On estimation of the Orey index for a class of Gaussian processes

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by  Kestutis Kubilius

On estimation of the Orey index for a class of Gaussian
processes.
3 K. Kubilius
1
0
2
Vilnius universityInstituteof Mathematics and Informatics,

n
Akademijos 4, LT-08663, Vilnius, Lithuania
a
J

4
1
Abstract

]
R Orey suggested the definition of some index for Gaussian processes with stationary incre-
ments which determines various properties of the sample paths of this process. We give an
P
extension of the definition of the Orey index for a second order stochastic processes which
.
h may not have stationary increments and estimate the Orey index for Gaussian process from
at discrete observations of its sample paths.
m
Keywords: Gaussian process, Hurst index, fractional Brownian motion, incremental vari-
[
ance function

1
v
1 Introduction
3
8
8 ThefractionalBrownianmotion(fBm)isapopularmodelinfinancialmathematics,economics
2 and natural sciences. As is well known the fBm BH is the only continuous Gaussian process
.
1 whichisselfsimilarwithstationaryincrementsanddependingonindex0<H <1. Moreover,
0 a fBm with Hurst index H is H¨older up toorder H.
3 For a real mean zero Gaussian process with stationary increments, Orey suggested the
1 following definition of index.
:
v Definition 1 (see [10], [8]) Let X be a real-valued mean zero Gaussian stochastic process
i
X with stationary increments and continuous in quadratic mean. Let σX be the incremental
variance of X given by σ2(h)=E[X(t+h)−X(t)]2 for t,h>0. Define
r X
a
hβ lnσ (h)
β :=inf β>0: lim =0 =limsup X (1)
∗ h↓0 σX(h) h↓0 lnh
n o
b
and
hβ lnσ (h)
β∗ :=sup β>0: lim =+∞ =liminf X . (2)
h↓0 σX(h) h↓0 lnh
n o
If β =β∗ then Xbhas the Orey index β .
∗ X
If Gaussian process with stationary increments has Orey index then almost all sample paths
b b
satisfy a H¨older condition of order γ for each γ ∈ (0,β ) (see Section 9.4 of Cramer and
X
Leadbetter[5]). ForfBmBH withtheHurstindex0<H <1theOreyindexβ =H. Sowe
X
havea class of Gaussian processes with stationary increments dependingon Orey index β .
X
Recently there havesbeen two extensions of fBm which preservemany properties of fBm,
but have no stationary increments except for particular parameter values. One of them is a
so called sub-fractional Brownian motion (sfBm) and another one is a bifractional Brownian
motion(bifBm). ThusitisverynaturaltoextendthedefinitionoftheOreyindexforGaussian
processessuchthattherewasapossibilitytoconsiderprocesseswhichmaynothavestationary
increments and are H¨olderup to theOrey index.
E-mail: [email protected]
1
We shall give such extension of the Orey index. As it will be proved, processes sfBm and
bifBm satisfy this extended definition of the Orey index and they are H¨older up to the Orey
index. Moreover,forfBm,sfBm,andbifBm,theOreyindexcoincideswiththeirself-similarity
parameter. Therefore it is enough to construct and consider the asymptotic behavior of an
estimate of the Orey index instead of estimating parameters of each of the processes under
consideration.
Many authors have already considered the asymptotic behavior of the first- and second-
order quadratic variations of Gaussian processes. The conditions in these papers were ex-
pressed in terms of covariance of a Gaussian process and depended on some parameter γ ∈
(0,2). IfaGaussianprocesshastheOreyindexthenconditionsonacovariancefunctionmay
expressed by means of it. As it will be shown below, the Orey index can be obtained for
some well-known Gaussian processes. Moreover, if we wanted to consider stochastic differen-
tial equations (SDE) driven by processes with a bounded p-variation, we should know when
the Riemann-Stieltjes (RS) integral is defined. For Gaussian processes the Orey index helps
to obtain these conditions.
The purposeof this paper is to give an extension of the definition of the Orey index for a
second order stochastic processes which may not have stationary increments and to estimate
the Orey index for Gaussian process from discrete observations of its sample paths.
Norvaiˇsa[9]extendsthedefinitionoftheOreyindexforasecondorderstochasticprocesses
which may not have stationary increments. He showed that sfBm and bifBm satisfies this
extended definition of theOrey index. In this paper we shall give a different extension of the
definition of theOrey index. This newdefinition will bemore convenientfor our purposes.
The paper is organized in the following way. Section 2 contains thedefinition of theOrey
indexforthesecondorderstochasticprocess. Theconditionswhenthesecondorderstochastic
process hastheOreyindexare also given. Forsome well-known Gaussian processes which do
not have stationary increments the Orey index is obtained. Section 3 contains the results on
an almost sure asymptotic behavior of the second-order quadratic variations of a Gaussian
process. Herewealso verifyobtainedconditions forthesamewell-known Gaussian processes.
2 Orey index for the second order stochastic pro-
cesses
LetX ={X(t): t∈[0,T]}beasecondorderstochasticprocesswiththeincrementalvariance
function σ2 defined on [0,T]2:=[0,T]×[0,T]with values
X
σ2(s,t):=E[X(t)−X(s)]2, (s,t)∈[0,T]2.
X
Denote byΨ a class of continuous functions ϕ: (0,T]→[0,∞) such that lim ϕ(h)=0
h↓0
and lim [h·L3(h)] = 0, where L(h) = ϕ(h)/h → ∞, h ↓ 0. For example, we can take
h↓0
ϕ(h)=h·|lnh|α or ϕ(h)=h1−β,where α>0, 0<β <1/3. Set
hγ
γ :=inf γ >0: lim sup =0 , (3)
∗ (cid:26) h↓0ϕ(h)6s6T−hσX(s,s+h) (cid:27)
hγ
γ :=inf γ >0: lim =0 (4)
∗ h↓0 σX(0,h)
n o
and e
hγ
γ∗ :=sup γ >0: lim inf =+∞ , (5)
(cid:26) h↓0ϕ(h)6s6T−hσX(s,s+h) (cid:27)
hγ
γ∗ :=sup γ >0: lim =+∞ , (6)
h↓0 σX(0,h)
n o
where ϕ∈Ψ. Noteethat 06γ∗ 6γ 6+∞ and 06γ∗ 6γ 6+∞.
∗ ∗
We give thefollowing extension of the Orey index.
Definition 2 Let X = {X(et): te ∈ [0,T]} be a second order stochastic process with the
incremental variance function σ2 such that sup σ (s,s + h) → 0 as h → 0. If
X 06s6T−h X
γ =γ =γ∗ =γ∗ forany function ϕ∈Ψ, then we say that the process X has the Orey index
∗ ∗
γ =γ =γ =γ∗ =γ∗.
X ∗ ∗
e e
e e
2
Remark 3 FromourdefinitionoftheOreyindexwegetthedefinitionoftheOreyindexfora
real-valued mean zero Gaussian stochastic process with stationary increments and continuous
in quadratic mean.
Let us introducenotions
lnσ (s,s+h) lnσ (0,h)
γ :=limsup sup X and γ :=limsup X , (7)
∗ lnh ∗ lnh
h↓0 ϕ(h)6s6T−h h↓0
lnσ (s,s+h) lnσ (0,h)
γb∗ :=liminf inf X and γ∗ :=liminf X . (8)
h↓0 ϕ(h)6s6T−h lnh h↓0 lnh
Wehavbethatγ =γ γ∗ =γ∗. ItfollowsfromRemark3and(1)and(2). Nowwecompare
∗ ∗
quantities γ∗ and γ with γ∗ and γ , respectively, for a second order stochastic process X.
∗ ∗
e e
Lemma 4 Let X = {X(t): t ∈ [0,T]} be a second order stochastic process with the incre-
mental varbiance fubnction σ2 such that
X
sup σ (s,s+h)−→0 as h↓0. (9)
X
ϕ(h)6s6T−h
If 0<γ∗ 6γ <+∞, then γ∗ =γ∗, γ =γ .
∗ ∗ ∗
Proof. Theproofofthelemmarepeatstheoutlinesoftheproofoflimitsofthelogarithmic
e e b b
ratios (see AnnexA.4 in [11]). For completeness we give this proof in Appendix.
Assumethatforsomeγ ∈(0,1)thesecondorderstochasticprocessX satisfiesconditions:
(C1) σ (0,δ)=O(δγ), as δ↓0;
X
(C2) thereexist a constant κ>0 such that
σ (t,t+h)
Λ(δ):= sup sup X −1 −→0 as δ↓0
κhγ
ϕ(δ)6t6T−δ0<h6δ(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
for every function ϕ∈Ψ. (cid:12) (cid:12)
For (s,t)∈[0,T]2 set
σ2(s,t)
c2(s,t):= X −1. (10)
κ2|t−s|2γ
It follows from (C1) and (C2) that for any ϕ∈Ψ
sup σ2(s,s+h)6 sup σ2(s,s+h)+ sup σ2(s,s+h)
X X
06s6T−h 06s6ϕ(h) ϕ(h)6s6T−h
64 sup σ2(0,δ)+κ2h2γ sup |c2(s,s+h)|+1
X
06δ6ϕ(h)+h ϕ(h)6s6T−h
(cid:16) (cid:17)
6O (ϕ(h))2γ +κ2h2γ Λ2(h)+2Λ(h)+1 −→0 as h↓0. (11)
Thus theprocess X is continu(cid:0)ousin qu(cid:1)adratic m(cid:2)ean for all s∈[0,T(cid:3)−h].
Theorem 5 Assume that for some constant γ ∈(0,1) the second order stochastic process X
satisfies conditions (C1) and (C2). Then the Orey index is equal to γ .
X
Proof. By Lemma 4 it suffice to show that γ =γ∗ =γ and γ =γ∗ =γ .
∗ X ∗ X
For simplicity, we shall omit index X for γ. Observe first that condition (C1) implies
γ∗ =γ∗ =γ. Really, b b
lnσ (0,h) ln(O(hγ)/hγ)
X =γ+ −→γ as h↓0.
lnh lnh
It remains to prove γ∗ = γ . By conditions (C1) and (C2) it follows that there exists
∗
δ such that for δ 6 δ < 1 inequalities σ (s,s+δ) 6 1/2 and Λ(δ) < 1/2 holds for all
0 0 X
0 6 s 6 T −δ0. Supposbe thatbthese inequalities are fulfill in the course of the proof of this
theorem.
For (s,t)∈[0,T]2 set
σ (s,t)
b(s,t):= X −1.
κ|t−s|γ
3
Assumethat−1/2<b(s ,s +δ )60forsome fixeds ∈[ϕ(δ ),T −δ ]. Furthermore,it
0 0 0 0 0 0
is known that −2x6ln(1−x)6−x for 06x61/2. Then by inequality abovewe get
lnσ (s ,s +δ )=ln(κδγ)+ln(1+b(s ,s +δ ))=ln(κδγ)+ln(1−(−b(s ,s +δ )))
X 0 0 0 0 0 0 0 0 0 0 0
6ln(κδγ)+b(s ,s +δ )6ln(κδγ)+Λ(δ )
0 0 0 0 0 0
and
lnσ (s ,s +δ )>ln(κδγ)+2b(s ,s +δ )=ln(κδγ)−2|b(s ,s +δ )|
X 0 0 0 0 0 0 0 0 0 0 0
>ln(κδγ)−2Λ(δ )
0 0
for any ϕ∈Ψ.
It is known that |ln(1+x)|6x for x>0. Assume that 06b(s ,s +δ )<1/2 for some
0 0 0
fixed s ∈[ϕ(δ ),T −δ ], then
0 0 0
lnσ (s ,s +δ )=ln(κδγ)+ln(1+b(s ,s +δ ))6ln(κδγ)+b(s ,s +δ )
X 0 0 0 0 0 0 0 0 0 0 0
6ln(κδγ)+Λ(δ )
0 0
and
lnσ (s ,s +δ )=ln(κδγ)+ln(1+b(s ,s +δ ))>ln(κδγ)−2|b(s ,s +δ )|
X 0 0 0 0 0 0 0 0 0 0 0
>ln(κdγ)−2Λ(δ )
0 0
for any ϕ∈Ψ. Thus for every s∈[ϕ(δ ),T −δ ] we obtain
0 0
ln(κδγ)−2Λ(δ )6lnσ (s,s+δ )6ln(κδγ)+Λ(δ ).
0 0 X 0 0 0
Consequently,
lnκ Λ(δ ) lnσ (s,s+δ ) lnσ (s,s+δ )
γ+ − 0 6 inf X 0 6 sup X 0
lnδ0 |lnδ0| ϕ(δ0)6s6T−δ0 lnδ0 ϕ(δ0)6s6T−δ0 lnδ0
lnκ Λ(δ )
6γ+ +2 0 .
lnδ |lnδ |
0 0
and both sides of theabove inequality goes to γ as δ →0. Thusγ =γ∗ =γ .
0 ∗ X
2.1 Subfractional Brownian motion b b
We shall provethat sfBm satisfies conditions (C1) and (C2).
Definition 6 ([1])Asub-fractional BrownianmotionwithindexH,H ∈(0,1),isamean
zero Gaussian stochastic process SH =(SH,t>0) with covariance function
t
1
G (s,t):=s2H +t2H − (s+t)2H +|s−t|2H .
H 2
(cid:2) (cid:3)
The incremental variance function of sfBm is of the following form
σ2 (s,t)=E|SH −SH|2 =|t−s|2H +(s+t)2H −22H−1(t2H +s2H). (12)
SH t s
Since for any 06s6t6T inequalities (see [1])
(t−s)2H 6σ2 (s,t)6(2−22H−1)(t−s)2H, if 0<H <1/2, (13)
SH
(2−22H−1)(t−s)2H 6σ2 (s,t)6(t−s)2H, if 1/2<H <1 (14)
SH
holds, then condition (C1) is satisfied.
From incremental variance function (12) we get
σ2 (s,s+h)=h2H +f (h),
SH s
where
f (h):=(2s+h)2H −22H−1 s2H +(s+h)2H .
s
Note that (cid:2) (cid:3)
f (0)=f′(0)=0.
s s
4
By Taylor formula we obtain
h h
f (h)=f (0)+f′(0)h+ f′′(x)(h−x)dx= f′′(x)(h−x)dx
s s s s s
Z0 Z0
h
=2H(2H−1) (2s+x)2H−2−22H−1(s+x)2H−2 (h−x)dx.
Z0
(cid:2) (cid:3)
From inequality
1 s+x 2−2H
22H−1(s+x)2H−2−(2s+x)2H−2 = 22H−1−
(s+x)2−2H 2s+x
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:2) 1 (cid:3)s 2−2H 1
= 22H−1− 1− 6 22H−1−2−1 ,
(s+x)2−2H 2s+x (s+x)2−2H
(cid:20) (cid:18) (cid:19) (cid:21)
(cid:2) (cid:3)
it follows that for s>0
h h−x 1
|f (h)|6(22H −1) dx6 (22H −1)s2H−2h2
s (s+x)2−2H 2
Z0
and
σ2 (s,s+h) |f (h)|
sup sup SH −1 = sup sup s
h2H h2H
ϕ(δ)6s6T−δ0<h6δ(cid:12) (cid:12) ϕ(δ)6s6T−δ0<h6δ
(cid:12) (cid:12)
(cid:12) (cid:12) 22H−1δ2−2H 22H−1
(cid:12) (cid:12)6 sup 6
s2−2H (L(δ))2−2H
ϕ(δ)6s6T−δ
for every ϕ∈Ψ,where L(h)=ϕ(h)/h. So we get condition (C2) with κ=1.
Remark 7 In condition (C2) the function ϕ(δ) we could not change by δ or 0. Really, let
H >1/2. Then
sup sup |h−2Hf (h)|> sup sup |h−2Hf (h)|
s s
06s6T−δ06h6δ δ6s6T−δ06h6δ
h 22H−1 1
=2H(2H−1) sup sup − (h−x)dx
h2H(s+x)2−2H h2H(2s+x)2−2H
δ6s6T−δ06h6δZ0 h i
h 22H−1−1
>2H(2H−1) sup sup (h−x)dx
h2H(2s+x)2−2H
δ6s6T−δ06h6δZ0
(22H−1−1)h2−2H
>H(2H−1) sup sup
(2s+h)2−2H
δ6s6T−δ06h6δ
δ2−2H
=H(2H−1)(22H−1−1) sup
(2s+δ)2−2H
δ6s6T−δ
=H(2H−1)(22H−1−1)32H−2.
2.2 Bifractional Brownian motion
Definition 8 ([7])AbifractionalBrownian motionBHK =(BHK,t>0)withparameters
t
H ∈(0,1) and K ∈(0,1] is a centered Gaussian process with covariance function
R (t,s)=2−K (t2H +s2H)K−|t−s|2HK , s,t>0.
HK
The incremental variance funct(cid:0)ion of bifBm is of thefollow(cid:1)ing form
σ2 (s,t)=E|BH,K−BH,K|2 =21−K |t−s|2HK−(t2H +s2H)K +t2HK+s2HK.
BH,K t s
Let H ∈(0,1) and K ∈(0,1]. Then (cid:2) (cid:3)
2−K|t−s|2HK 6σ2 (s,t)621−K|t−s|2HK (15)
BH,K
for all s,t∈[0,∞) (see [7]). Thuscondition (C1) holds.
Since
σ2 (s,s+h)=21−K(h2HK−f (h))
BH,K s
5
with
f (h):= s2H +(s+h)2H K−2K−1 s2HK+(s+h)2HK ,
s
then fs(0)=fs′(0)=0 and(cid:2)byTaylor formu(cid:3)la we obta(cid:2)in (cid:3)
σ2 (s,s+h) h
BHK −1=−h−2HK f′′(x)(h−x)dx,
21−Kh2HK s
Z0
where
f′′(x)=4K(K−1)H2 s2H +(s+x)2H K−2(s+x)2(2H−1)
s
+2HK(2H−(cid:2)1) s2H +(s+x)(cid:3)2H K−1(s+x)2H−2
−2KHK(2HK−(cid:2) 1)(s+x)2HK−2(cid:3).
Note that for H >1/2
(s+x)2(2H−1) (s+x)2H 2−K
= (s+x)2HK−2 6(s+x)2HK−2.
[s2H +(s+x)2H]2−K s2H +(s+x)2H
(cid:20) (cid:21)
Thus for s>0
4 4
sup |f′′(x)|6 1 + 1
s s2−2HK {H>1/2} (2s2H)2−Ks2(1−2H) {H<1/2}
06x6h
2 2 8
+ + 6
(2s2H)1−Ks2−2H s2−2HK s2−2HK
and
σ2 (s,s+h) 8δ2−2HK 8
sup sup BHK −1 6 sup 6
21−Kh2H s2−2HK (L(δ))2−2HK
ϕ(δ)6s6T−δ0<h6δ(cid:12) (cid:12) ϕ(δ)6s6T−δ
(cid:12) (cid:12)
(cid:12) (cid:12)
for every ϕ∈Ψ. So cond(cid:12)ition (C2) holds. (cid:12)
2.3 Ornstein-Uhlenbeck process
The fractional Ornstein-Uhlenbeck (fO-U) process of the first kind is the unique solution of
the following stochastic differential equation
t
X =x −µ X ds+θBH, t6T, (16)
t 0 s t
Z0
with µ,θ>0, where BH, 0<H <1, is a fBm. It has explicit solution
t
X =x e−µt+θ e−µ(t−u)dBH,
t 0 u
Z0
where the integral exists as a Riemann-Stieltjes integral for all t>0 (see, e.g., [4]).
First of all we verify condition (C1). From [4] we knowthat
t t
eµudBH =eµtBH −µ eµuBHdu.
u t u
Z0 Z0
Thus
t 2 t
X2 62x2+2θ2 eµudBH 62x2+4θ2 e2µt(BH)2+µ2t e2µu(BH)2du
t 0 u 0 t u
(cid:18)Z0 (cid:19) (cid:18) Z0 (cid:19)
and
supEX2 62x2+4θ2e2µTT2H(1+µ2T2).
t 0
t6T
From (16) we get
h 2
σ2(0,h)62µ2E X dt +2θ2E(BH)262µ2h2supEX2+2θ2h2H 6Ch2H.
X t h t
(cid:18)Z0 (cid:19) t6h
6
This provesa condition (C1).
The incremental variance function of X has thefollowing form
t+h 2 t+h
σ2(t,t+h)=µ2E X ds −2µθE [BH(t+h)−BH(t)] X ds
X s s
(cid:18)Zt (cid:19) (cid:18) Zt (cid:19)
+θ2σ2 (t,t+h).
BH
Cauchy-Schwarzinequality yields
t+h t+h 1/2
E [BH(t+h)−BH(t)] X ds 6E1/2[BH(t+h)−BH(t)]2 h EX2ds
s s
(cid:18) Zt (cid:19) (cid:18) Zt (cid:19)
1/2
6hH+1 sup EX2 .
s
t6s6t+h
(cid:16) (cid:17)
Thus for every ϕ∈Ψ
σ2(t,t+h) 1/2
sup sup X −1 6θ−2δ1−H δ1−Hµ2supEX2+2µθ supEX2 −→0
θ2h2H t t
ϕ(δ)6t6T−δ0<h6δ(cid:12) (cid:12) t6T t6T
(cid:12) (cid:12) h (cid:16) (cid:17) i
(cid:12) (cid:12)
as δ↓0. Conditio(cid:12)n (C2) follow fro(cid:12)m theinequality
σ (t,t+h) σ2(t,t+h) σ (t,t+h) σ2(t,t+h)
X −1 = X −1 X +1 6 X −1 .
θhH θ2h2H θhH θ2h2H
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12).(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
2.4 (cid:12)Fractional B(cid:12) ro(cid:12)wnian brid(cid:12)ge(cid:12) (cid:12) (cid:12) (cid:12)
The fractional Brownian bridgeis definedin [0,T]by
t2H +T2H −|t−T|2H
XH =BH − BH.
t t 2T2H T
where BH, 0<H <1, is a fBm in [0,T].
Nowweverifycondition (C1). TheincrementalvariancefunctionofXH hasthefollowing
form
1
σ2 (t,t+h)=h2H − f2(h)
XH 4T2H t
where
f2(h):= (t+h)2H −t2H −|t+h−T|2H +|t−T|2H 2.
t
Thus (cid:2) (cid:3)
σ2 (t,t+h)6h2H. (17)
XH
So condition (C1) is satisfied.
Assume H <1/2. Since
(t+h)2H −t2H 6h2H and (T −t−h)2H −(T −t)2H 6h2H
then for every(cid:12)(cid:12) ϕ∈Ψ (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)
σ2 (t,t+h) 1 f2(h)
sup sup XH −1 = sup sup t 6T−2Hδ2H.
h2H 4T2H h2H
ϕ(δ)6t6T−δ0<h6δ(cid:12) (cid:12) ϕ(δ)6t6T−δ0<h6δ
(cid:12) (cid:12)
(cid:12) (cid:12)
Assume H >1/2. T(cid:12)hen ft(0)=0 and(cid:12)byTaylor formula we obtain
σ2 (t,t+h) 1 h 2
XH −1=− f′(x)dx ,
h2H 4T2Hh2H t
(cid:18)Z0 (cid:19)
where
f′(x)=2H (t+x)2H−1−(T −t−x)2H−1 .
t
Thus for every ϕ∈Ψ and H >1/2(cid:2)we get (cid:3)
σ2 (t,t+h) δ2−2H
sup sup XH −1 6 ·4H2T4H−2=H2T2H−2δ2−2H.
h2H 4T2H
ϕ(δ)6t6T−δ0<h6δ(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
7
3 The convergence of the second order quadratic
variation of process X along irregular partition
Let π = {0 = tn < tn < ··· < tn = T}, T > 0, be a sequence of partitions of the interval
n 0 1 Nn
[0,T] and (N ) is an increasing sequence of natural numbers. Such sequence of partitions is
n
called irregular. Define
m = max ∆nt, p = min ∆nt, ∆nt=tn−tn .
n k n k k k k−1
16k6Nn 16k6Nn
Usually in practiceobservations of theprocess areavailable at discrete regular timeinter-
vals. However, it may happen that part of theobservations are lost, resulting in observations
at irregular time intervals.
Definition 9 A sequence of partitions (πn)n∈N is regular if we have mn = pn = TNn−1 for
all n∈N or, equivalently, tn = kT for all n∈N and all k∈{0,...,N }.
k Nn n
Definition 10 The second order quadratic variations of Gaussian processes X along the par-
titions (πn)n∈N with Orey index γ is defined by
Nn−1 ∆n t(∆(2)nX)2
V(2)(X,2)=2 k+1 ir,k ,
πn (∆nt)γ+1/2(∆n t)γ+1/2[∆nt+∆n t]
Xk=1 k k+1 k k+1
where
∆(2)nXn =∆ntX(tn )+∆n tX(tn )−(∆nt+∆n t)X(tn).
ir,k k k k+1 k+1 k−1 k k+1 k
If the sequence(πn)n∈N is regular thenone has
Nn−1
V(2)(X,2)=(T−1N )2γ−1 ∆(2)X 2, ∆(2)X =X(tn )−2X(tn)+X(tn ).
Nn n n,k n,k k+1 k k−1
Xk=1 (cid:0) (cid:1)
TostudythealmostsureconvergenceofthesecondorderquadraticvariationofX weneed
additional assumptions on thesequence (πn)n∈N.
Definition 11 (see [2]) Let (ℓk)k>1 be a sequence of real numbers in the interval (0,∞). We
say that (πn)n∈N is a sequence of partitions with asymptotic ratios (ℓk)k>1 if it satisfies the
following assumptions:
1. There exists c>1 such that m 6cp for all n.
n n
2. lim sup ∆k−1,nt −ℓ =0.
n→∞ 16k6Nn(cid:12) ∆nkt k(cid:12)
The set L = {ℓ1;ℓ2;..(cid:12)(cid:12).;ℓk;...} wil(cid:12)(cid:12)l be called the range of the asymptotic ratios of the
sequence (πn)n∈N. (cid:12) (cid:12)
Itisclearthatifthesequence(πn)n∈N isregular, thenit isasequencewith asymptoticratios
ℓ =1for all k>1.
k
Definition 12 (see [2]) The function g : (0,∞) → R is invariant on L if for all ℓ,ℓˆ∈ L,
g(ℓ)=g(ℓˆ).
For example, let L={α,α−1} be theset containing two real positive numbersand let
1+λ2γ−1−(1+λ)2γ−1
g(λ)= .
λγ−1/2
The function g is invariant on L.
Proposition 13 Let X = {X(t) : t ∈ [0,T]}, T > 0, be a mean zero second order process
satisfying conditions (C1) and (C2). Let (πn)n∈N be a sequence of partitions with asymptotic
ratios (ℓk)k>1 and range of the asymptotic ratios L. If the function g is invariant on L or the
sequence of functions ℓ (t) converges uniformly to ℓ(t) on the interval [0,T], where
n
Nn−1
ℓn(t)= ℓk1[tnk,tnk+1)(t),
Xk=1
8
then
T
lim EV(2)(X,2)=2κ2 g(ℓ(t))dt,
n→∞ πn Z0
where
1+λ2γ−1−(1+λ)2γ−1
g(λ)= .
λγ−1/2
Proof. Rewrite the expectation of the increments of the second order irregular variation
in thefollowing way
E(∆(2)nX)2 =(∆nt)2σ2(tn,tn )+(∆n t)2σ2(tn ,tn)
ir,k k X k k+1 k+1 X k−1 k
+∆nt·∆n t σ2(tn,tn )−σ2(tn ,tn )+σ2(tn ,tn)
k k+1 X k k+1 X k−1 k+1 X k−1 k
=[∆nkt+∆nk+1t](cid:2)∆nkt·σX2(tnk,tnk +∆nk+1t)+∆nk+1t·σX2(tnk−1,t(cid:3)nk−1+∆nkt)
−∆nkt·∆nk+1t(cid:2)·σX2(tnk−1,tnk−1+∆nkt+∆nk+1t) (cid:3)
=I(1)−I(2)+I(3),
k k k
where
I(1) :=[∆nt+∆n t] ∆nt σ2(tn,tn )−κ2(∆n t)2γ
k k k+1 k X k k+1 k+1
+∆nk+1t σX2(t(cid:8)nk−1,t(cid:2)nk)−κ2(∆nkt)2γ , (cid:3)
I(2) :=∆nt·∆n(cid:2) t σ2(tn ,tn )−κ2(∆n(cid:3)(cid:9)t+∆n t)2γ ,
k k k+1 X k−1 k+1 k k+1
I(3) :=κ2[∆nt+∆n(cid:2) t]∆nt·∆n t (∆n t)2γ−1+(∆nt(cid:3))2γ−1−(∆nt+∆n t)2γ−1 .
k k k+1 k k+1 k+1 k k k+1
Set (cid:8) (cid:9)
∆nt
µn =[∆nt+∆n t](∆n t)γ+1/2(∆nt)γ+1/2 and ℓn = k .
k k k+1 k+1 k k ∆n t
k+1
Then
I(1) =κ2[∆nt+∆n t]∆nt·∆n t (∆n t)2γ−1c2(tn,tn )+(∆nt)2γ−1c2(tn ,tn)
k k k+1 k k+1 k+1 k k+1 k k−1 k
=κ2µnk (ℓnk)1/2−γc2(tnk,tnk+1)+(cid:2)(ℓnk)γ−1/2c2(tnk−1,tnk) , (cid:3)
I(2) =κ2µn(cid:2)(∆nt)1/2−γ(∆n t)1/2−γ(∆nt+∆n t)2γ−1c2(cid:3)(tn ,tn )
k k k k+1 k k+1 k−1 k+1
=κ2µn(ℓn)1/2−γ(1+ℓn)2γ−1c2(tn ,tn )
k k k k−1 k+1
and
I(3) =κ2µn (ℓn)1/2−γ+(ℓn)γ−1/2−(ℓn)1/2−γ(1+ℓn)2γ−1
k k k k k k
=κ2µnk((cid:0)ℓnk)1/2−γ 1+(ℓnk)2γ−1−(1+ℓnk)2γ−1 , (cid:1)
where the function c2(s,t) is defined i(cid:2)n (10). Wefurtherobserve t(cid:3)hat
τn+1∆n t·E(∆(2)nX)2 Nn−1 ∆n t·E(∆(2)nX)2
EV(2)(X,2)=2 k+1 ir,k +2 k+1 ir,k
πn µn µn
Xk=1 k k=Xτn+2 k
τn+1∆n t·E(∆(2)nX)2 Nn−1
=2 k+1 ir,k +2κ2 ∆n t·J(1)
µn k+1 k
Xk=1 k k=Xτn+2
Nn−1
+2κ2 ∆n t·J(2), (18)
k+1 k
k=Xτn+2
where τ =[ϕ(m )N ], [a] is an integer part of a real numbera,
n n n
J(1) =(ℓn)1/2−γ c2(tn,tn )+ℓ2γ−1c2(tn ,tn)−(1+ℓn)2γ−1c2(tn ,tn ) ,
k k k k+1 k−1 k k k−1 k+1
J(2) =(ℓn)1/2−γ(cid:2)1+ℓ2γ−1−(1+ℓn)2γ−1 . (cid:3)
k k k k
(cid:2) (cid:3)
9
Now we estimate thefirst term of equality (18). Note that
ϕ(m )
τ 6 n 6cL(m ), 2p2γ+2 6µn 62m2γ+2, (19)
n p n n k n
n
τn+1 m
∆tn 6ϕ(m ) n +m 62cϕ(m ). (20)
k+1 n p n n
n
Xk=1
By conditions (C1), (C2), and inequalities (19), (20) we get
τn+1∆n t·E(∆(2)nX)2
2 k+1 ir,k
µn
Xk=1 k
8c3ϕ(m ) 32c3ϕ(m )
6 n max σ2(tn ,tn)6 n sup σ2(0,tn)
p2nγ 16k6τn+2 X k−1 k p2nγ 16k6τn+2 X k
6 32c3ϕ(mn) max O (tn)2γ = 32c3ϕ(mn)O (cL(m )+2)m 2γ
p2nγ 16k6τn+2 k p2nγ n n
632c3ϕ(m )O cL(m )(cid:0)+2 2γ(cid:1) (cid:0)(cid:0) (cid:1) (cid:1)
n n
as m ↓0. From theproper(cid:0)t(cid:0)ies of functio(cid:1)n ϕ(cid:1) we obtain that theright hand sideof theabove
n
inequality tendsto zero as m ↓0.
n
Next, since [ϕ(m )N ]+1>ϕ(m ), for thesecond term of equality (18) we get
n n n
Nn−1
2κ2 ∆n t·J(1)
k+1 k
k=Xτn+2
Nn−1
62κ2 max c2(tn,tn ) ∆n t (ℓn)1/2−γ+(ℓn)γ−1/2
k k+1 k+1 k k
τn+16k6Nn−1
(cid:12) (cid:12)k=Xτn+2 (cid:2) (cid:3)
(cid:12) (cid:12) Nn−1
+2κ2 max c2(tn ,tn ) ∆n t·(ℓn)1/2−γ(1+ℓn)2γ−1
k−1 k+1 k+1 k k
τn+26k6Nn−1
(cid:12) (cid:12)k=Xτn+2
62κ2T sup (cid:12) sup c2(t,(cid:12)t+h) max (ℓn)1/2−γ+(ℓn)γ−1/2
k k
ϕ(mn)6t6T−mn0<h6mn 16k6Nn
(cid:12) (cid:12) (cid:2) (cid:3)
+2κ2T sup sup(cid:12) c2(t,t+(cid:12)2h) max (ℓn)1/2−γ(1+ℓn)2γ−1
k k
ϕ(mn)6t6T−2mn0<h6mn 16k6Nn
(cid:12) (cid:12) (cid:2) (cid:3)
62κ2T Λ2(m )+2Λ(m ) max(cid:12) (ℓn)1/2−γ(cid:12)+(ℓn)γ−1/2
n n k k
16k6Nn
+2κ2(cid:2)T Λ2(2m )+2Λ(2(cid:3)m ) m(cid:2)ax (ℓn)1/2−γ(1+ℓn)(cid:3)2γ−1
n n k k
16k6Nn
64κ2Tc Λ(cid:2)2(m )+2Λ(m ) +(cid:3)2κ2T(1+(cid:2)c)c Λ2(2m )+2Λ(2m(cid:3) ) .
n n n n
Thus thesecond t(cid:2)erm of equality (18(cid:3)) tendsto zero a(cid:2)s n→∞. (cid:3)
Itstillremainstoinvestigateasymptoticbehaviorofthethirdtermofequality(18). Ifthe
function g is invariant on L, then
Nn−1 Nn−1 1+ℓ2γ−1−(1+ℓn)2γ−1
2κ2 ∆n t·J(2) =2κ2 ∆n t k k
k+1 k k+1 (ℓn)γ−1/2
k=Xτn+1 k=Xτn+1 k
τn
=2κ2g(ℓ)T −2κ2g(ℓ) ∆n t−→2κ2g(ℓ)T as n→∞
k+1
Xk=1
for all ℓ∈L by the inequality (20). If the sequence of functions ℓ (t) converges uniformly to
n
ℓ(t) on theinterval [0,T], then
Nn−1 T τn
2κ2 ∆n t·J(2) =2κ2 g(ℓ (t))dt− ∆n t·J(2)
k+1 k n k+1 k
k=Xτn+1 Z0 Xk=1
T
−→2κ2 g(ℓ(t))dt
Z0
10