of statistical variational or spectral analysis methods, as well as various modifications of regression methods. In this case, parametric criteria are used that depend on individual parameters of the distribution density of the observed process, and not on the density itself, which significantly limits the possibilities of increasing the estimation accuracy. Target. In this regard, for stochastic processes of a general form - described by nonlinear stochastic differential equations, it is necessary to develop a method of terminal filtering according to a criterion that takes into account the finiteness of the observation time interval and depends on the posterior distribution density - the most informative characteristic of the observed process (and not on its individual parameters). Results. The proposed solution to the problem of high-precision terminal filtering of stochastic processes - their optimal estimation over a finite observation time interval - is based on the use of a terminal criterion that depends directly on the posterior distribution density and takes into account the finiteness of the observation time interval. When describing the observed stochastic processes, their most general representation was used - nonlinear stochastic differential equations, which significantly expands the field of application of the results obtained in comparison, for example, with time series. The general solution to the problem of optimal terminal filtering is obtained using the Pontryagin maximum principle, the solution to the problem of suboptimal filtering, which significantly reduces computational costs, is based on the method of invariant immersion. Practical significance. A numerical example is considered to illustrate the effectiveness of the developed method. The proposed approach can be widely used in various fields of scientific and technical research: radio engineering, Earth sensing, satellite navigation, astronomy, seismology, geodesy, etc.