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

Анотація

Abstract 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> {\xi(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\in[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> {\{\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau),u\in[0,T),\,j\in\mathbb{Z}\}} 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> {L_{2}([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> <jats:inline-graphic

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