Minimax Interpolation Problem for Random Processes with Stationary Increments

Abstract

The problem of mean-square optimal estimation of the linear functional A T ξ = ∫ T 0 a(t)ξ(t)dt that depends on the unknown values of a continuous time random process ξ(t), t ∈ R, with stationary nth increments from observations of the process ξ(t) at time points t ∈ R \ [0; T ] is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formulas for calculation the value of the mean-square error and spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty where spectral density of the process is exactly known. In the case of spectral uncertainty where spectral density of the process is not exactly known, but a class of admissible spectral densities is given, relations that determine the least favourable spectral density and the minimax spectral characteristic are specified.Keywords Random process with stationary increments, minimax-robust estimate, mean-square error, least favourable spectral density, minimax spectral characteristic AMS 2010 subject classifications Primary: 60G10, 60G25, 60G35, Secondary: 62M20, 93E10, 93E11

Interpolation Problem for Multidimensional Stationary Processes with Missing Observations

“…is the indicator function of the set D = D F × D G . A solution of the problem (18) is characterized by the condition…”

“…This condition makes it possible to find the least favourable spectral densities in some special classes of spectral densities D (see books by Ioffe and Tihomirov [13], Pshenichnyj [35], Rockafellar [36]). Note, that the form of the functional ∆(h(F 0 , G 0 ); F, G) is convenient for application of the Lagrange method of indefinite multipliers for finding solution to the problem (18). Making use of the method of Lagrange multipliers and the form of subdifferentials of the indicator functions we describe relations that determine the least favourable spectral densities in some special classes of spectral densities (see books by Moklyachuk [22,23], Moklyachuk and Masyutka [28] for additional details).…”

“…Lemma 3.2 Let (F 0 , G 0 ) be a solution of the optimization problem (18). The spectral densities F 0 (λ), G 0 (λ) are the least favorable in the class D = D F × D G and the spectral characteristic h 0 = h(F 0 , G 0 ) is the minimax of the optimal linear estimate of the functional…”

Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

“…The minimax interpolation problem for the linear functional A N ξ = ∑ N k=0 a(k)ξ(k) which depends on the unknown values of the sequence ξ(k) based on observations with and without noise was investigated in papers [19,20], and for the linear functional Aξ = ∫ ∞ 0 a(t)ξ(t)dt which depends on the unknown values of a random process ξ(t) in the paper [24].…”

Filtering Problem for Random Processes with Stationary Increments

“…In particular, the minimax-robust filtering problem for such sequences is investigated in papers by Luz and Moklyachuk (2013b;2014b). Random processes with stationary increments with continuous time are considered in the articles by Luz and Moklyachuk (2014a;2015a), where the authors investigated the extrapolation and the interpolation problems.…”