# Estimation problem for continuous time stochastic processes with
periodically correlated increments

Maksym Luz, Mikhail Moklyachuk BNP Paribas Cardif in Ukraine, Kyiv, Ukraine,
<EMAIL_ADDRESS>of Probability Theory, Statistics and Actuarial
Mathematics, Taras Shevchenko National University of Kyiv, Kyiv 01601,
Ukraine<EMAIL_ADDRESS>
(28th August 2024)

###### Abstract

We deal with the problem of optimal estimation of the linear functionals
constructed from unobserved values of a continuous time stochastic process
with periodically correlated increments based on past observations of this
process. To solve the problem, we construct a corresponding to the process
sequence of stochastic functions which forms an infinite dimensional vector
stationary increment sequence. In the case of known spectral density of the
stationary increment sequence, 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.

Keywords:Periodically Correlated Increments, Minimax-Robust Estimate, Mean
Square Error

AMS 2010 subject classifications. Primary: 60G10, 60G25, 60G35, Secondary:
62M20, 62P20, 93E10, 93E11

## Introduction

In this article, we study the prediction problem for the continuous time
stochastic processes $\xi(t)$, $t\in\mathbb{R}$, with periodically correlated
increments $\xi^{(d)}(t,\tau T)=\Delta_{T\tau}^{d}\xi(t)$ of order $d$ and
period $T$, where $\Delta_{s}\xi(t)=\xi(t)-\xi(t-s)$. The resent studies, for
example, by Basawa et al. [1], Dudek et al. [4], Reisen et al. [28], show a
constant interest to the non-stationary models and robust methods of
estimation.

Kolmogorov [13], Wiener [32] and Yaglom [34] developed effective methods of
solution of interpolation, extrapolation (prediction) and filtering problems
for stationary stochastic sequences and processes. For a particular problem,
they found an estimate $\widetilde{x}(t)$ constructed from available
observations that minimizes the mean square error
$\Delta(\widetilde{x}(t),f)=\mathsf{E}|x(t)-\widetilde{x}(t)|^{2}$ in the case
where the spectral density $f(\lambda)$ of the stationary process or sequence
$x(t)$ is exactly known and fixed. Such estimates are called optimal linear
estimates within this article.

The developed classical estimation methods are not directly applicable in
practice since the exact spectral structure of the processes is not usually
available. In this case the estimated spectral densities can be considered as
the true ones. However, Vastola and Poor [31] showed with the help of the
concrete examples, that such substitution can result in a significant increase
of the estimate error. Therefore it is reasonable to consider the estimates,
called minimax-robust, which minimize the maximum of the mean-square errors
for all spectral densities from a given set of admissible spectral densities
simultaneously. The minimax-robust method of extrapolation was proposed by
Grenander [8] who considered an estimation of the functional
$Ax=\int_{0}^{1}a(t)x(t)dt$ as a game between two players, one of which
minimizes $\Delta(\widetilde{A}x,f)$ by $\widetilde{A}\zeta$ and another one
maximizes it by $f$. The game has a saddle point under proper conditions:

$\max_{\begin{subarray}{c}f\end{subarray}}\min_{\begin{subarray}{c}\widetilde{A}x\end{subarray}}\mathsf{\Delta}(\widetilde{A}x,f)=\min_{\begin{subarray}{c}\widetilde{A}x\end{subarray}}\max_{\begin{subarray}{c}f\end{subarray}}\Delta(\widetilde{A}x,f)=\nu.$

For more details see the further study by Franke and Poor [5] and the survey
paper by Kassam and Poor [11]. A wide range of results has been obtained by
Moklyachuk [19, 20, 22]. These results have been extended on the vector-valued
stationary processes and sequences by Moklyachuk and Masyutka [24].

The concept of stationarity admits some generalizations, a combination of two
of which – stationary $d$th increments and periodical correlation – is in
scope of this article. Random processes with stationary $d$th increments
$x(t)$ were introduced by Yaglom and Pinsker [27]. The increment sequence
$x^{(d)}(t,\tau)=\Delta_{\tau}^{d}x(t)$ generated by such process is
stationary by the variable $t$, namely, the mathematical expectations
$\mathsf{E}\xi^{(n)}(t,\tau)$ and
$\mathsf{E}\xi^{(n)}(t+s,\tau_{1})\xi^{(n)}(t,\tau_{2})$ do not depend on $t$.
Yaglom and Pinsker [27] described the spectral representation of such process
and the spectral density canonical factorization, and they also solved the
extrapolation problem for these processes. The minimax-robust extrapolation,
interpolation and filtering problems for stochastic processes with stationary
increments were investigated by Luz and Moklyachuk [16].

Dubovetska and Moklyachuk [3] derived the classical and minimax-robust
estimates for another generalization of stationary processes – periodically
correlated (cyclostationary) processes, introduced by Gladyshev [7]. The
correlation function $K(t,s)={\textsf{E}}{x(t)\overline{x(s)}}$ of such
processes is a $T$-periodic function: $K(t,s)=K(t+T,s+T)$, which implies a
time-dependent spectrum. Periodically correlated processes are widely used in
signal processing and communications (see Napolitano [26] for a review of
recent works on cyclostationarity and its applications).

In this article, we deal with the problem of the mean-square optimal
estimation of the linear functionals $A\xi=\int_{0}^{\infty}a(t)\xi(t)dt$ and
$A_{NT}{\xi}=\int_{0}^{(N+1)T}a(t)\xi(t)dt$ which depend on the unobserved
values of a continuous time stochastic process $\xi(t)$ with periodically
stationary $d$-th increments from its observations at points $t<0$. The
similar problems for discrete time processes have been studied by Kozak and
Moklyachuk [14], Luz and Moklyachuk [17, 18]. In section 2, we describe a
presentation of a continuous time periodically stationary process as a
stationary H-valued sequence. This approach is extended on the periodically
stationary increments in the subsection 3.1. In subsection 3.2, the maximum
mean square error of the estimate is derived. The classical prediction (or
extrapolation) problem is solved in subsection 3.3. Particularly, formulas for
calculating the mean-square errors and the spectral characteristics of the
optimal linear estimates of the functionals $A\xi$ and $A_{NT}{\xi}$ are
derived under the condition of spectral certainty. In subsection 4, we present
our results on minimax-robust prediction for the studied processes: relations
that determine the least favourable spectral densities and the minimax
spectral characteristics derived for some classes of spectral densities.

##  1 Continuous time periodically correlated processes and generated vector
stationary sequences

In this section we present a brief review of periodically correlated processes
and describe an approach to be applied in the next section to develop a
spectral theory for periodically correlated increment process.

######  Definition 1.1 (Gladyshev [7]).

A mean-square continuous stochastic process $\zeta:\mathbb{R}\to
H=L_{2}(\Omega,\mathcal{F},\mathbb{P})$, with $\textsf{E}\zeta(t)=0,$ is
called periodically correlated (PC) with period $T$, if its correlation
function $K(t,s)={\textsf{E}}{\zeta(t)\overline{\zeta(s)}}$ for all
$t,s\in\mathbb{R}$ and some fixed $T>0$ is such that

$K(t,s)={\textsf{E}}{\zeta(t)\overline{\zeta(s)}}={\textsf{E}}{\zeta(t+T)\overline{\zeta(s+T)}}=K(t+T,s+T).$

For a periodically correlated stochastic process $\zeta(t)$, one can construct
the following sequence of stochastic functions [2], [23]

$\\{\zeta_{j}(u)=\zeta(u+jT),u\in[0,T),j\in\mathbb{Z}\\}.$ (1)

Sequence (1) forms a $L_{2}([0,T);H)$-valued stationary sequence
$\\{\zeta_{j},j\in\mathbb{Z}\\}$ with the correlation function

$B_{\zeta}(l,j)=\langle\zeta_{l},\zeta_{j}\rangle_{H}=\int_{0}^{T}\textsf{E}[\zeta(u+lT)\overline{\zeta(u+jT)}]du=\int_{0}^{T}K_{\zeta}(u+(l-j)T,u)du=B_{\zeta}(l-j),$

where $K_{\zeta}(t,s)=\textsf{E}{\zeta}(t)\overline{{\zeta}(s)}$ is the
correlation function of the PC process $\zeta(t)$. If we chose the following
orthonormal basis in the space $L_{2}([0,T);\mathbb{R})$

$\\{\widetilde{e}_{k}=\frac{1}{\sqrt{T}}e^{2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}u/T},k=1,2,3,\dots\\},\quad\langle\widetilde{e}_{j},\widetilde{e}_{k}\rangle=\delta_{kj},$
(2)

the stationary sequence $\\{\zeta_{j},j\in\mathbb{Z}\\}$ can be represented in
the form

$\zeta_{j}=\sum_{k=1}^{\infty}\zeta_{kj}\widetilde{e}_{k},$

where

$\zeta_{kj}=\langle\zeta_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}\zeta_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

The sequence $\\{\zeta_{j},j\in\mathbb{Z}\\},$ or the corresponding to it a
vector sequence

$\\{\vec{\zeta}_{j}=(\zeta_{kj},k=1,2,\dots)^{\top},j\in\mathbb{Z}\\},$

is called a generated by the process $\\{\zeta(t),t\in\mathbb{R}\\}$ vector
stationary sequence. Components $\\{\zeta_{kj}\\}:k=1,2,\dots;j\in\mathbb{Z}$
of the generated stationary sequence $\\{\zeta_{j},j\in\mathbb{Z}\\}$ satisfy
the following relations [10], [19]

$\textsf{E}{\zeta_{kj}}=0,\quad\|{\zeta}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\textsf{E}|\zeta_{kj}|^{2}\leq
P_{\zeta}=B_{\zeta}(0),\quad\textsf{E}\zeta_{kl}\overline{\zeta}_{nj}=\langle
R_{\zeta}(l-j)\widetilde{e}_{k},\widetilde{e}_{n}\rangle.$

The correlation function $R_{\zeta}(j)$ of the generated stationary sequence
$\\{\zeta_{j},j\in\mathbb{Z}\\}$ are correlation operator functions. The
correlation operator $R_{\zeta}(0)=R_{\zeta}$ is a kernel operator and its
kernel norm satisfies the following limitations:

$\|{\zeta}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\langle
R_{\zeta}z\widetilde{e}_{k},\widetilde{e}_{k}\rangle\leq P_{\zeta}.$

The generated stationary sequence $\\{\zeta_{j},j\in\mathbb{Z}\\}$ has the
spectral density function $f(\lambda)=\\{f_{kn}(\lambda)\\}_{k,n=1}^{\infty}$,
that is positive valued operator functions of variable $\lambda\in[-\pi,\pi)$,
if its correlation function $R_{\zeta}(j)$ can be represented in the form

$\langle
R_{\zeta}(j)\widetilde{e}_{k},\widetilde{e}_{n}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij\lambda}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{n}\rangle d\lambda.$

We finish our review by the statement, that For almost all
$\lambda\in[-\pi,\pi)$ the spectral density $f(\lambda)$ is a kernel operator
with an integrable kernel norm

$\sum_{k=1}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{k}\rangle
d\lambda=\sum_{k=1}^{\infty}\langle
R_{\zeta}\widetilde{e}_{k},\widetilde{e}_{k}\rangle=\|{\zeta}_{j}\|^{2}_{H}\leq
P_{\zeta}.$

##  2 Extrapolation problem for stochastic processes with periodically
correlated $d$th increments

###  2.1 Stochastic processes with periodically correlated $d$th increments

For a given stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$, consider the
stochastic $d$th increment process

$\xi^{(d)}(t,\tau)=(1-B_{\tau})^{d}\xi(t)=\sum_{l=0}^{d}(-1)^{l}{d\choose
l}\xi(t-l\tau),$ (3)

with the step $\tau\in\mathbb{R}$, generated by the stochastic process
$\xi(t)$. Here $B_{\tau}$ is the backward shift operator:
$B_{\tau}\xi(t)=\xi(t-\tau)$, $\tau\in\mathbb{R}$.

We prefer to use the notation $\xi^{(d)}(t,\tau)$ instead of widely used
$\Delta_{\tau}^{d}\xi(t)$ to avoid a duplicate with the mean square error
notation.

######  Definition 2.1.

A stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$ is called a _stochastic
process with periodically stationary (periodically correlated) increments_
with the step $\tau\in\mathbb{Z}$ and the period $T>0$ if the mathematical
expectations

$\displaystyle\mathsf{E}\xi^{(d)}(t+T,\tau T)$ $\displaystyle=$
$\displaystyle\mathsf{E}\xi^{(d)}(t,\tau T)=c^{(d)}(t,\tau T),$
$\displaystyle\mathsf{E}\xi^{(d)}(t+T,\tau_{1}T)\xi^{(d)}(s+T,\tau_{2}T)$
$\displaystyle=$ $\displaystyle D^{(d)}(t+T,s+T;\tau_{1}T,v_{2}T)$
$\displaystyle=$ $\displaystyle D^{(d)}(t,s;\tau_{1}T,\tau_{2}T)$

exist for every $t,s\in\mathbb{R}$, $\tau_{1},\tau_{2}\in\mathbb{Z}$ and for
some fixed $T>0$.

The functions $c^{(d)}(t,\tau T)$ and $D^{(d)}(t,s;\tau_{1}T,\tau_{2}T)$ from
the Definition 2.1 are called the _mean value_ and the _structural function_
of the stochastic process $\xi(t)$ with periodically stationary (periodically
correlated) increments.

For the stochastic process $\\{\xi(t),t\in\mathbb{R}\\}$ with periodically
correlated increments $\xi^{(d)}(t,\tau T)$ and the integer step $\tau$, we
follow the procedure described in the Section 1 and construct a sequence of
stochastic functions

$\\{\xi^{(d)}_{j}(u):=\xi^{(d)}_{j,\tau}(u)=\xi^{(d)}_{j}(u+jT,\tau
T),\,\,u\in[0,T),j\in\mathbb{Z}\\}.$ (4)

Sequence (4) forms a $L_{2}([0,T);H)$-valued stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ with the structural function

$\displaystyle B_{\xi^{(d)}}(l,j)$ $\displaystyle=$
$\displaystyle\langle\xi^{(d)}_{l},\xi^{(d)}_{j}\rangle_{H}=\int_{0}^{T}\textsf{E}[\xi^{(d)}_{j}(u+lT,\tau_{1}T)\overline{\xi^{(d)}_{j}(u+jT,\tau_{2}T)}]du$
$\displaystyle=$
$\displaystyle\int_{0}^{T}D^{(d)}(u+(l-j)T,u;\tau_{1}T,\tau_{2}T)du=B_{\xi^{(d)}}(l-j).$

Making use of the orthonormal basis (2) the stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ can be represented in the form

$\xi^{(d)}_{j}=\sum_{k=1}^{\infty}\xi^{(d)}_{kj}\widetilde{e}_{k},$ (5)

where

$\xi^{(d)}_{kj}=\langle\xi^{(d)}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}\xi^{(d)}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

We call this sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\},$ or the
corresponding to it vector sequence

$\\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},j\in\mathbb{Z}\\},$

_an infinite dimension vector stationary increment sequence_ generated by the
increment process $\\{\xi^{(d)}(t,\tau T),t\in\mathbb{R}\\}$. Further, we will
omit the word vector in the notion generated vector stationary increment
sequence.

Components $\\{\xi^{(d)}_{kj}\\}:k=1,2,\dots;j\in\mathbb{Z}$ of the generated
stationary increment sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ are such
that, [10], [19]

$\textsf{E}{\xi^{(d)}_{kj}}=0,\quad\|\xi^{(d)}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\textsf{E}|\xi^{(d)}_{kj}|^{2}\leq
P_{\xi^{(d)}}=B_{\xi^{(d)}}(0),$

and

$\textsf{E}\xi^{(d)}_{kl}\overline{\xi^{(d)}_{nj}}=\langle
R_{\xi^{(d)}}(l-j;\tau_{1},\tau_{2})\widetilde{e}_{k},\widetilde{e}_{n}\rangle.$

The _structural function_
$R_{\xi^{(d)}}(j):=R_{\xi^{(d)}}(j;\tau_{1},\tau_{2})$ of the generated
stationary increment sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ is a
correlation operator function. The correlation operator
$R_{\xi^{(d)}}(0)=R_{\xi^{(d)}}$ is a kernel operator and its kernel norm
satisfies the following limitations:

$\|\xi^{(d)}_{j}\|^{2}_{H}=\sum_{k=1}^{\infty}\langle
R_{\xi^{(d)}}\widetilde{e}_{k},\widetilde{e}_{k}\rangle\leq P_{\xi^{(d)}}.$

Suppose that the structural function $R_{\xi^{(d)}}(j)$ admits a
representation

$\langle
R_{\xi^{(d)}}(j;\tau_{1},\tau_{2})\widetilde{e}_{k},\widetilde{e}_{n}\rangle=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{ij\lambda}(1-e^{-i\tau_{1}\lambda})^{d}(1-e^{i\tau_{2}\lambda})^{d}\frac{1}{\lambda^{2d}}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{n}\rangle d\lambda.$

Then $f(\lambda)=\\{f_{kn}(\lambda)\\}_{k,n=1}^{\infty}$ is a _spectral
density function_ of the generated stationary increment sequence
$\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$. It is a positive valued operator
functions of variable $\lambda\in[-\pi,\pi)$, and for almost all
$\lambda\in[-\pi,\pi)$ it is a kernel operator with an integrable kernel norm

$\sum_{k=1}^{\infty}\frac{1}{2\pi}\int_{-\pi}^{\pi}(1-e^{-i\tau_{1}\lambda})^{d}(1-e^{i\tau_{2}\lambda})^{d}\frac{1}{\lambda^{2d}}\langle
f(\lambda)\widetilde{e}_{k},\widetilde{e}_{k}\rangle
d\lambda=\sum_{k=1}^{\infty}\langle
R_{\xi^{(d)}}\widetilde{e}_{k},\widetilde{e}_{k}\rangle=\|{\zeta}_{j}\|^{2}_{H}\leq
P_{\xi^{(d)}}.$ (6)

###  2.2 Extrapolation problem: the greatest value of the mean-square error

Consider the problem of the mean square optimal linear estimation of the
functionals

$A{\xi}=\int_{0}^{\infty}a(t)\xi(t)dt,\quad
A_{NT}{\xi}=\int_{0}^{(N+1)T}a(t)\xi(t)dt$

which depend on the unknown values of the stochastic process $\xi(t)$ with
periodically correlated $d$th increments. Estimates are based on observations
of the process $\xi(t)$ at points $t<0$.

######  Lemma 2.1 (Luz and Moklyachuk [16]).

Any linear functional

$A\xi=\int_{0}^{\infty}a(t)\xi(t)dt$

allows the representation

$A\xi=B\xi-V\xi,$

where

$B\xi=\int_{0}^{\infty}b^{\tau}(t)\xi^{(d)}(t,\tau T)dt,\quad V\xi=\int_{-\tau
Td}^{0}v^{\tau}(t)\xi(t)dt,$

and

$\displaystyle v^{\tau}(t)$ $\displaystyle=$
$\displaystyle\sum_{l=\left[-\frac{t}{\tau
T}\right]^{\prime}}^{d}(-1)^{l}{d\choose l}b^{\tau}(t+l\tau T),\quad
t\in[-\tau Td;0),$ (7) $\displaystyle b^{\tau}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}a(t+\tau Tk)d(k)=D^{\tau
T}\mathbf{a}(t),\,\,t\geq 0,$ (8)

where $[x]^{\prime}$ denotes the least integer number among numbers that are
equal to or greater than $x$, coefficients $\\{d(k):k\geq 0\\}$ are determined
by the relation

$\sum_{k=0}^{\infty}d(k)x^{k}=\left(\sum_{j=0}^{\infty}x^{j}\right)^{d},$

$D^{\tau T}$ is the linear transformation acting on an arbitrary function
$x(t)$, $t\geq 0$, as follows:

$D^{\tau T}\mathbf{x}(t)=\sum_{k=0}^{\infty}x(t+\tau Tk)d(k).$

######  Corollary 2.1.

A linear functional

$A_{NT}\xi=\int_{0}^{(N+1)T}a(t)\xi(t)dt$

allows the representation

$A_{NT}\xi=B_{NT}\xi-V_{NT}\xi,$

where

$B_{NT}\xi=\int_{0}^{(N+1)T}b^{\tau,N}(t)\xi^{(d)}(t,\tau T)dt,\quad
V_{NT}\xi=\int_{-\tau Td}^{0}v^{\tau,N}(t)\xi(t)dt,$

and

$\displaystyle v^{\tau,N}(t)$ $\displaystyle=$
$\displaystyle\sum_{l=\left[-\frac{t}{\tau
T}\right]^{\prime}}^{\min\left\\{\left[\frac{{(N+1)T}-t}{\tau
T}\right],d\right\\}}(-1)^{l}{d\choose l}b^{\tau,N}(t+l\tau T),\quad
t\in[-\tau Td;0),$ (9) $\displaystyle b^{\tau,N}(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\left[\frac{{(N+1)T}-t}{\tau T}\right]}a(t+\tau
Tk)d(k)=D^{\tau T,N}\mathbf{a}(t),\,t\in[0;{(N+1)T}],$ (10)

the linear transformation $D^{\tau T,N}$ acts on an arbitrary function $x(t)$,
$t\in[0;{(N+1)T}]$, as follows:

$D^{\tau T,N}\mathbf{x}(t)=\sum_{k=0}^{\left[\frac{{(N+1)T}-t}{\tau
T}\right]}x(t+\tau Tk)d(k).$

Let $\widehat{A}\xi$ denote the mean square optimal linear estimate of the
functional ${A}\xi$ from observations of the process $\xi(t)$ at points $t<0$
and let $\widehat{B}\xi$ denote the mean square optimal linear estimate of the
functional ${B}\xi$ from observations of the stochastic dth increment process
$\xi^{(d)}(t,\tau T)$ at points $t<0$.

Let $\Delta(f;\widehat{A}\xi)=\mathsf{E}|A\xi-\widehat{A}\xi|^{2}$ and
$\Delta(f;\widehat{B}\xi)=\mathsf{E}|B\xi-\widehat{B}\xi|^{2}$ denote the mean
square errors of the estimates $\widehat{A}\xi$ and $\widehat{B}\xi$
respectively/Ṡince values $\xi(t)$ at points $t\in[-\tau Tn;0)$ are known, the
following equality comes from Lemma 2.1:

$A\xi=B\xi-V\xi,\quad\widehat{A}\xi=\widehat{B}\xi-V\zeta,$ (11)

and

$\Delta\left(f;\widehat{A}\xi\right)=\mathsf{E}\left|A\xi-\widehat{A}\xi\right|^{2}=\mathsf{E}\left|B\xi-V\xi-\widehat{B}\xi+V\xi\right|^{2}=\mathsf{E}\left|B\xi-\widehat{B}\xi\right|^{2}=\Delta\left(f;\widehat{B}\xi\right).$

The similar relations for the functional $A_{NT}\xi$ are valid:

$A_{NT}\xi=B_{NT}\xi-V_{NT}\xi,\quad\widehat{A}_{NT}\xi=\widehat{B}_{NT}\xi-
V_{NT}\zeta,$ (12)

and

$\displaystyle\Delta\left(f;\widehat{A}_{NT}\xi\right)$ $\displaystyle=$
$\displaystyle\mathsf{E}\left|A_{NT}\xi-\widehat{A}_{NT}\xi\right|^{2}=\mathsf{E}\left|B_{NT}\xi-
V_{NT}\xi-\widehat{B}_{NT}\xi+V_{NT}\xi\right|^{2}$ $\displaystyle=$
$\displaystyle\mathsf{E}\left|B_{NT}\xi-\widehat{B}_{NT}\xi\right|^{2}=\Delta\left(f;\widehat{B}_{NT}\xi\right).$

Thus, to find the mean square optimal linear estimation of the functionals
$A\xi$, $A_{NT}\xi$ we have to find estimates of the functionals $B\xi$,
$B_{NT}\xi$.

The functional $B_{NT}\xi$ can be represented in the form

$B_{NT}\xi=\int_{0}^{(N+1)T}b^{\tau,N}(t)\xi^{(d)}(t,\tau
T)dt=\sum_{j=0}^{N}\int_{0}^{T}b^{\tau,N}_{j}(u)\xi^{(d)}_{j}(u)du,$

where

$b^{\tau,N}_{j}(u)=b^{\tau,N}(u+jT),\quad\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau
T),\,\,u\in[0,T).$

Making use of the decomposition (5) of the generated stationary increment
sequence $\\{\xi^{(d)}_{j},j\in\mathbb{Z}\\}$ and the solution of equation

$(-1)^{k}\left[\frac{k}{2}\right]+(-1)^{m}\left[\frac{m}{2}\right]=0$ (13)

of two variables $(k,m)$, which is given by pairs $(1,1)$, $(2l+1,2l)$ and
$(2l,2l+1)$ for $l=2,3,\dots$, the functional $B_{NT}\xi$ can be rewritten in
the form [23]

$\displaystyle B_{NT}\xi$ $\displaystyle=$
$\displaystyle\int_{0}^{(N+1)T}b^{\tau,N}(t)\xi^{(d)}(t,\tau
T)dt=\sum_{j=0}^{N}\int_{0}^{T}b^{\tau,N}_{j}(u)\xi^{(d)}_{j}(u)du$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{N}\frac{1}{T}\int_{0}^{T}\left(\sum_{k=1}^{\infty}b^{\tau,N}_{kj}e^{2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}u/T}\right)\left(\sum_{m=1}^{\infty}\xi^{(d)}_{mj}e^{2\pi
i\\{(-1)^{m}\left[\frac{m}{2}\right]\\}u/T}\right)du$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N}\sum_{k=1}^{\infty}\sum_{m=1}^{\infty}b^{\tau,N}_{kj}\xi^{(d)}_{mj}\frac{1}{T}\int_{0}^{T}e^{2\pi
i\left\\{(-1)^{k}\left[\frac{k}{2}\right]+(-1)^{m}\left[\frac{m}{2}\right]\right\\}u/T}du$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{N}\sum_{k=1}^{\infty}b^{\tau,N}_{kj}\xi^{(d)}_{kj}=\sum_{j=0}^{N}{(\vec{b}^{\tau,N}_{j})}^{\top}\vec{\xi}^{(d)}_{j}=B_{N}\vec{\xi},$

where the infinite dimensional vectors $\vec{\xi}^{(d)}_{j}$ and
$\vec{b}^{\tau,N}_{j}$ are defined as follows:

$\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},$

and

$\vec{b}^{\tau,N}_{j}=(b^{\tau,N}_{kj},k=1,2,\dots)^{\top}=(b^{\tau,N}_{1j},b^{\tau,N}_{3j},b^{\tau,N}_{2j},\dots,b^{\tau,N}_{2k+1,j},b^{\tau,N}_{2k,j},\dots)^{\top}.$

Here

$b^{\tau,N}_{kj}=\langle
b^{\tau,N}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}b^{\tau,N}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,$

where

$b^{\tau,N}_{j}(u)=\sum_{l=0}^{\left[\frac{{(N+1)T}-u-jT}{\tau
T}\right]}a(u+jT+\tau Tl)d(l)=D^{\tau T,N}a(u),\,u\in[0;t),\,j=0,1,\ldots,N,$

and

$b^{\tau,N}_{kj}=\sum_{k=0}^{\left[\frac{N-j}{\tau}\right]}a_{kj+\tau
l}d(k),\,j=0,1,\ldots,N,$

where

$a_{kj}=\langle
a_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}a_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,\quad
k=1,2,\dots,\,j=0,1,\ldots,N.$

Assume, that coefficients $\\{\vec{b}^{\tau,N}_{j},j=0,1,\dots,N\\}$, that
determine the functional $B_{NT}\xi$, satisfy the conditions

$\|\vec{b}^{\tau,N}_{j}\|<\infty,\quad\|\vec{b}^{\tau,N}_{j}\|^{2}=\sum_{k=1}^{\infty}|{b}^{\tau,N}_{kj}|^{2},\quad
j=0,1,\dots,N.$ (14)

It follows from the condition (14) that the functional

$B_{NT}\xi=\sum_{j=0}^{N}\sum_{k=1}^{\infty}b^{\tau,N}_{kj}\xi^{(d)}_{kj}=\sum_{j=0}^{N}{(\vec{b}^{\tau,N}_{j})}^{\top}\vec{\xi}^{\,(d)}_{j}=B_{N}\vec{\xi}$

has a finite second moment.

Let us calculate the greatest value of the mean-square error
$\Delta(\xi,\hat{B}_{NT})$ of estimate $\hat{B}_{NT}\xi$ of the functional
$B_{NT}\xi$. Denote by $\Lambda$ the set of all linear estimates of the
functional $B_{NT}\xi$, based on an observation of the process $\xi(t)$ at
points $t<0$.

Let $\mathbf{Y}$ denote the class of the mean-square continuous PC increment
processes $\xi^{(d)}(t,\tau T)$, such that $\textsf{E}\xi^{(d)}(t,\tau
T)=0,\textsf{E}|\xi^{(d)}(t,\tau T)|^{2}\leq P_{\xi^{d}}$. Let
$\mathbf{Y}_{R}$ denote the class of all regular stationary sequences, that
satisfy the condition $\|\xi^{(d)}_{j}\|^{2}_{H}\leq P_{\xi^{(d)}}$. Then we
can formulate the following theorem (see [19, 21]).

######  Theorem 2.1.

Let the coefficients $\\{\vec{b}^{\tau,N}_{j},j=0,1,\dots,N\\}$, which
determine the functional $B_{NT}\xi$, satisfy condition (14). The function
$\Delta(\xi,\hat{B}_{NT})$ has a saddle point on the set
$\mathbf{Y}\times\Lambda$:

$\mathop{\min}\limits_{\hat{B}_{NT}\in\Lambda}\mathop{\max}\limits_{\zeta\in\mathbf{Y}}\Delta(\xi,\hat{B}_{NT})=\mathop{\max}\limits_{\zeta\in\mathbf{Y}}\mathop{\min}\limits_{\hat{B}_{NT}\in\Lambda}\Delta(\xi,\hat{B}_{NT})=P_{\xi}\cdot\nu_{N}^{2},$

where $\nu_{N}^{2}$ is the greatest eigenvalue of the self-adjoint compact
operator $Q_{N}$ in the space $\ell_{2}$, determined by the matrix
$\\{Q_{N}(p,q)\\}_{p,q=0}^{D}$, which is constructed from the block-matrices
$Q_{N}(p,q)=\\{Q_{kn}^{D}(p,q)\\}_{k,n=1}^{\infty}$ with elements

$Q_{kn}^{N}(p,q)=\sum_{s=0}^{min(N-p,N-q)}b^{\tau,N}_{k,s+p}\cdot\overline{b^{\tau,N}_{n,s+q}},$
(15) $k,n=1,2,\dots,\quad p,q=0,1,\dots,N.$

The least favorable stochastic sequence $\vec{\xi}^{(d)}_{j}$ in the class
$\mathbf{Y}$ for the optimal estimate of the functional $B_{NT}\xi$ is an one-
sided moving average sequence of order $N$ of the form

$\vec{\xi}^{(d)}_{j}=\sum_{s=j-N}^{j}g(j-s)\vec{\varepsilon}(s),$

where $g_{N}=(g(0),g(1),\dots,g(N))^{\top}$ is the eigenvector, that
corresponds to $\nu_{N}^{2}$, which is constructed from matrices
$g(p)=\\{g_{km}(p)\\}_{k=\overline{1,\infty}}^{m=\overline{1,M}}$ and is
determined by the condition
$\|g_{N}\|^{2}=\sum_{p=0}^{D}\|g(p)\|^{2}=P_{\zeta}$,
$\vec{\varepsilon}(s)=\\{\varepsilon_{m}(s)\\}_{m=1}^{M}$ is a vector valued
stationary stochastic sequence with orthogonal values.

The similar representations can be obtained for the functional $B\xi$ [23]:

$B\xi=\int_{0}^{\infty}b^{\tau}(t)\xi^{(d)}(t,\tau
T)dt=\sum_{j=0}^{\infty}{(\vec{b}^{\tau}_{j})}^{\top}\vec{\xi}^{(d)}_{j}=B\vec{\xi},$

where the infinite dimensional vectors $\vec{\xi}^{(d)}_{j}$ and
$\vec{b}^{\tau}_{j}$ are defined as follows:

$\displaystyle\vec{\xi}^{(d)}_{j}$ $\displaystyle=$
$\displaystyle(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},$
$\displaystyle\vec{b}^{\tau}_{j}$ $\displaystyle=$
$\displaystyle(b^{\tau}_{kj},k=1,2,\dots)^{\top}=(b^{\tau}_{1j},b^{\tau}_{3j},b^{\tau}_{2j},\dots,b^{\tau}_{2k+1,j},b^{\tau}_{2k,j},\dots)^{\top}.$

Here

$\displaystyle b^{\tau}_{kj}$ $\displaystyle=$ $\displaystyle\langle
b^{\tau}_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}b^{\tau}_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv,$ $\displaystyle
b^{\tau}_{j}(u)$ $\displaystyle=$ $\displaystyle\sum_{l=0}^{\infty}a(u+jT+\tau
Tl)d(l)=D^{\tau}a_{j}(u),\quad u\in[0,T),\,j=0,1,\ldots,$

and

$\displaystyle b^{\tau}_{k,j}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}a_{k,j+\tau l}d(l),\quad k,j=0,1,\ldots,$
$\displaystyle a_{kj}$ $\displaystyle=$ $\displaystyle\langle
a_{j},\widetilde{e}_{k}\rangle=\frac{1}{\sqrt{T}}\int_{0}^{T}a_{j}(v)e^{-2\pi
i\\{(-1)^{k}\left[\frac{k}{2}\right]\\}v/T}dv.$

Assume, that coefficients $\\{\vec{b}^{\tau}_{j},j=0,1,\dots\\}$, that
determine the functional $B\xi$, satisfy the conditions

$\sum_{j=0}^{\infty}\|\vec{b}^{\tau}_{j}\|<\infty,\quad\sum_{j=0}^{\infty}(j+1)\|\vec{b}^{\tau}_{j}\|<\infty,\quad\|\vec{b}^{\tau}_{j}\|^{2}=\sum_{k=1}^{\infty}|{b}^{\tau}_{kj}|^{2}.$
(16)

It follows from the condition (16) that the functional

$B\xi=\sum_{j=0}^{\infty}\sum_{k=1}^{\infty}b^{\tau}_{kj}\xi^{(d)}_{kj}=\sum_{j=0}^{\infty}{(\vec{b}^{\tau}_{j})}^{\top}\vec{\xi}^{\,(n)}_{j}$

has a finite second moment. The following theorem provides the greatest value
of the mean-square error $\Delta(\xi,\hat{B})$ of the estimate $\hat{B}\xi$ of
the functional $B\xi$ (see [19, 21]).

######  Theorem 2.2.

Let the coefficients $\\{\vec{b}^{\tau}_{j},j=0,1,\dots\\}$, which determine
the functional $B\xi$, satisfy condition (16). The function
$\Delta(\xi,\hat{B})$ has a saddle point on the set $\mathbf{Y}\times\Lambda$:

$\mathop{\min}\limits_{\hat{B}\in\Lambda}\mathop{\max}\limits_{\zeta\in\mathbf{Y}}\Delta(\xi,\hat{B})=\mathop{\max}\limits_{\zeta\in\mathbf{Y}}\mathop{\min}\limits_{\hat{B}\in\Lambda}\Delta(\xi,\hat{B})=P_{\xi}\cdot\nu^{2},$

where $\nu^{2}$ is the greatest eigenvalue of the self-adjoint compact
operator $Q$ in the space $\ell_{2}$, determined by the matrix
$\\{Q(p,q)\\}_{p,q=0}^{\infty}$, which is constructed from the block-matrices
$Q(p,q)=\\{Q_{kn}(p,q)\\}_{k,n=1}^{\infty}$ with elements

$Q_{kn}(p,q)=\sum_{s=0}^{\infty}b^{\tau}_{k,s+p}\cdot\overline{b^{\tau}_{n,s+q}},$
(17) $k,n=1,2,\dots,\quad p,q=0,1,\dots$

The least favorable stochastic sequence $\vec{\xi}^{(d)}_{j}$ in the class
$\mathbf{Y}$ for the optimal estimate of the functional $B\xi$ is an one-sided
moving average sequence of the form

$\vec{\xi}^{(d)}_{j}=\sum_{s=-\infty}^{j}g(j-s)\vec{\varepsilon}(s),$

where $g=(g(0),g(1),\dots)^{\top}$ is the eigenvector, that corresponds to
$\nu^{2}$, which is constructed from matrices
$g(p)=\\{g_{km}(p)\\}_{k=\overline{1,\infty}}^{m=\overline{1,M}}$ and is
determined by the condition
$\|g\|^{2}=\sum_{p=0}^{\infty}\|g(p)\|^{2}=P_{\zeta}$,
$\vec{\varepsilon}(s)=\\{\varepsilon_{m}(s)\\}_{m=1}^{M}$ is a vector valued
stationary stochastic sequence with orthogonal values.

###  2.3 Classical extrapolation problem

Let the spectral density $f(\lambda)$ satisfy the _minimality condition_

$\int_{-\pi}^{\pi}\text{Tr}\left[\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}f^{-1}(\lambda)\right]d\lambda<\infty.$
(18)

This is the necessary and sufficient condition under which the mean square
errors of the estimates of the functionals $A\vec{\xi}$ and $A_{N}\vec{\xi}$
are not equal to $0$.

Consider the Hilbert space $H=L_{2}(\Omega,\mathcal{F},\mathsf{P})$ of random
variables $\gamma$ with zero mean value, $\mathsf{E}\gamma=0$, finite
variance, $\mathsf{E}|\gamma|^{2}<\infty$, and endowed with the inner product
$(\gamma_{1};\gamma_{2})=\mathsf{E}\gamma_{1}\overline{\gamma_{2}}$. With the
generated increment sequence $\\{\xi^{(d)}_{j}\\}$, we can associate the
closed linear subspace
$H^{0-}(\vec{\xi}^{(d)})=\overline{span}\\{\xi^{(d)}_{kj}:k=1,2,\ldots;j=-1,-2,-3,\ldots\\}$.

Consider also the closed linear subspace

$L_{2}^{0-}(f)=\overline{span}\\{e^{i\lambda
j}(1-e^{-i\lambda\tau})^{d}(i\lambda)^{-d}{\delta}_{n}:n=1,2,\ldots,\infty,\,j\leqslant-1\\},\quad{\delta}_{n}=\\{\delta_{kn}\\}_{k=1}^{\infty},$

of the Hilbert space $L_{2}(f)$ of vector-valued functions, endowed with the
inner product $\langle
g_{1};g_{2}\rangle=\int_{-\pi}^{\pi}(g_{1}(\lambda))^{\top}f(\lambda)\overline{g_{2}(\lambda)}d\lambda$.
Here $\delta_{kn}$ are Kronecker symbols. The relation

$\xi^{(d)}_{kj}=\int_{-\pi}^{\pi}e^{i\lambda
j}(1-e^{-i\lambda\tau})^{d}\dfrac{1}{(i\lambda)^{d}}dZ_{k}(\lambda),\quad
k=1,2,\dots,$ (19)

implies a map between the subspaces spaces $H^{0-}(\vec{\xi}^{(d)})$ and
$L_{2}^{0-}(f)$. Here $Z_{k}(\lambda)$ are the components of a vector-valued
stochastic process
$\vec{Z}_{\xi^{(d)}}(\lambda)=\\{Z_{k}(\lambda)\\}_{k=1}^{\infty}$ with
uncorrelated increments on $[-\pi,\pi)$ connected with the spectral function
$F(\lambda)$ by the relation

$\mathsf{E}(Z_{k}(\lambda_{2})-Z_{k}(\lambda_{1}))(\overline{Z_{n}(\lambda_{2})-Z_{n}(\lambda_{1})})=F_{kn}(\lambda_{2})-F_{kn}(\lambda_{1})$

for

$-\pi\leq\lambda_{1}<\lambda_{2}<\pi,\quad k,n=1,2,\dots,\infty.$

Define the Fourier coefficients of the function
$\lambda^{2d}|1-e^{i\lambda\tau}|^{-2d}f^{-1}(\lambda)$

$F^{\tau}_{j,l}=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-i\lambda(j-l)}\frac{\lambda^{2d}}{|1-e^{i\lambda\tau}|^{2d}}f^{-1}(\lambda)d\lambda,\quad
j,l\geq 0.$

Define the vector

$\vec{a}_{j}=(a_{kj},k=1,2,\dots)^{\top}=(a_{1j},a_{3j},a_{2j},\dots,a_{2k+1,j},a_{2k,j},\dots)^{\top}.$

Then Lemma 2.1 implies

$\vec{b}_{j}=\sum_{m=j}^{\infty}\mathsf{diag}_{\infty}(d_{\tau}(m-j))\vec{a}_{m}=(D^{\tau}{\mathbf{a}})_{j},\quad
j=0,1,2,\dots,$ (20)

where
$\mathbf{a}=((\vec{a}_{0})^{\top},(\vec{a}_{1})^{\top},(\vec{a}_{2})^{\top},\ldots)^{\top}$,
the coefficients $\\{d_{\tau}(k):k\geq 0\\}$ are determined by the
relationship

$\sum_{k=0}^{\infty}d_{\tau}(k)x^{k}=\left(\sum_{j=0}^{\infty}x^{\tau
j}\right)^{d},$

$D^{\tau}$ is a linear transformation determined by a matrix with infinite
dimension matrix entries $D^{\tau}(k,j),k,j=0,1,2,\dots$ such that
$D^{\tau}(k,j)=\mathsf{diag}_{\infty}(d_{\tau}(j-k))$ if $0\leq k\leq j$ and
$D^{\tau}(k,j)=\mathsf{diag}_{\infty}(0)$ for $0\leq j<k$;
$\mathsf{diag}_{\infty}(x)$ denotes an infinite dimension diagonal matrix with
the entry $x$ on its diagonal.

######  Theorem 2.3.

Let a stochastic process $x(t)$, $t\in\mathbb{R}$ with a periodically
stationary increments determine a generated stationary $d$th increment
sequence $\vec{\xi}^{(d)}_{m}$ with the spectral density matrix
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{\infty}$ satisfying minimality
condition (18). Let the coefficients $\vec{a}_{j},j\geqslant 0$ generated by
the function $a(t)$, $t\geq 0$, satisfy conditions (16) taking into account
(20). The optimal linear estimate $\widehat{A}\xi$ of the functional $A\xi$
based on observations of the process $\xi(t)$ on the interval $t<0$ is
calculated by the formula

$\widehat{A}\xi=\int_{-\pi}^{\pi}(\vec{h}_{\tau}(\lambda))^{\top}d\vec{Z}_{\xi^{(d)}}(\lambda)-\int_{-\tau
Td}^{0}v^{\tau}(t)\xi(t)dt,$ (21)

where $\vec{h}_{\tau}(\lambda)=\\{h_{k}(\lambda)\\}_{k=1}^{\infty}$ is the
spectral characteristic of the estimate $\widehat{A}\vec{\xi}$. The spectral
characteristic $\vec{h}_{\tau}(\lambda)$ and the value of the mean square
error $\Delta(f;\widehat{A}\xi)$ of the optimal estimate $\widehat{A}\xi$ are
calculated by the formulas

$(\vec{h}_{\tau}(\lambda))^{\top}=(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\frac{(-i\lambda)^{n}}{(1-e^{i\lambda\tau})^{d}}(\vec{C}_{\tau}(e^{i\lambda}))^{\top}f^{-1}(\lambda).$
(22)

and

$\displaystyle\Delta\left(f;\widehat{A}\xi\right)$ $\displaystyle=$
$\displaystyle\Delta\left(f;\widehat{B}\xi\,\right)=\mathsf{E}\left|B\xi-\widehat{B}\xi\,\right|^{2}$
(23) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(\vec{C}_{\tau}(e^{i\lambda}))^{\top}f(\lambda)\overline{\vec{C}_{\tau}(e^{i\lambda})}\frac{(i\lambda)^{d}}{(1-e^{-i\lambda\tau})^{d}}d\lambda$
$\displaystyle=$ $\displaystyle\left\langle
D^{\tau}\mathbf{a},\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a}\right\rangle.$

respectively, where

$\vec{B}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{b}^{\tau}_{j}e^{i\lambda
j},\quad\vec{C}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}(\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a})_{j}e^{i\lambda
j},$

$\mathbf{F}_{\tau}$ is a linear operator in the space $\ell_{2}$ which is
determined by a matrix with the infinite matrix entries
$(\mathbf{F}_{\tau})_{j,l}=F_{j,l}^{\tau}$, $j,l\geq 0$.

###### Proof.

Representation (21) for the estimate $\widehat{A}\xi$ comes from relation
(11). The functional $B\vec{\xi}$ allows the spectral representation

$B\vec{\xi}=\int_{-\pi}^{\pi}\left(\vec{B}_{\tau}(e^{i\lambda})\right)^{\top}\frac{(1-e^{-i\tau\lambda})^{d}}{(i\lambda)^{d}}d\vec{Z}_{\xi^{(d)}}(\lambda).$

Thus, the spectral characteristic $\vec{h}_{\tau}(\lambda)$ is characterized
by the following conditions, which determine a projection of the element
$\vec{B}_{\tau}(e^{i\lambda})\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}$
on the subspace $L_{2}^{0-}(f)$:

$\vec{h}_{\tau}(\lambda)\in L_{2}^{0-}\left(f\right),$ (24)
$\left(\vec{B}_{\tau}(e^{i\lambda})\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\vec{h}_{\tau}(\lambda)\right)\perp
L_{2}^{0-}\left(f\right).$ (25)

From the condition (25), we obtain the following relation which holds true for
all $j\leqslant-1$

$\displaystyle\int_{-\pi}^{\pi}\left(\vec{B}_{\tau}(e^{i\lambda})\frac{(1-e^{-i\tau\lambda})^{d}}{(i\lambda)^{d}}-\vec{h}_{\tau}(\lambda)\right)^{\top}f(\lambda)e^{-ij\lambda}\frac{(1-e^{i\tau\lambda})^{d}}{(-i\lambda)^{d}}d\lambda=0.$
(26)

Thus, the spectral characteristic of the estimate $\widehat{B}\vec{\xi}$ can
be represented in the form

$(\vec{h}_{\tau}(\lambda))^{\top}=(\vec{B}_{\tau}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\frac{(-i\lambda)^{n}(\vec{C}_{\tau}(e^{i\lambda}))^{\top}}{(1-e^{i\lambda\tau})^{d}}f^{-1}(\lambda),$

where

$\vec{C}_{\tau}(e^{i\lambda})=\sum_{j=0}^{\infty}\vec{c}^{\,\tau}_{j}e^{ij\lambda},$

coefficients
$\vec{c}_{j}^{\,\tau}=\\{c_{kj}^{\tau}\\}_{k=1}^{\infty},j\geqslant 0$ are
unknown and to be found.

Condition (24) implies the equestions

$\int_{-\pi}^{\pi}\left[(\vec{B}_{\tau}(e^{i\lambda}))^{\top}-\frac{\lambda^{2d}(\vec{C}_{\tau}(e^{i\lambda}))^{\top}}{(1-e^{-i\lambda\tau})^{d}(1-e^{i\lambda\tau})^{d}}f^{-1}(\lambda)\right]e^{-ij\lambda}d\lambda=0,j\geqslant
0.$ (27)

Making use of the introduced Fourier coefficients $F^{\tau}_{j,l}$, relation
(27) can be presented as a system of linear equations

$\vec{b}^{\tau}_{l}=\sum_{j=0}^{\infty}F_{l,j}^{\tau}\,\,{\vec{c}}^{\,\tau}_{j},\quad
l\geq 0,$

allowing the matrix form representation

$D^{\tau}\mathbf{a}=\mathbf{F}_{\tau}\mathbf{c}^{\tau},$ (28)

where
$\mathbf{c}^{\tau}=(({\vec{c}}^{\,\tau}_{0})^{\top},({\vec{c}}^{\,\tau}_{1})^{\top},({\vec{c}}^{\,\tau}_{2})^{\top},\ldots)^{\top}$.

Note that the operator $\mathbf{F}_{\tau}$ is invertible. The projection
problem for the element ${B}\vec{\xi}\in H$ on the closed convex set
$H^{0-}(\vec{\xi}^{(d)}_{\tau})$ has a unique solution for each non-zero
vector coefficients $\\{\vec{a}_{0},\vec{a}_{1},\vec{a}_{2},\ldots\\}$ under
conditions (16). Therefore, equation (28) has a unique solution for each
vector $\mathbf{b}^{\tau}=D^{\tau}\mathbf{a}$, which implies existence of the
inverse operator $\mathbf{F}^{-1}_{\tau}$. Now we can conclude, that the
coefficients ${\vec{c}}^{\,\tau}_{j}$, $j\geq 0$, can be calculated as

$\vec{c}^{\,\tau}_{j}=\left(\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a}\right)_{j},\quad
j\geq 0,$ (29)

where $\left(\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a}\right)_{j}$, $j\geq 0$,
is the $j$th infinite dimension vector element of the vector
$\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a}$.

The derived expressions and the coefficients ${\vec{c}}^{\,\tau}_{j}$, $j\geq
0$, prove formulas (22) for the spectral characteristic
$\vec{h}_{\tau}(\lambda)$ and (23) for the value of the mean square error of
the estimate $\widehat{A}\xi$. ∎

Consider the extrapolation problem for the functional $A_{NT}\xi$. We can
apply the results of Theorem 2.3 by putting $(\vec{a}_{j})^{\top}=0$ for
$j>N$. Corollary 2.1 implies

$\vec{b}^{d}_{j}=\sum_{m=j}^{N}\mathsf{diag}_{\infty}(d_{\tau}(m-j))\vec{a}(m)=(D^{\tau}_{N}{\mathbf{a}}_{N})_{j},\,j=0,1,\dots,N,$

where $D^{\tau}_{N}$ is the linear transformation determined by an infinite
matrix with the entries
$(D^{\tau}_{N})(k,j)=\mathsf{diag}_{\infty}(d_{\tau}(j-k))$ if $0\leq k\leq
j\leq N$, and $(D^{\tau}_{N})(k,j)=0$ if $j<k$ or $j,k>N$;
$\mathbf{a}_{N}=((\vec{a}_{0})^{\top},(\vec{a}_{1})^{\top},\ldots,(\vec{a}_{N})^{\top},\vec{0}\ldots)^{\top}$.

######  Theorem 2.4.

Let a stochastic process $\xi(t)$, $t\in\mathbb{R}$ with a periodically
stationary increments determine a generated stationary $d$th increment
sequence $\vec{\xi}^{(d)}_{m}$ with the spectral density matrix
$f(\lambda)=\\{f_{ij}(\lambda)\\}_{i,j=1}^{\infty}$ satisfying minimality
condition (18). The optimal linear estimate $\widehat{A}_{NT}\xi$ of the
functional $A_{NT}\xi$ based on observations of the process $\xi(t)$ on the
interval $t<0$ is calculated by the formula

$\widehat{A}_{NT}{\xi}=\int_{-\pi}^{\pi}(\vec{h}_{\tau,N}(\lambda))^{\top}d\vec{Z}_{\xi^{(d)}}(\lambda)-\int_{-\tau
Td}^{0}v^{\tau,N}(t)\xi(t)dt,$ (30)

The spectral characteristic
$\vec{h}_{\tau,N}(\lambda)=\\{h_{\tau,N,k}(\lambda)\\}_{k=1}^{\infty}$ and the
value of the mean square error $\Delta(f;\widehat{A}_{N}\xi)$ are calculated
by the formulas

$(\vec{h}_{\tau,N}(\lambda))^{\top}=(\vec{B}_{\tau,N}(e^{i\lambda}))^{\top}\frac{(1-e^{-i\lambda\tau})^{d}}{(i\lambda)^{d}}-\frac{(-i\lambda)^{n}}{(1-e^{i\lambda\tau})^{d}}(\vec{C}_{\tau,N}(e^{i\lambda}))^{\top}f^{-1}(\lambda).$
(31)

and

$\displaystyle\Delta(f,\widehat{A}_{NT}{\xi})$ $\displaystyle=$
$\displaystyle\Delta(f,\widehat{B}_{NT}{\xi})={\mathsf{E}}|B_{NT}{\xi}-\widehat{B}_{NT}{\xi}|^{2}$
(32) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(\vec{C}_{\tau,N}(e^{i\lambda}))^{\top}f(\lambda)\overline{\vec{C}_{\tau,N}(e^{i\lambda})}\frac{(i\lambda)^{d}}{(1-e^{-i\lambda\tau})^{d}}d\lambda$
$\displaystyle=$ $\displaystyle\left\langle
D^{\tau}_{N}{\mathbf{a}}_{N},\mathbf{F}_{\tau,N}^{-1}D^{\tau}_{N}{\mathbf{a}}_{N}\right\rangle.$

respectively, where

$\vec{B}_{\tau,N}(e^{i\lambda})=\sum_{j=0}^{N}(D^{\tau}_{N}{\mathbf{a}}_{N})_{j}e^{i\lambda
j},\quad\vec{C}_{\tau,N}(e^{i\lambda})=\sum_{j=0}^{\infty}(\mathbf{F}_{\tau,N}^{-1}D^{\tau}_{N}{\mathbf{a}}_{N})_{j}e^{i\lambda
j}$

$\mathbf{F}_{\tau,N}$ is a linear operator in the space $\ell_{2}$ which is
determined by a matrix with the infinite matrix entries
$(\mathbf{F}_{\tau,N})_{l,m}=F^{\tau}_{l,m}$, $l\geq 0$, $0\leq m\leq N$, and
$(\mathbf{F}_{\tau,N})_{l,m}=0$, $l\geq 0$, $m>N$.

##  3 Minimax (robust) method of extrapolation

The values of the mean square errors and the spectral characteristics of the
optimal estimates of the functionals ${A}\vec{\xi}$ and ${A}_{NT}\vec{\xi}$
depending on the unobserved values of a stochastic process ${\xi}(t)$, which
determine a periodically stationary stochastic increment process
${\xi}^{(d)}(t,\tau T)$ or the corresponding to it infinite dimension vector
stationary increment sequence
$\\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top}$,
$j\in\mathbb{Z}\\}$, with the spectral density matrix $f(\lambda)$, can be
calculated by formulas (22), (23) and (31), (32) respectively, provided the
spectral density $f(\lambda)$ of the stochastic process $\vec{\xi}(t)$ is
exactly known. If the spectral density is not known, but a set $\mathcal{D}$
of admissible spectral densities is defined, the minimax method of estimation
of the functionals depending on unobserved values of stochastic sequences with
stationary increments may be applied. This method consists in finding an
estimate that minimizes the maximal values of the mean square errors for all
spectral densities from a given class $\mathcal{D}$ of admissible spectral
densities simultaneously. The described method is formalized by the two
following definitions [21].

######  Definition 3.1.

For a given class of spectral densities $\mathcal{D}$ the spectral density
$f_{0}(\lambda)\in\mathcal{D}$ is called the least favourable in $\mathcal{D}$
for the optimal linear estimation of the functional $A\vec{\xi}$ if the
following relation holds true:

$\Delta(f_{0})=\Delta(h_{\tau}(f_{0});f_{0})=\max_{f\in\mathcal{D}}\Delta(h_{\tau}(f);f).$

######  Definition 3.2.

For a given class of spectral densities $\mathcal{D}$ the spectral
characteristic $h^{0}(\lambda)$ of the optimal linear estimate of the
functional $A\xi$ is called minimax-robust if the following conditions are
satisfied

$h^{0}(\lambda)\in H_{\mathcal{D}}=\bigcap_{f\in\mathcal{D}}L_{2}^{0-}(f),$
$\min_{h\in
H_{\mathcal{D}}}\max_{f\in\mathcal{D}}\Delta(h;f)=\sup_{f\in\mathcal{D}}\Delta(h^{0};f).$

Taking into account the introduced definitions and the derived relations we
can verify that the following lemmas hold true.

######  Lemma 3.1.

The spectral density $f_{0}(\lambda)\in\mathcal{D}$ satisfying the minimality
condition (18) is the least favourable density in the class $\mathcal{D}$ for
the optimal linear extrapolation of the functional $A\xi$ based on
observations of the process $\xi(t)$ on the interval $t<0$ if the operator
$\mathbf{F}_{\tau}^{0}$ defined by the Fourier coefficients of the function

$\lambda^{2d}|1-e^{i\lambda\tau}|^{-2d}f_{0}^{-1}(\lambda),$ (33)

determines a solution to the constrained optimization problem

$\max_{f\in\mathcal{D}}\left(\left\langle
D^{\tau}\mathbf{a},\mathbf{F}_{\tau}^{-1}D^{\tau}\mathbf{a}\right\rangle\right)=\left\langle
D^{\tau}\mathbf{a},(\mathbf{F}^{0}_{\tau})^{-1}D^{\tau}\mathbf{a}\right\rangle.$
(34)

The minimax spectral characteristic $h^{0}=h_{\tau}(f^{0})$ is calculated by
formula (22) if $h_{\tau}(f^{0})\in H_{\mathcal{D}}$.

For more detailed analysis of properties of the least favorable spectral
densities and the minimax-robust spectral characteristics, we observe that the
minimax spectral characteristic $h^{0}$ and the least favourable spectral
density $f_{0}$ form a saddle point of the function $\Delta(h;f)$ on the set
$H_{\mathcal{D}}\times\mathcal{D}$ [21]. The saddle point inequalities

$\Delta(h;f_{0})\geq\Delta(h^{0};f_{0})\geq\Delta(h^{0};f)\quad\forall
f\in\mathcal{D},\forall h\in H_{\mathcal{D}}$

hold true if $h^{0}=h_{\tau}(f_{0})$, $h_{\tau}(f_{0})\in H_{\mathcal{D}}$ and
$f_{0}$ is a solution of the constrained optimization problem

$\widetilde{\Delta}(f)=-\Delta(h_{\tau}(f_{0});f)\to\inf,\quad
f\in\mathcal{D},$ (35)

where the functional $\Delta(h_{\tau}(f_{0});f)$ is calculated by the formula

$\Delta(h_{\tau}(f_{0});f)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{(-i\lambda)^{d}}{(1-e^{i\lambda\tau})^{d}}(\vec{C}_{\tau}^{0}(e^{i\lambda}))^{\top}f_{0}^{-1}(\lambda)f(\lambda)f_{0}^{-1}(\lambda)\overline{\vec{C}_{\tau}^{0}(e^{i\lambda})}\frac{(i\lambda)^{d}}{(1-e^{-i\lambda\tau})^{d}}d\lambda$

where

$\vec{C}_{\tau}^{0}(e^{i\lambda})=\sum_{j=0}^{\infty}((\mathbf{F}^{0}_{\tau})^{-1}D^{\tau}\mathbf{a})_{j}e^{i\lambda
j}$

.

The constrained optimization problem (35) is equivalent to the unconstrained
optimization problem

$\Delta_{\mathcal{D}}(f)=\widetilde{\Delta}(f)+\delta(f|\mathcal{D})\to\inf,$

where $\delta(f|\mathcal{D})$ is the indicator function of the set
$\mathcal{D}$, namely $\delta(f|\mathcal{D})=0$ if $f\in\mathcal{D}$ and
$\delta(f|\mathcal{D})=+\infty$ if $f\notin\mathcal{D}$. A solution $f_{0}$ of
the unconstrained optimization problem is characterized by the condition
$0\in\partial\Delta_{\mathcal{D}}(f_{0})$, which is the necessary and
sufficient condition under which the point $f_{0}$ belongs to the set of
minimums of the convex functional $\Delta_{\mathcal{D}}(f)$ [5, 21, 29]. This
condition makes it possible to find the least favourable spectral densities in
some special classes of spectral densities $D$.

The form of the functional $\widetilde{\Delta}(f)$ allows us to apply the
Lagrange method of indefinite multipliers for investigating the constrained
optimization problem (35).

###  3.1 Least favorable spectral density in classes $\mathcal{D}_{0}$

Consider the extrapolation problem for the functional $A{\xi}$ which depends
on unobserved values of a process $\xi(t)$ with periodically stationary
increments based on observations of the process at points $t<0$ under the
condition that the sets of admissible spectral densities
$\mathcal{D}_{0}^{k},k=1,2,3,4$ are defined as follows:

$\displaystyle\mathcal{D}_{0}^{1}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f(\lambda)d\lambda=P\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{2}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}{\rm{Tr}}\,[f(\lambda)]d\lambda=p\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{3}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{|\lambda|^{2d}}f_{kk}(\lambda)d\lambda=p_{k},k=1,2,3,\ldots\right.\bigg{\\}},$
$\displaystyle\mathcal{D}_{0}^{4}$ $\displaystyle=$
$\displaystyle\bigg{\\{}f(\lambda)\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{|1-e^{i\lambda\tau}|^{2d}}{|\lambda|^{2d}}\left\langle
B_{1},f(\lambda)\right\rangle d\lambda=p\right.\bigg{\\}},$

where $p,p_{k},k=1,2,3,\ldots$ are given numbers, $P,B_{1},$ are given
positive-definite Hermitian matrices.

From the condition $0\in\partial\Delta_{\mathcal{D}}(f_{0})$ we find the
following equations which determine the least favourable spectral densities
for these given sets of admissible spectral densities.

For the set $\mathcal{D}_{0}^{1}$ of admissible spectral densities, we have
equation

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)\vec{\alpha}\cdot\vec{\alpha}^{*}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(36)

where $\vec{\alpha}$ is a vector of Lagrange multipliers.

For the set $\mathcal{D}_{0}^{2}$ of admissible spectral densities, we have
equation

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\alpha^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)^{2},$
(37)

where $\alpha^{2}$ is a Lagrange multiplier.

For the set $\mathcal{D}_{0}^{3}$ of admissible spectral densities, we have
equation

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)\left\\{\alpha_{k}^{2}\delta_{kl}\right\\}_{k,l=1}^{\infty}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(38)

where $\alpha_{k}^{2}$ are Lagrange multipliers, $\delta_{kl}$ are Kronecker
symbols.

For the set $\mathcal{D}_{0}^{4}$ of admissible spectral densities, we have
equation

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\alpha^{2}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)B_{1}^{\top}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(39)

where $\alpha^{2}$ is a Lagrange multiplier.

The following theorem holds true.

######  Theorem 3.1.

Let minimality condition (18) hold true. The least favorable spectral
densities $f_{0}(\lambda)$ in the classes $\mathcal{D}_{0}^{k}$, $k=1,2,3,4$,
for the optimal linear extrapolation of the functional $A{\xi}$ from
observations of the process ${\xi}(t)$ on the interval $t<0$ are determined by
the equations (36), (37), (38), (39), the constrained optimization problem
(34) and restrictions on densities from the corresponding classes
$\mathcal{D}_{0}^{k}$, $k=1,2,3,4$. The minimax-robust spectral characteristic
of the optimal estimate of the functional $A{\xi}$ is determined by the
formula (22).

###  3.2 Least favorable spectral density in classes $\mathcal{D}_{1\delta}$

Consider the extrapolation problem for the functional $A{\xi}$ which depends
on unobserved values of the process $\xi(t)$ with periodically stationary
increments based on its observations at points $t<0$ under the condition that
the sets of admissible spectral densities ${D_{1\delta}^{k}},k=1,2,3,4$ are
defined as follows:

$\displaystyle\mathcal{D}_{1\delta}^{1}$ $\displaystyle=$
$\displaystyle\left\\{f(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f_{ij}(\lambda)-f_{ij}^{1}(\lambda)\right|d\lambda\leq\delta_{i}^{j},i,j=\overline{1,\infty}\right\\}.$
$\displaystyle\mathcal{D}_{1\delta}^{2}$ $\displaystyle=$
$\displaystyle\left\\{f(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\rm{Tr}}(f(\lambda)-f_{1}(\lambda))\right|d\lambda\leq\delta\right\\};$
$\displaystyle\mathcal{D}_{1\delta}^{3}$ $\displaystyle=$
$\displaystyle\left\\{f(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f_{kk}(\lambda)-f_{kk}^{1}(\lambda)\right|d\lambda\leq\delta_{k},k=\overline{1,\infty}\right\\};$
$\displaystyle\mathcal{D}_{1\delta}^{4}$ $\displaystyle=$
$\displaystyle\left\\{f(\lambda)\biggl{|}\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},f(\lambda)-f_{1}(\lambda)\right\rangle\right|d\lambda\leq\delta\right\\};$

Here $f_{1}(\lambda)=\\{f_{ij}^{1}(\lambda)\\}_{i,j=1}^{\infty}$ is a fixed
spectral density, $B_{2}$ is a given positive-definite Hermitian matrix,
$\delta,\delta_{k},k=\overline{1,\infty}$,
$\delta_{i}^{j},i,j=\overline{1,\infty}$, are given numbers.

From the condition $0\in\partial\Delta_{\mathcal{D}}(f_{0})$, we find the
following equations which determine the least favourable spectral densities
for these given sets of admissible spectral densities.

For the set $\mathcal{D}_{1\delta}^{1}$ of admissible spectral densities, we
have equations

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)\left\\{\beta_{ij}\gamma_{ij}(\lambda)\right\\}_{i,j=1}^{\infty}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(40)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f_{ij}^{0}(\lambda)-f_{ij}^{1}(\lambda)\right|d\lambda=\delta_{i}^{j},$
(41)

where $\beta_{ij}$ are Lagrange multipliers, functions
$\left|\gamma_{ij}(\lambda)\right|\leq 1$ and

$\gamma_{ij}(\lambda)=\frac{f_{ij}^{0}(\lambda)-f_{ij}^{1}(\lambda)}{\left|f_{ij}^{0}(\lambda)-f_{ij}^{1}(\lambda)\right|}:\;f_{ij}^{0}(\lambda)-f_{ij}^{1}(\lambda)\neq
0,\;i,j=\overline{1,\infty}.$

For the set $\mathcal{D}_{1\delta}^{2}$ of admissible spectral densities, we
have equations

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\beta^{2}\gamma_{2}(\lambda)\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)^{2},$
(42)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|{\mathrm{Tr}}\,(f_{0}(\lambda)-f_{1}(\lambda))\right|d\lambda=\delta,$
(43)

where $\beta^{2}$ are Lagrange multipliers, the function
$\left|\gamma_{2}(\lambda)\right|\leq 1$ and

$\gamma_{2}(\lambda)={\mathrm{sign}}\;({\mathrm{Tr}}\,(f_{0}(\lambda)-f_{1}(\lambda))):\;{\mathrm{Tr}}\,(f_{0}(\lambda)-f_{1}(\lambda))\neq
0.$

For the set $\mathcal{D}_{1\delta}^{3}$ of admissible spectral densities, we
have equations

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)\left\\{\beta_{k}^{2}\gamma^{2}_{k}(\lambda)\delta_{kl}\right\\}_{k,l=1}^{\infty}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(44)
$\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|f_{kk}^{0}(\lambda)-f_{kk}^{1}(\lambda)\right|d\lambda=\delta_{k},$
(45)

where $\beta_{k}^{2}$ are Lagrange multipliers, $\delta_{kl}$ are Kronecker
symbols, functions $\left|\gamma^{2}_{k}(\lambda)\right|\leq 1$ and

$\gamma_{k}^{2}(\lambda)={\mathrm{sign}}\;(f_{kk}^{0}(\lambda)-f_{kk}^{1}(\lambda)):\;f_{kk}^{0}(\lambda)-f_{kk}^{1}(\lambda)\neq
0,\;k=\overline{1,\infty}.$

For the set $\mathcal{D}_{1\delta}^{4}$ of admissible spectral densities, we
have equations

$\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)\left(\vec{C}_{\tau}^{0}(e^{i\lambda})\right)^{*}=\beta^{2}\gamma_{2}^{\prime}(\lambda)\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right)B_{2}^{\top}\left(\frac{|1-e^{i\lambda\tau}|^{2d}}{\lambda^{2d}}f_{0}(\lambda)\right),$
(46) $\frac{1}{2\pi}\int_{-\pi}^{\pi}\left|\left\langle
B_{2},f_{0}(\lambda)-f_{1}(\lambda)\right\rangle\right|d\lambda=\delta,$ (47)

where $\beta^{2}$ are Lagrange multipliers, function
$\left|\gamma_{2}^{\prime}(\lambda)\right|\leq 1$ and

$\gamma_{2}^{\prime}(\lambda)={\mathrm{sign}}\;\left\langle
B_{2},f_{0}(\lambda)-f_{1}(\lambda)\right\rangle:\;\left\langle
B_{2},f_{0}(\lambda)-f_{1}(\lambda)\right\rangle\neq 0.$

The derived results are summarized in the following theorem.

######  Theorem 3.2.

Let minimality condition (18) hold true. The least favorable spectral
densities $f_{0}(\lambda)=\\{f_{ij}^{0}(\lambda)\\}_{i,j=1}^{\infty}$ in the
classes $\mathcal{D}_{1\delta}^{k},k=1,2,3,4$ for the optimal linear
extrapolation of the functional $A{\xi}$ from observations of the process
${\xi}(t)$ at points $t<0$ are determined by equations (40) – (41), (42) –
(43), (44) – (45), (46) – (47), respectively, the constrained optimization
problem (34) and restrictions on the densities from the corresponding classes
$\mathcal{D}_{1\delta}^{k},k=1,2,3,4$. The minimax-robust spectral
characteristic of the optimal estimate of the functional $A{\xi}$ is
determined by the formula (22).

##  4 Conclusions

In this article, we dealt with continuous time stochastic processes with
periodically correlated $d$-th increments. These stochastic processes form a
class of non-stationary stochastic processes that combine periodic structure
of covariation functions of processes as well as integrating one.

We derived solutions of the problem of estimation of the linear functionals
constructed from the unobserved values of a continuous time stochastic process
with periodically correlated $d$-th increments. Estimates are based on
observations of the process at points $t<0$. We obtained the estimates by
representing the process under investigation as a vector-valued sequence with
stationary increments. Based on the solutions for these type of sequences, we
solved the corresponding problem for the defined class of continuous time
stochastic processes. The problem is investigated in the case of spectral
certainty, where spectral densities of sequences are exactly known. In this
case we propose an approach based on the Hilbert space projection method. We
derive formulas for calculating the spectral characteristics and the mean-
square errors of the optimal estimates of the functionals. In the case of
spectral uncertainty where the spectral densities are not exactly known while,
instead, some sets of admissible spectral densities are specified, the
minimax-robust method is applied. We propose a representation of the mean
square error in the form of a linear functional in $L_{1}$ with respect to
spectral densities, which allows us to solve the corresponding constrained
optimization problem and describe the minimax-robust estimates of the
functionals. Formulas that determine the least favorable spectral densities
and minimax-robust spectral characteristics of the optimal linear estimates of
the functionals are derived for a wide list of specific classes of admissible
spectral densities.

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