Robust Extrapolation Problem for Stochastic Processes with Stationary Increments

Minimax Interpolation Problem for Random Processes with Stationary Increments

The problem of mean-square optimal estimation of the linear functional AT 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. Formu...

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...

Minimax-robust prediction 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(m) with stationary nth increments is considered. Estimates are obtained which are based on observations of the sequence ξ(m) + η(m) at points of time m = −1,−2, . . ., where the sequence η(m) is stationary and uncorrelated with...

Representations of Gaussian processes with stationary increments

Bipower variation for Gaussian processes with stationary increments

Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati and others.