Minimax interpolation of continuous time stochastic processes with periodically correlated increments observed with noise

We deal with the problem of optimal estimation of linear functionals constructed from the missed values of a continuous time stochastic process <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ξ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {ξ(t)} with periodically stationary increments at points <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>;</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> t∈[0;(N+1)T] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:mrow> <m:msubsup> <m:mi>ξ</m:mi> <m:mi>j</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msubsup> <m:mi>ξ</m:mi> <m:mi>j</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>⁢</m:mo> <m:mi>T</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>τ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="4.2pt">,</m:mo> <m:mrow> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mi>ℤ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{ξ⁽ᵈ⁾ⱼ(u)=ξ⁽ᵈ⁾ⱼ(u+jT,τ),u∈[0,T),\,j\}} is constructed. It forms an <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>;</m:mo> <m:mi>H</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L₂([0,T);H)} -valued stationary increment sequence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:msubsup> <m:mi>ξ</m:mi> <m:mi>j</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi>ℤ</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{ξ⁽ᵈ⁾ⱼ,j\}} or corresponding to it an (infinite-dimensional) vector stationary increment sequence <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msubsup> <m:mover accent="true"> <m:mi>ξ</m:mi> <m:mo stretchy="false">→</m:mo> </m:mover> <m:mi>j</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msubsup> <m:mi>ξ</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>⁢</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>d</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⊤</m:mo> </m:msup> <m:mo rspace="4.2pt">,</m:mo> <m:mi>j</m:mi> <m:mo>∈</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{ξ⃗⁽ᵈ⁾ⱼ=(ξ⁽ᵈ⁾ₖⱼ,k=1,2,)^{},\,j\}} . In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.