Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Filtering of stochastic processes having periodically correlated increments

Maksym Luz, Mikhail Moklyachuk

Abstract

We deal with the problem of the mean square optimal estimation of linear transformations of the unobserved values of a continuous time stochastic process with periodically correlated increments. Estimates are based on observations of the process with a continuous time stochastic noise process which is periodically correlated increments as well. To solve the problem, we transform the processes to infinite dimensional vector valued stationary sequences. We obtain formulas for calculating the mean square errors and the spectral characteristics of the optimal estimates of the transformations. Formulas determining the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of transformations are derived.

Filtering of stochastic processes having periodically correlated increments

Abstract

We deal with the problem of the mean square optimal estimation of linear transformations of the unobserved values of a continuous time stochastic process with periodically correlated increments. Estimates are based on observations of the process with a continuous time stochastic noise process which is periodically correlated increments as well. To solve the problem, we transform the processes to infinite dimensional vector valued stationary sequences. We obtain formulas for calculating the mean square errors and the spectral characteristics of the optimal estimates of the transformations. Formulas determining the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of transformations are derived.
Paper Structure (7 sections, 15 theorems, 161 equations)

This paper contains 7 sections, 15 theorems, 161 equations.

Table of Contents

  1. Stochastic processes with periodically correlated $d$th increments
  2. Hilbert space projection method of filtering
  3. Filtering based on factorizations of the spectral densities
  4. Minimax filtering based on observations with periodically stationary increment noise
  5. Least favorable spectral densities and minimax-robust spectral characteristic
  6. Least favorable spectral densities in the class $\mathcal{D}_f^0\times\mathcal{D}_g^0$
  7. Semi-uncertain filtering problem in classes $\mathcal{D}_0$ of least favorable spectral density

Key Result

Theorem 1.1

A stationary increment sequence $\xi^{(d)}_{j}$ is uniquely represented in the form where $\xi^{(d)}_{R,kj}, k=1,\ldots,\infty$, is a regular stationary increment sequence and $\xi^{(d)}_{S,kj}, k=1,\dots,\infty$, is a singular stationary increment sequence. The increment sequences $\xi^{(d)}_{R,kj}$ and $\xi^{(d)}_{S,kj}$ are orthogonal for all $j\in\mathbb{Z}$. They are defined b

Theorems & Definitions (31)

  • Definition 1.1: Gladyshev Glad1963
  • Definition 1.2: Luz and Moklyachuk Luz_Mokl_extra_cont_PCI
  • Remark 1.1
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 21 more