Robust Extrapolation Problem for Stochastic
Processes with Stationary Increments
Maksym Luz
,
Mikhail Moklyachuk
∗,
Department of Probability Theory, Statistics and Actuarial Mathematics,
Taras Shevchenko National University of Kyiv, Kyiv 01601, Ukraine
∗Corresponding Author: [email protected]
Copyright c⃝2014 Horizon Research Publishing All rights reserved.
Abstract
The problem of optimal estimation of the linear functionals Aξ = ∫0∞a(t)ξ(t)dt andATξ =
∫T
0 a(t)ξ(t)dt depending on the unknown values
of stochastic process ξ(t), t ∈ R, with stationary
nth increments from observations of the process at points t < 0 is considered. Formulas for calculating the mean square error and the spectral characteristic of optimal linear estimates of the functionals are derived in the case where the spectral density of the process is exactly known. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the process is not exactly known, but a set of admissible spectral densities is given.
Keywords
Stochastic process with stationary increments; minimax-robust estimate; mean square error; least favorable spectral density; minimax-robust spectral characteristic1
Introduction
Estimation of unknown values of stochastic processes is an important part of the theory of stochastic pro-cesses. Effective methods of solution of the linear extrap-olation, interpolation and filtering problems for station-ary stochastic processes were developed by Kolmogorov [1], Wiener[2], Yaglom [3, 4]. The further results one can find in book by Rozanov [5]. Yaglom [6, 7] de-veloped a theory of non-stationary processes whose in-crements of order µ ̸= 0 define a stationary process. The spectral representation for stationary increments and canonical factorization for spectral densities were re-ceived, the problem of linear extrapolation of unknown value of stationary stochastic increment from observa-tion of the process was solved. Further results for such stochastic processes were presented by Pinsker [8], Ya-glom and Pinsker [9]. See books by YaYa-glom [3, 4] for more relative results and references.
The mean square optimal estimation problems for
problem for the linear functionalANξ=
∑N
k=0a(k)ξ(k)
which depends on unknown (missed) values of a stochas-tic sequence ξ(m) with stationarynth increments from observations of the sequence with and without noise.
This article is dedicated to the mean square optimal estimates of the linear functionals
Aξ=
∫ ∞
0
a(t)ξ(t)dt, ATξ=
∫ T
0
a(t)ξ(t)dt
which depend on the unknown values of a stochastic pro-cessξ(t) with stationarynth increments. Estimates are based on observations of the processξ(t) at pointst <0. The estimation problem for processes with stationary in-crements is solved in the case of spectral certainty where the spectral density of the process is known as well as in the case of spectral uncertainty where the spectral density of the process is not known, but a set of admis-sible spectral densities is given. Formulas are derived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimates of functionalsAξandATξin the case of spectral certainty
where spectral density of the process is known. Formu-las that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case of spectral uncertainty for concrete classes of admissible spectral densities.
2
Stationary
stochastic
incre-ment process. Spectral
repre-sentation
Definition 1 For a given stochastic process {ξ(t), t ∈ R} a process
ξ(n)(t, τ) = (1−Bτ)nξ(t) = n
∑
l=0
(−1)lCnlξ(t−lτ), (1)
where Bτ is a backward shift operator with step τ ∈R,
such that Bτξ(t) =ξ(t−τ), is called the stochasticnth
increment with step τ∈R.
For the stochasticnth increment processξ(n)(t, τ) the
following relations hold true:
ξ(n)(t,−τ) = (−1)nξ(n)(t+nτ, τ), (2)
ξ(n)(t, kτ) =∑(k−1)n
l=0 Alξ
(n)(t−lτ, τ), ∀k∈N, (3)
where coefficients {Al, l = 0,1,2, . . . ,(k−1)n} are
de-termined by the representation
(1 +x+. . .+xk−1)n=
(k∑−1)n
l=0
Alxl.
Definition 2 The stochastic nth increment process
ξ(n)(t, τ)generated by stochastic process{ξ(t), t∈R} is
wide sense stationary if the mathematical expectations
Eξ(n)(t0, τ) =c(n)(τ),
Eξ(n)(t0+t, τ1)ξ(n)(t0, τ2) =D(n)(t, τ1, τ2)
exist for all t0, τ, t, τ1, τ2 and do not depend ont0. The
functionc(n)(τ)is called the mean value of thenth
incre-ment and the function D(n)(t, τ
1, τ2)is called the
struc-tural function of the stationary nth increment (or the structural function ofnth order of the stochastic process
{ξ(t), t∈R}).
The stochastic process{ξ(t), t∈R}which determines the stationary nth increment process ξ(n)(t, τ) by for-mula (1) is called stochastic process with stationary nth increments.
Theorem 1 The mean value c(n)(τ)and the structural
function D(n)(m, τ
1, τ2) of a stochastic stationary nth
increment processξ(n)(t, τ)can be represented in the
fol-lowing forms
c(n)(τ) =cτn, (4)
D(n)(t, τ1, τ2) =
=
∫ ∞
−∞
eiλt(1−e−iτ1λ)n(1−eiτ2λ)n(1 +λ 2)n
λ2n dF(λ), (5)
where c is a constant, F(λ) is a left-continuous nonde-creasing bounded function with F(−∞) = 0. The con-stant c and the function F(λ) are determined uniquely by the increment processξ(n)(t, τ).
From the other hand, a functionc(n)(τ)which has the
form(4)with a constantcand a functionD(n)(m, τ 1, τ2)
which has the form(5) with a function F(λ)which sat-isfies the indicated conditions are the mean value and the structural function of some stochastic stationary nth increment process ξ(n)(t, τ).
Using representation (5) of the structural function of the stationary nth increment processξ(n)(t, τ) and the
Karhunen theorem (see Karhunen [31]), we get the fol-lowing spectral representation of the stationary nth in-crement processξ(n)(t, τ):
ξ(n)(t, τ) =
∫ ∞
−∞
eitλ(1−e−iλτ)n(1 +iλ)
n
(iλ)n dZ(λ), (6)
where Z(λ) is an orthogonal stochastic measure on R
connected with thespectral functionF(λ) by the relation
EZ(A1)Z(A2) =F(A1∩A2)<∞. (7)
Denote by H(ξ(n)) the subspace of the Hilbert space
H =L2(Ω,F,P) of the second order stochastic variables
which is generated by elements{ξ(n)(t, τ) :t, τ∈R}and
letHt(ξ(n)),t∈R, be a subspace ofH(ξ(n)) generated
by elements {ξ(n)(u, τ) : u≤t, τ > 0}. Let S(ξ(n)) be
defined by the relationship
S(ξ(n)) = ∩
t∈R
Ht(ξ(n)).
Since the spaceS(ξ(n)) is a subspace of the Hilbert space
H(ξ(n)), the spaceH(ξ(n)) admits the decomposition
H(ξ(n)) =S(ξ(n))⊕R(ξ(n)),
where R(ξ(n)) is an orthogonal complement of the
sub-spaceS(ξ(n)) in the spaceH(ξ(n)).
In the following we will consider increments ξ(n)(t, τ)
Definition 3 A stationary nth increment process
ξ(n)(t, τ) is called regular if H(ξ(n)) = R(ξ(n)). It is
called singular ifH(ξ(n)) =S(ξ(n)).
Theorem 2 A wide-sense stationary stochastic incre-ment process ξ(n)(t, τ) admits a unique representation in the form
ξ(n)(t, τ) =ξr(n)(t, τ) +ξs(n)(t, τ), (8)
where{ξr(n)(t, τ) :t∈R} is a regular increment process
and {ξs(n)(t, τ) : t ∈ R} is a singular increment
pro-cess. Moreover, the increment processes ξr(n)(z, τ) and
ξs(n)(t, τ)are orthogonal for allt, z∈R.
Components of the representation (8) are constructed in the following way:
ξs(n)(t, τ) =E[ξ(n)(t, τ)|S(ξ(n))], ξ(rn)(t, τ) =ξ(n)(t, τ)−ξs(n)(t, τ).
Let{η(t) :t ∈R} be a stochastic process with inde-pendent increments such that E|η(t)−η(s)|2 = |t−s|
and Hz(ξ(n)) = Hz(η) for all z ∈ R, where the
sub-space Hz(η) of the space H is generated by values
{η(u) : u≤ z} of the process η(t). Defined stochastic process is calledinnovation process.
Theorem 3 A stochastic stationary increment process
ξ(n)(t, τ)is regular if and only if there exists an
innova-tion process {η(t) :t ∈ R} and a function {φ(n)(t, τ) :
t≥0},∫0∞|φ(n)(t, τ)|2dt <∞, such that
ξ(n)(t, τ) =
∫ ∞
0
φ(n)(u, τ)dηu(t−u), (9)
Conclusion 1 Using theorems 2 and 3 one can
con-clude that a wide-sense stationary stochastic increment process admits a unique representation in the form
ξ(n)(t, τ) =ξ(sn)(t, τ) +
∫ ∞
0
φ(n)(u, τ)dηu(t−u), (10)
where ∫0∞|φ(n)(t, τ)|2dt < ∞ and η(t), t ∈ R, is an
innovation process.
Let the stationarynth increment processξ(n)(t, τ)
ad-mit the canonical representation (9). In this case the spectral functionF(λ) of the stationary increment pro-cessξ(n)(t, τ) has spectral densityf(λ) which admits the
canonical factorization
f(λ) =|Φ(λ)|2, Φ(λ) =
∫ ∞
0
e−iλtφ(t)dt. (11) Let us define
Φτ(λ) =
∫ ∞
0
e−iλtφ(n)(t, τ)dt=
∫ ∞
0
e−iλtφτ(t)dt,
whereφτ(t) =φ(n)(t, τ) is the function from the
repre-sentation (9). Defined function Φτ(λ), which is a Fourier
transform of the functionφ(n)(t, τ), is related with
spec-tral densityf(λ) of the stochastic process ξ(n)(t, τ) by
relations
|Φτ(λ)|
2
=|1−e
−iλτ|2n(1 +λ2)n
λ2n f(λ), (12)
Φτ(λ) =
(1−e−iλτ)n(1 +iλ)n
(iλ)n Φ(λ). (13)
The one-sided moving average representation (9) is used for finding the optimal mean square estimate of the unknown values of a process ξ(t) based on observations of the process at pointst <0.
3
Hilbert
space
projection
method
of
extrapolation
of
linear functionals
Let a stochastic process {ξ(t), t∈R} defines nth in-crement process ξ(n)(t, τ) with an absolutely
continu-ous spectral function F(λ) which has spectral density
f(λ). Without loss of generality we will assume that the mean value of the increment processξ(n)(t, τ) equals to 0. Let the stationary increment process ξ(n)(t, τ) ad-mit the one-sided moving average representation (9) and the spectral density f(λ) admits the canonical factor-ization (11). Consider the case where the step τ > 0. Let the values of the process ξ(t) be known fort < 0. The problem is to find the mean square optimal lin-ear estimates of functionals ATξ =
∫T
0 a(t)ξ(t)dt and
Aξ = ∫0∞a(t)ξ(t)dt which depend on unknown values
ξ(t),t≥0.
In order to solve the stated problem we will present the process ξ(t), t ≥ 0, as a sum of its increments
ξ(n)(t, τ),t≥0,τ >0, and its initial valuesξ0.
Partic-ularly, whenτ∗> t∗≥0, a relation
ξ(t∗) =ξ(n)(t∗, τ∗) +
n
∑
l=1
(−1)lCnlξ(t∗−lτ∗)
comes from (1), where ξ0 = {ξ(t∗ − lτ∗) : l = 1,2, . . . , n} ⊂ {ξ(t) : t ≤ 0} are known observations. The following lema describes a representation of the functional Aξ that depends on known initial values of the processξ(t) and its incrementsξ(n)(t, τ) fort≥0 in
the case of arbitrary stepτ >0.
Lema 1 A linear functionalAξ=∫0∞a(t)ξ(t)dtadmits a representation Aξ=Bξ−V ξ, where
Bξ=
∫ ∞
0
bτ(t)ξ(n)(t, τ)dt, V ξ=
∫ 0
−τ n
vτ(t)ξ(t)dt,
vτ(t) = n
∑
l=[−t τ]
′
(−1)lCnlbτ(t+lτ), t∈[−τ n; 0), (14)
bτ(t) =
∞
∑
k=0
a(t+τ k)d(k) =Dτa(t), t≥0, (15) where[x]′ denotes the least integer number among num-bers that are greater or equal to x, {d(k) : k ≥ 0} are coefficients determined by the relation ∑∞k=0d(k)xk =
(∑∞
j=0x
j)n, Dτ is a linear transformation which acts
on an arbitrary functionx(t),t >0, by formula
Dτx(t) =
∞
∑
k=0
Proof. From (1) we can obtain the formal equation
ξ(t) = 1 (1−Bτ)n
ξ(n)(t, τ) = ∞
∑
j=0
d(j)ξ(n)(t−τ j, τ),
(17) and the relations
∫ ∞
0
a(t)ξ(t)dt=−
∫ 0
−τ n
vτ(t)ξ(t)dt+
+
∫ ∞
0 (∞
∑
k=0
a(t+τ k)d(k)
)
ξ(n)(t, τ)dt,
∫ ∞
0
bτ(k)ξ(n)(t, τ)dt=
=
∫ 0
−τ n
ξ(t)
n
∑
l=[−t τ]
′
(−1)lCnkbτ(t+τ l)dt+
+
∫ ∞
0
ξ(t)
n
∑
l=0
(−1)lCnkbτ(t+τ l)dt.
From two last relations we can get the representation of the functionalAξand relations (15), (14).
Conclusion 2 The linear functionalATξadmits a
rep-resentationATξ=BTξ−VTξ, where
BTξ=
∫ T
0
bτ,T(t)ξ(n)(t, τ)dt, VTξ=
∫ 0
−τ n
vτ,T(t)ξ(t)dt,
and functions bτ,T(t), t ∈ [0, T], and vτ,T(t), t ∈
[−τ n; 0), are defined by formulas (14) and(15) respec-tively under the conditiona(t) = 0fort > T.
We will suppose that the following restrictions on the functionbτ(t) hold true
∫ ∞
0
|bτ(t)|dt <∞,
∫ ∞
0
t|bτ(t)|2dt <∞. (18)
Under these conditions the functionalBξhas the second moment and the operatorBτ defined below is compact.
Since the functions a(t) and bτ(t) are related by (15),
the following conditions hold true
∫ ∞
0
|Dτa(t)|dt <∞,
∫ ∞
0
t|Dτa(t)|2dt <∞. (19)
Let Aξb denote the mean square optimal linear es-timate of the functional Aξ from observations of the process ξ(t) for t < 0 and let Bξb denote the mean square optimal linear estimate of the functionalBξfrom observations of the stochastic nth increment process
ξ(n)(t, τ) for t < 0. Let ∆(f,Aξb ) := E|Aξ−Aξb |2 de-note the mean square error of the estimate Aξb and let ∆(f,Bξb ) :=E|Bξ−Bξb |2 denote the mean square error
of the estimateBξb . Since valuesξ(t) fort∈[−τ n; 0) are known, the following equality comes from lema 1:
b
Aξ=Bξb −V ξ. (20) Thus
∆(f,Aξb ) =E|Aξ−Aξb |2=E|Aξ+V ξ−Bξb |2=
=E|Bξ−Bξb |2= ∆(f,Bξb ).
Denote by L02−(f) the subspace of the Hilbert space
L2(f) generated by the set of functions
h(λ) = (1−e−iλτ)n(1 +iλ)
n
(iλ)n
∫ ∞
0
h(t)e−iλtdt.
Every linear estimate Bξb of the functionalBξ admits a representation
b
Bξ=
∫ ∞
−∞
hτ(λ)(1−e−iλτ)n
(1 +iλ)n
(iλ)n dZ(λ), (21)
wherehτ(λ) is the spectral characteristic of the estimate
b
Bξ. The spectral characteristic of the optimal estimate provides the minimum value of the mean square error ∆(f,Bξb ).
Let the stochastic increment ξ(n)(t, τ) admits the
canonical representation (9). Then the functional Bξ
can be presented by formulas
Bξ=
∫ ∞
0 ∫ ∞
0
bτ(t)φ(u, τ)dηu(t−u)dt=
=
∫ 0
−∞
∫ ∞
0
bτ(t)φ(t−u, τ)dtdη(u)+
+
∫ ∞
0 ∫ ∞
u
bτ(t)φ(t−u, τ)dtdη(u).
As the relation H0(ξ(n)) =H0(η) holds true and
incre-ments of the process η(t) are orthogonal, the optimal estimateBξb of the functionalBξ is calculated as
b
Bξ=
∫ 0
−∞
∫ ∞
0
bτ(t)φτ(t−u)dtdη(u) =
+
∫ 0
−∞
Bφτ(t−u)dη(u) =
∫ ∞
−∞
B∗φτ(λ)dη∗(λ), (22)
whereB∗φτ(λ) andη∗(λ) are inverse Fourier transforms
of the function Bφτ(u) =
∫∞
0 bτ(t)φτ(t−u)dt, u < 0,
and the process η(u) respectively. Yaglom[6] showed that
η∗(λ) =
∫ λ
−∞
dZ(p)
Φ(p). (23)
So we need to findB∗φτ(λ).
B∗φτ(λ) =
∫ 0
−∞
eiλs
∫ ∞
0
bτ(t)φτ(t−s)dtds=
=
∫ ∞
0
eiλtbτ(t)
∫ ∞
0
e−iλ(s+t)φτ(t+s)dsdt=
=
∫ ∞
0
eiλtbτ(t)
∫ ∞
t
e−iλzφτ(z)dzdt=
=Bτ(λ)Φτ(λ)−
∫ ∞
0
eiλtbτ(t)
∫ t
0
e−iλzφτ(z)dzdt=
=Bτ(λ)Φτ(λ)−
∫ ∞
0
eiλy
∫ ∞
0
bτ(y+z)φτ(z)dzdy. (24)
Substituting the expressions (23) and (24) in (22) and using (13) one can obtain the following formulas for cul-culating the spectral characteristic of the optimal esti-mate Bξb :
Bτ(λ) =
∫ ∞
0
bτ(t)eiλtdt, rτ(λ) =
∫ ∞
0
eiλt(Bτφτ)(t)dt,
whereBτ is a linear operator inL
2([0,∞)) space which
defined by the relation (Bτφτ)(t) =
∫ ∞
0
bτ(t+u)φτ(u)du.
Hereφτ(u) =φ(n)(u, τ) is the function from the moving
average representation (9). The operatorBτ is compact providing (18).
The value of the mean square error ∆(f,Bξb ) can be calculated by the formula
∆(f,Bξb ) =E|Bξ−Bξb |2= = 1
2π
∫ ∞
−∞|
rτ(λ)|2dλ=||Bτφτ||2. (26)
Theorem 4 Let a stochastic process{ξ(t), m∈Z} de-termine a stationary stochastic nth increment process
ξ(n)(t, τ) with absolutely continuous spectral function
F(λ)and spectral densityf(λ)which admits the canon-ical factorization (11). The optimal linear estimate Bξb
of the functional Bξ which depends on the unobserved values {ξ(n)(t, τ) : t ≥0}, τ > 0, from observations of
the process ξ(t) for t <0 can be calculated by formula (21). The spectral characteristic hµ(λ) of the optimal
linear estimate Bξb can be calculated by formula (25). The value of the mean square error∆(f,Bξb )can be cal-culated by formula (26).
Using Theorem 4 and representation (8), we can obtain the optimal estimate of an unobserved value
ξ(n)(u, τ),τ > 0, at the pointu≥0 from observations
of the process ξ(t) for t <0. The singular component
ξs(n)(u, τ) of representation (8) of the process has
error-less estimate. We will use formula (25) to obtain the spectral characteristic hu,τ(λ) of the optimal estimate
b
ξ(n)(t, τ) of the regular componentξ(n)
r (u, τ) of the
pro-cess. Consider a functionBτ(λ) =eiλu. It follows from
the derived formulas that the spectral characteristic of the estimate
b
ξ(n)(u, τ) =ξ(sn)(u, τ)+ +
∫ ∞
−∞
hu,τ(λ)(1−e−iλτ)n
(1 +iλ)n
(iλ)n dZ(λ) (27)
can be calculated by the formula
hu,τ(λ) =eiλu−Φ−τ1(λ)
∫ u
0
φτ(y)e−iλydy. (28)
The value of the mean square error can be calculated by the formula
∆(f,ξb(n)(u, τ)) = 1 2π
∫ u
0
|φτ(y)|2dy. (29)
The following statement holds true.
Conclusion 3 The optimal linear estimateξb(n)(u, τ)of
the valueξ(n)(u, τ),τ >0, at the point u≥0 of the
in-crement process ξ(n)(t, τ) from observations of the
pro-cessξ(t),t <0, can be calculated by formula (27). The spectral characteristichu,τ(λ)of the optimal linear
esti-mate ξb(n)(u, τ) can be calculated by formula (28). The
value of mean square error ∆(f,ξb(n)(u, τ))of the
opti-mal linear estimate can be calculated by formula (29).
Making use relation (20) we can find the optimal es-timateAξb of the functionalAξfrom observations of the processξ(t) fort <0. These estimate can be presented in the following form:
b
Aξ=−
∫ 0
−τ n
vτ(t)ξ(t)dt+
+
∫ ∞
−∞
h(τa)(λ)(1−e−iλτ)n(1 +iλ)
n
(iλ)n dZ(λ), (30)
where the function vτ(t), t = [−τ n,0), is defined by
relation (14). Using the relationship (15) between the functions a(t) and bτ(t),t ≥0, we obtain the following
equation:
(Bτφτ)(t) =
∫ ∞
0
Dτa(t+u)φ(u, τ)du=
=Dτ
∫ ∞
0
a(t+u)φ(u, τ)du=Dτ(Aφτ)(t),
where the linear operator A is defined by the function
a(t),t≥0, in the following way: (Aφτ)(t) =
∫ ∞
0
a(t+u)φτ(u)du.
Thus the spectral characteristic and the value of the mean square error of the optimal estimate Aξb can be calculated by the formulas
h(τa)(λ) =Aτ(λ)−r(τa)(λ)Φ−
1
τ (λ), (31)
Aτ(λ) =
∫ ∞
0
Dτa(t)eiλtdt,
r(τa)(λ) =
∫ ∞
0
Dτ(Aφτ)(t)eiλtdt. (32)
∆(f,Aξb ) =E|Aξ−Aξb |2= = 1
2π
∫ ∞
−∞|
r(τa)(λ)|2dλ=||DτAφτ||2. (33)
The following theorem holds true.
Theorem 5 Let a stochastic process {ξ(t), t ∈ R} de-termine a stationary stochastic nth increment process
ξ(n)(t, τ) with absolutely continuous spectral function
F(λ)and spectral densityf(λ)which admits the canon-ical factorization (11). The optimal linear estimate Aξb
of the functional Aξ of unobserved values ξ(t), t ≥ 0, from observations of the process ξ(t) for t < 0 can be calculated by formula (30). The spectral characteristic
h(µa)(λ) of the optimal linear estimateAξb can be
calcu-lated by formula (31). The value of the mean square error∆(f,Aξb )of the optimal linear estimate can be cal-culated by formula (33).
Consider now the problem of the mean square optimal estimation of the functionalATξ. The optimal estimate
of the functional can be calculated by formula
b
ATξ=−
∫ 0
−τ n
vτ,T(t)ξ(t)dt+
+
∫ ∞
−∞
h(τ,Ta)(λ)(1−e−iλτ)n(1 +iλ)
n
where the function vτ,T(t), t ∈[−τ n; 0), can be
calcu-lated by formulas
vτ,T(t) =
min{[T−t τ ],n}
∑
l=[−t τ]
′
(−1)lCnlbτ,T(lτ +t), t∈[−τ n; 0),
bτ,T(t) =
[T−t τ ]
∑
k=0
a(t+τ k)d(k) =DτTa(t), t∈[0;T].
Here Dτ
T is a linear transformation which acts on an
arbitrary functionx(t),t∈[0, T], as
DτTx(t) =
[T−t τ ]
∑
k=0
x(t+τ k)d(k).
The spectral characteristic h(τ,Ta)(λ) and the value of the mean square error ∆(f,AbTξ) of the estimate AbTξ
can be calculated by formulas
h(τ,Ta)(λ) =Aτ,T(λ)−r
(a)
τ,T(λ)Φ−
1
τ (λ), (35)
Aτ,T(λ) =
∫ T
0
DτTa(t)eiλtdt,
r(τ,Ta)(λ) =
∫ T
0
DτT(ATφτ)(t)eiλtdt, (36)
whereAT is a linear operator inL2([0,∞)) space defined
by formula
(ATφτ)(t) =
∫ T−t
0
a(t+u)φτ(u)du,
and linear opperator Dτ
TATφτ in L2([0,∞)) space is
defined by formula
DτT(ATφτ)(t) =
[T−t τ ]
∑
k=0
∫ T−t−τ k
0
a(u+t+τ k)φτ(u)d(k)du;
∆(f,AbTξ) =E|ATξ−AbTξ|2=
= 1 2π
∫ ∞
−∞|
r(τ,Ta)(λ)|2dλ=||DτTATφτ||2. (37)
Consequently, the following theorem holds true.
Theorem 6 Let a stochastic process {ξ(t), t ∈ R} de-termine a stationary stochastic nth increment process
ξ(n)(t, τ) with absolutely continuous spectral function
F(λ)and spectral densityf(λ)which admits the canoni-cal factorization (11). The optimal linear estimateAbTξ
of the functional ATξ of unobserved values ξ(t),t ≥0,
from observations of the process ξ(t) for t < 0 can be calculated by formula (34). The spectral characteristic
h(τ,Ta)(λ)of the optimal linear estimateAbTξcan be
calcu-lated by formula (35). The value of mean square error ∆(f,AbTξ)can be calculated by formula (37).
Consider the case whereτ > u≥0. In this case the optimal mean square estimate of the valueξ(u) at the
point u ≥ 0 from observations ξ(t) for t < 0 can be calculated by formula
b
ξ(u) =
T
∑
l=1
(−1)l+1Cnlξ(u−lτ)+
+
∫ ∞
−∞
hu,τ(λ)(1−e−iλτ)n
(1 +iλ)n
(iλ)n dZ(λ). (38)
The spectral characteristichu,τ(λ) and the value of the
mean square error ∆(f,ξb(u)) = ∆(f,ξb(n)(u, τ)) of the
estimate ξb(u) can be calculated by formulas (28) and (29) respectively.
Consequently, the following statement holds true.
Conclusion 4 Let τ > u ≥ 0. The optimal mean
square estimate ξb(u) of the unknown value of the ele-mentξ(u)from observations of the processξ(t)fort <0 can be calculated by formula (38). The spectral charac-teristic hu,τ(λ) of the optimal linear estimate ξb(u) can
be calculated by formula (28). The value of mean square error∆(f,ξb(u))can be calculated by formula (29).
Remark 1 Using relation(12) we can find a relation-ship between functions φτ(t), t ≥ 0, and φ(t), t ≥ 0.
Since
∫ ∞
−∞
ln|1−e
−iλτ|2n(1 +λ2)n
λ2n
1 +1λ2dλ <∞
for alln≥1andτ >0, there are functionsωτ(t),t≥0,
and
Ωτ(λ) =
∫ ∞
0
ωτ(λ)e−iλtdt
such that
||ωτ(λ)||2=
1 2π
∫ ∞
0
|ωτ(λ)|2dt <∞,
|1−e−iλτ|2n(1 +λ2)n
λ2n =|Ωτ(λ)|
2
and the following relation holds true:
Φτ(λ) = Ωτ(λ)Φ(λ). (39)
From (39)using the inverse Fourier transform we get
φτ(t) =
∫ ∞
−∞
eiλtΩτ(λ)Φ(λ)dλ=
=
∫ ∞
0
φ(x)
∫ ∞
−∞
eiλ(t−x)Ωτ(λ)dλdx=
=
∫ t
0
ωτ(t−x)φ(x)dx.
Therefore, the functions φτ(t), t ≥0, and φ(t), t ≥0,
from the space L2([0,∞))are related by the relation
φτ(t) =Wτφ(t) =
∫ t
0
ωτ(t−x)φ(x)dx, (40)
where Wτ is a linear operator in the space L
2([0,∞))
defined by the function ωτ(t), t ≥ 0, from the space
L2([0,∞)). In the case where the functions φτ(t) and
φ(t)are defined on segment[0, T], which meansφτ(t) =
4
Minimax-robust method of
ex-trapolation
The proposed formulas may be employed under the condition that the spectral density f(λ) of the consid-ered stochastic process ξ(t) with stationary nth incre-ments is known. The value of the mean square error ∆(h(τa)(f);f) := ∆(f,Aξb ) and the spectral
characteris-tich(τa)(f) of the optimal linear estimateAξb of the
func-tional Aξ which depends of unknown values ξ(t) can be calculated by formulas (31) and (33), the value of mean square error ∆(h(τ,Ta)(f);f) := ∆(f,AbTξ) and the
spectral characteristich(τ,Ta)(f) of the optimal linear es-timate AbTξof the functional Aξ which depends of
un-known valuesξ(t),t≥0, can be calculated by formulas (35) and (37). In the case where the spectral density is not known, but a setDof admissible spectral densities is given, the minimax (robust) approach to estimation of the functionals of the unknown values of a stochas-tic process with stationary increments is reasonable. In other words we are interesting in finding an estimate that minimizes the maximum of the mean square errors for all spectral densities from a given classDof admis-sible spectral densities simultaneously.
Definition 4 For a given class of spectral densitiesDa spectral density f0(λ)∈ D is called least favorable in D
for the optimal linear estimate the functionalAξ if the following relation holds true:
∆(f0) = ∆(h(τa)(f0);f0) = max
f∈D∆(h
(a)
τ (f);f).
Definition 5 For a given class of spectral densities D a spectral characteristic h0(λ) of the optimal linear es-timate of the functional Aξ is called minimax-robust if there are satisfied conditions
h0(λ)∈HD = ∩
f∈D
L02−(f),
min
h∈HD
max
f∈D∆(h;f) = supf∈D
∆(h0;f).
Analyzing the derived formulas and using the intro-duced definitions we can conclude that the following statements are true.
Lema 2 Spectral density f0(λ) ∈ D which admits the
canonical factorization(11) is the least favorable in the class of admissible spectral densities D for the optimal linear estimation of the functionalAξ if
f0(λ) =
∫ ∞
0
φ0(t)e−iλtdt
2
, (41)
where φ0(t), t ∈ [0;∞) is a solution to the conditional
extremum problem
||DτAφτ||2→max, f(λ) =
∫0∞φ(t)e−iλtdt
2
∈ D.
(42)
Lema 3 Spectral density f0(λ) ∈ D which admits the
canonical factorization (11)is the least favorable in the class of admissible spectral densities D for the optimal linear estimation of the functionalATξ if
f0(λ) =
∫ T
0
φ0(t)e−iλtdt
2
, (43)
where φ0(t), t ∈ [0;T], is a solution to the conditional
extremum problem
||DτTATφτ||2→max, f(λ) =
∫ T
0
φ(t)e−iλtdt
2
∈ D.
(44) If h(τa)(f0) ∈HD, the minimax-robust spectral
char-acteristic can be calculated ash0=h(τa)(f0).
The minimax-robust spectral characteristich0and the least favorable spectral density f0 form a saddle point of the function ∆(h;f) on the setHD× D. The saddle point inequalities
∆(h;f0)≥∆(h0;f0)≥∆(h0;f) ∀f ∈ D,∀h∈HD
hold true ifh0=h(a)
τ (f0) andh
(a)
τ (f0)∈HD, wheref0
is a solution to the conditional extremum problem
e
∆(f) =−∆(h(τa)(f0);f)→inf, f ∈ D, (45)
∆(h(τa)(f0);f) = 1 2π
∫ ∞
−∞
|rτ(λ)|2
f0(λ) f(λ)dλ.
Here rτ is determined by formula (32) or (36) with
f(λ) =f0(λ). The conditional extremum problem (45) is equivalent to the unconditional extremum problem
∆D(f) =∆(e f) +δ(f|D)→inf,
where δ(f|D) is the indicator function of the set D. Solution f0 to this unconditional extremum problem
is characterized by the condition 0 ∈ ∂∆D(f0) (see
Pshenichnyi[32]), where ∂∆D(f0) is the subdifferential
of the functional ∆D(f0) at point f0. With the help
of the condition 0∈∂∆D(f0) we can find the least
fa-vorable spectral densities in some special classes of spec-tral densities (see books by Moklyachuk[16], Moklyachuk and Masyutka[17] for more details).
5
Least favorable spectral
densi-ties in the class
D
0Consider the problem of the optimal estimation of functionals Aξ andATξof unknown values ξ(t),t ≥0,
of the stochastic process ξ(t) with stationary nth in-crements in the case where the spectral density is not known, but the following set of spectral densities is given
D0= {
f(λ)| 1 2π
∫ ∞
−∞
f(λ)dλ≤P0 }
.
It follows from the condition 0 ∈∂∆D(f0) for D=D0
that the least favorable density satisfies the equation
where ψ(λ)≤0 and ψ(λ) = 0 iff0(λ)>0. Therefore,
the least favorable density in the classD0for the optimal
linear estimation of the functionalAξ can be presented in the form
f0(λ) =c
∫ ∞
0
Dτ(Aφ0τ)(t)eiλtdt
2
, (46)
where the unknown function cφ0
τ(t) can be calculated
using factorization (11), equation (40), condition (42) and condition∫−∞∞ |φ0(λ)|2dλ= 2πP0.
Consider the equation
DτAWτφ=αφ, α∈C. (47)
For each solution of this equation such that ||φ||2 = 1
2π
∫∞
−∞|φ(λ)|2dλ=P0the following relation holds true:
f0(λ) =
∫ ∞
0
φ(t)e−iλtdt
2
=
=c
∫ ∞
0
Dτ(AWτφ)(t)eiλtdt
2
.
Denote by ν0P0 the maximum value of ||DτAWτφ||2
on the set of those solutions φ of equation (47), which satisfy condition ||φ||2 = P
0 and define canonical
fac-torization (11) of the spectral density f0(λ). Letν+ 0P0
be the maximum value of ||DτAWτφ||2 on the set of
those φ which satisfy condition ||φ||2 = P
0 and define
canonical factorization (11) of the spectral densityf0(λ)
defined by (46).
The derived equations and conditions give us a possi-bility to verify the validity of following statement.
Theorem 7 If there exists a solution φ0 = φ0(t) of
equation(47)which satisfies conditions||φ0||2=P 0and
ν0P0=ν0+P0=||DτAWτφ0||2, the spectral density(41)
is the least favorable density in the classD0 for the
op-timal estimation of the functionalAξof unknown values
ξ(t), t ≥ 0, of the stochastic process ξ(t) with station-ary nth increments. The increment ξ(n)(t, τ) admits a one-sided moving average representation. If ν0 < ν0+,
the density (46) which admits the canonical factoriza-tion (11) is the least favorable in the class D0. The
function cφτ = cφτ(t) is determined by equality (40),
condition (42) and condition ∫−∞∞ |φ(λ)|2dλ = 2πP 0.
The minimax-robust spectral characteristic is calculated by formulas(31),(32)substitutingf(λ)by f0(λ).
Consider the problem of optimal estimation of the functional ATξ. In this case the least favorable
spec-tral density is determined by the relation
f0(λ) =
c
∫ T
0
DτT(ATφ0τ)(t)e iλtdt
2
. (48)
Define a linear operatorAbT in the spaceL2([0,∞)) by
relation
(AbTφτ)(t) =
∫ t
0
a(T−t+u)φτ(u)du. (49)
Taking into consideration (40), we have the following equality
r(τ,Ta)(λ)
2
=
∫ T
0
DτT(ATWτφ)(t)eiλtdt
2
=
=
∫ T
0
DτT(AbTWτφ)(t)e−iλtdt
2
, (50) where the linear operator Dτ
TAbTφτ in the space
L2([0,∞)) is calculated by formula
DτT(AbTφτ)(t) =
[t τ]
∑
k=0 ∫ t−τ k
0
a(T−t+u+τ k)φτ(u)d(k)du.
Therefore each solution φ = φ(t), t ∈ [0, T], of the equation
DτTATWτφ=αφ, α∈C, (51)
or the equation
DτTAbTWτφ=βφ, β∈C, (52)
such that ||φ||2=P0, satisfies the following equality
f0(λ) =
∫ N
0
φ(t)e−iλtdt
2
=cr(τ,Ta)(λ)
2
.
Denote by ν0TP0 the maximum value of
||DτTATWτφ||2 = ||DτTAbTWτφ||2 on the set of
solutions φT of the equation (51) or the equation (52),
which satisfy condition ||φ||2 = P
0 and determine the
canonical factorization (11) of the spectral density
f0(λ) ∈ D
0. Let ν0N+P0 be the maximum value of
||Dτ
TATWτφ||2 on the set of those φ which satisfy
condition ||φ||2 = P
0 and determine the canonical
factorization (11) of the spectral density f0(λ) defined
by (48).
The following statement holds true.
Theorem 8 If there exists a solution φ0 = φ0(t),
t ∈ [0;T] of equation (51) or equation (52) such that
||φ0||2 = P0 and ν0TP0 = ν0T+P0 = ||DτTATWτφ0||2,
the spectral density (43) is least favorable in the class
D0 for the optimal estimation of the functionalATξ of
unknown values ξ(t), t ∈ [0;T], of the stochastic pro-cessξ(t)with stationarynth increments. The increment
ξ(n)(t, τ)admits a one-sided moving average
representa-tion. If νT
0 < ν
T+
0 , the density (48) which admits the
canonical factorization (11)is the least favorable in the class D0. The function cφτ = cφτ(t), t ∈ [0;T], is
determined by equation (40), condition(44) and condi-tion ∫−∞∞ |φ(λ)|2dλ= 2πP
0. The minimax-robust
spec-tral characteristic is calculated by formulas (35), (36) substituting f(λ)byf0(λ).
6
Least favorable spectral
densi-ties in the class
D
uvConsider the case where the spectral density f(λ) is not known, but the following set of spectral densities is given
Du v =
{
f(λ)|v(λ)≤f(λ)≤u(λ),
∫ ∞
−∞
f(λ)dλ= 2πP0 }
wherev(λ) andu(λ) are some given (fixed) spectral den-sities. It follows from the condition 0 ∈ ∂∆D(f0) for
D = Du
v that the least favorable density f0(λ) in the
class Dvu for the optimal linear estimation of the func-tionalAξ is of the form
f0(λ) = max
{
v(λ),min
{
u(λ),s0τ(λ)2
}}
, (53)
s0τ(λ) =c
∫ ∞
0
Dτ(Aφ0τ)(t)eiλtdt,
where the unknown function cφ0τ(t) can be calculated
using factorization (11), equestion (40), conditions (42) and∫−∞∞ |φ0(λ)|2dλ= 2πP
0.
Denote byνuvP0the maximum value of||DτAWτφ||2
on the set of those solutions φ of equation (47) which satisfy condition||φ||2=P0, inequalities
v(λ)≤
∫ ∞
0
φ(t)e−iλtdt
2
≤u(λ)
and determine the canonical factorization (11) of the spectral densityf(λ). Letν+
uvP0be the maximum value
of||DτAWτφ||2on the set of thoseφwhich satisfy
con-dition||φ||2=P
0and determine the canonical
factoriza-tion (11) of the spectral densityf0(λ) defined by (53).
The derived equations and conditions give us a possi-bility to verify the validity of the following statement.
Theorem 9 If there exists a solution φ0 = φ0(t) of equation(47)which satisfies conditions||φ0||2=P0 and
νuvP0 = νuv+P0 = ||DτAWτφ0||2, the spectral density
(41) is the least favorable in the class Du
v for the
op-timal estimation of the functional Aξ of unknown val-ues ξ(t), t ≥0, of the stochastic processξ(t) with sta-tionarynth increments. The incrementξ(n)(t, τ)admits
one-sided moving average representation. Ifνuv < νuv+,
the density(53)which admits the canonical factorization (11) is the least favorable in the class Du
v. The
func-tion cφτ = cφτ(t) is determined by equality (40),
con-ditions(42)and ∫−∞∞ |φ(λ)|2dλ= 2πP
0. The
minimax-robust spectral characteristic is calculated by formulas (31),(32)substituting f(λ) byf0(λ).
Consider the problem of the optimal estimation of the functionalATξ. In this case the least favorable spectral
density is determined by the relation
f0(λ) = max
{
v(λ),min
{
u(λ),s0τ,T(λ)2
}}
. (54)
s0τ,T(λ) =c
∫ T
0
DτT(ATφτ)(t)eiλtdt
Denote by νT
uvP0 the maximum value of
||Dτ
TATWτφ||2 = ||DτTAbTWτφ||2 on the set of
those solutions φ of equations (51) and (52), which satisfy condition||φ||2=P
0, inequalities
v(λ)≤
∫ T
0
φ(t)e−iλtdt
2
≤u(λ)
and define the canonical factorization (11) of the spec-tral density f(λ). Let νuvT+P0 be the maximum value
of ||Dτ
TATWτφ||2 on the set of those φ which satisfy
condition||φ||2 =P
0 and define canonical factorization
(11) of the spectral densityf0(λ) determined by (54).
The following statement holds true.
Theorem 10 If there exists a solution φ0 = φ0(t),
t ∈ [0;T], of equation (51) or equation (52) which sat-isfies conditions ||φ0||2 = P
0 and νuvT P0 = νuvT+P0 =
||DτTATWτφ0||2, spectral density(43)is least favorable
in the class Duv for the optimal estimation of the
func-tional ATξ of unknown values ξ(t), t ∈ [0;T], of the
stochastic process ξ(t) with stationary nth increments. The increment ξ(n)(t, τ) admits one-sided moving
av-erage representation. If νT
uv < νuvT+, the density (54),
which admits the canonical factorization (11), is least favorable in the class Du
v. The function cφτ = cφτ(t),
t ∈ [0;T], is determined by equation (40), conditions (44) and ∫−∞∞ |φ(λ)|2dλ = 2πP
0. The minimax-robust
spectral characteristic is calculated by formulas (35), (36)substituting f(λ) byf0(λ).
7
Least favorable spectral
densi-ties in the class
D
δConsider the problem of the optimal estimation of the functionals Aξ andATξof unknown values ξ(t),t ≥0,
of the stochastic process ξ(t) with stationary nth in-crements in the case where the spectral density is not known, but the following set of spectral densities is given
Dδ =
{
f(λ)| 1 2π
∫ ∞
−∞|
f(λ)−v(λ)|dλ≤δ
}
,
wherev(λ) is a bounded spectral density. It comes from the condition 0 ∈ ∂∆Dδ(f0) that the least favorable
spectral densities in the class Dδ for optimal linear
ex-trapolation of the functionalAξcan be presented in the form
f0(λ) = max
{
v(λ),c
∫ ∞
0
Dτ(Aφ0τ)(t)eiλtdt
2
}
,
(55) where unknown function cφ0
τ(t) is calculated using the
factorization (11), relation (40), condition (42) and con-dition
1 2π
∫ ∞
−∞|
φ0(λ)|2dλ=δ+ 1 2π
∫ ∞
−∞
v(λ)dλ=P1.
Define byνδP1 the maximum value of ||DτAWτφ||2
on the set of thoseφwhich belongs to the set of solutions of equation (47), satisfy equation||φ||2=P
1, inequality
v(λ)≤
∫ ∞
0
φ(t)e−iλtdt
2
and determine the canonical factorization (11) of the spectral densityf(λ). Letνδ+P1 be the maximum value
of ||DτAWτφ||2 on the set of those φ, which satisfy
condition ||φ||2 =P
1 and determine the canonical
fac-torization (11) of the spectral density f0(λ) defined by
(55). The following statement holds true.
Theorem 11 If there exists a solution φ0 = φ0(t) of equation(47)which satisfies conditions||φ0||2=P1and
νδP0 = νδ+P1 = ||DτAWτφ0||2, the spectral density
(41) is the least favorable in the class Dδ for the
opti-mal extrapolation of the functionalAξof unknown values
