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

The problem of optimal estimation of the linear functionals Aξ = ∑∞ k=0 a(k)ξ(−k) and AN ξ = ∑N k=0 a(k)ξ(−k) which depend on the unknown values of a stochastic sequence ξ(k) with stationary nth increments is considered. Estimates are based on observations of the sequence ξ(k) + η(k) at points of time k = 0,−1,−2, . . ., where the sequence η(k) is stationary and uncorrelated with the sequence ξ(k). Formulas for calculating the mean-square errors and spectral characteristics of the optimal estimates of the functionals are proposed in the case of spectral certainty, where spectral densities of sequences ξ(k) and η(k) are exactly known. Minimax (robust) method of estimation is applied in the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities. The filtering problem for a class of cointegrated sequences is investigated.