Non-Stationary Stochastic Processes Estimation: Vector Stationary Increments, Periodically Stationary Multi-Seasonal Increments

The problem of forecasting future values of economic and physical processes, the problem of restoring lost information, cleaning signals or other data observations from noise, is magnified in an information-laden word. Methods of stochastic processes estimation depend on two main factors.

The first factor is construction of a model of the process being investigated.

The second factor is the available information about the structure of the process under consideration. In this book, we propose results of the investigation of the problem of mean square optimal estimation (extrapolation, interpolation, and filtering) of linear functionals

depending on unobserved values of stochastic sequences and processes

with periodically stationary and long memory multiplicative seasonal increments.

Formulas for calculating the mean square errors and the spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where

spectral structure of the considered sequences and processes are exactly known.

In the case where spectral densities of the sequences and processes are not known exactly while some sets of admissible spectral densities are given, we apply the minimax-robust method of estimation.

1. Dr Maksym Luz is a Head of Actuary & Chief Risk Officer at BNP Paribas Cardif in Ukraine.

He is an author/co-author of more than 20 papers including the book ’’Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences’’, Wiley - ISTE, 2019.

2. Prof. Dr. Mikhail Moklyachuk is a Professor at the Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Ukraine.

He is author/co-author of more than 200 papers and 14 books, including ’’Robust estimates for functionals of stochastic processes’’, Kyiv University, 2008; ’’Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences’’, Wiley - ISTE,2019; Estimation of Stochastic Processes with Missing Observations’’, Nova Science Publishers, 2019; ’’Estimates of Periodically Correlated Isotropic Random Fields’’, Nova Science Publishers, 2018; ’’Convex Optimization: Introductory Course’’, ISTE-Wiley, 2020; ’’Stochastic Processes: Fundamentals and Emerging Applications} (Editor)’’, Nova Science Publishers, 2023.

Professor Moklyachuk was elected an academician of the Academy of Higher School of Ukraine (2016). He has received the State Prize of Ukraine in the field of education (2012), Taras Shevchenko prize (Kyiv University best textbook award, 1999) for the textbook ``Variational Calculus. Extremum Problems’’. He is Editor-in-Chief, journal ＂Bulletin of the Taras Shevchenko National University of Kyiv. Series: physics and mathematics＂, member of editorial board, journals ＂Statistics, Optimization and Information Computing＂, ＂Stochastic Modeling and Applications＂.