Multipower variation for Brownian semistationary processes

0.37 MB·English

by  Ole E. Barndorff-Nielsen

Bernoulli 17(4), 2011, 1159–1194
DOI: 10.3150/10-BEJ316
Multipower variation for Brownian
2
semistationary processes
1
0
2
OLE E. BARNDORFF-NIELSEN1, JOSE´ MANUEL CORCUERA2 and
n
a MARK PODOLSKIJ3
J
1Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK–8000
4
Aarhus C, Denmark. E-mail: [email protected]

2Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain.
]
T E-mail: [email protected]
S 3Department of Mathematics, ETH Zu¨rich, HG G32.2, 8092 Zu¨rich, Switzerland.
. E-mail: [email protected]
h
t
a
In this paper we study the asymptotic behaviour of power and multipower variations of pro-
m
cesses Y:
[
t
1 Yt=Z g(t−s)σsW(ds)+Zt,
−∞
v
8 where g:(0,∞)→R is deterministic, σ>0 is a random process, W is the stochastic Wiener
6
measureandZ isastochasticprocessinthenatureofadriftterm.Processesofthistypeserve,
8
in particular, to model data of velocity increments of a fluid in a turbulence regime with spot
0
intermittency σ. The purpose of this paper is to determine the probabilistic limit behaviour of
.
1 the(multi)powervariationsofY asabasisforstudyingpropertiesoftheintermittencyprocessσ.
0 NotablytheprocessesY areingeneralnotofthesemimartingalekindandtheestablishedtheory
2
of multipower variation for semimartingales does not suffice for deriving the limit properties.
1
Asakeytool fortheresults,ageneral centrallimit theorem fortriangular Gaussian schemesis
:
v formulated and proved. Examples and an application tothe realised variance ratio are given.
i
X Keywords: central limit theorem; Gaussian processes; intermittency;non-semimartingales;
r turbulence;volatility; Wienerchaos
a
1. Introduction
Themotivationforthedevelopmentoftheresultsreportedinthispaperhasbeentheneed
to construct tools for studying the probabilistic limit behaviour of (realised) quadratic
variationandothermultipowervariationsinrelationtotheclassofBrowniansemistation-
ary ( ) processes. This class, which was introduced in [12], consists of the processes
BSS
This is an electronic reprint of theoriginal article published by theISI/BS in Bernoulli,
2011, Vol. 17, No. 4, 1159–1194. This reprint differs from theoriginal in pagination and
typographic detail.
1350-7265 (cid:13)c 2011ISI/BS
1160 O.E. Barndorff-Nielsen, J.M. Corcuera and M. Podolskij
Y = Yt t R that are defined by
{ } ∈
t t
Y =µ+ g(t s)σ W(ds)+ q(t s)a ds, (1.1)
t s s
− −
Z−∞ Z−∞
where µ is a constant; W is a Brownian measure on R; g and q are non-negative deter-
ministic functions on R, with g(t)=q(t)=0 for t 0; and σ and a are cadlag processes.
≤
Whenσandaarestationary,soisY.HencethenameBrowniansemistationaryprocesses.
It is interesting to note that the fractional Ornstein–Uhlenbeck process is, in fact, also
representableinthe form(1.1). The sameis true ofawide classofstable pseudo-moving
averageprocesses; see Corollary 4.3 in [2].
The processes form the natural analogue, for stationarity related processes, to
BSS
the class of Brownian semimartingales
BSM
t t
Y =µ+ σ dW + a ds. (1.2)
t s s s
Z0 Z0
In the context of stochastic modelling in finance and in turbulence, the process σ em-
bodies the volatility or intermittency of the dynamics, whether the framework is that of
or . For detailed discussion of and the more general concept of tempo-
BSM BSS BSS
spatial ambit processes see [7–12]. Such processes are, in particular, able to reproduce
key stylized features of turbulent data.
Amaindifferencebetween and isthat,ingeneral,modelsofthe form
BSM BSS BSS
are not semimartingales (for a discussion of this, see Section 3 of [12]). In consequence,
various important techniques developed for semimartingales, such as the calculation of
quadratic variation by Itoˆ calculus and those of multipower variation, do not apply or
suffice in settings. The present paper addresses some of the issues that this raises.
BSS
The theory of multipower variation was primarily developed as a basis for inference
onσ under modelsand,moregenerally,Itoˆsemimartingales,withparticularfocus
BSM
on inference about the integrated squared volatility σ2+ given by
t
σ2+= σ2ds. (1.3)
t s
Z0
This quantity is likewise a focal point for the results discussed in the following.
Section2introducescommonnotationformultipowervariationandrecallssomebasic
properties of such quantities. A law of large numbers and a central limit theorem for
multipower variation of triangular arrays of Gaussian random variables are derived in
Section 3, and these limit results are drawn upon in Section 4 to establish probability
and central limit theorems for multipower variation for processes, with most of
BSS
the proofs postponed to Section 8. Section 5 presents several examples and Section 6
discusses an application concerning the limit behaviour of the realised variation ratio,
that is, the ratio of realised bipower variation to realised quadratic variation. Section 7
concludes and indicates some possible directions for further related work.
Asymptotic results 1161
2. Multipower variation
The concept of (realised) multipower variation was originally introduced in [13] in the
context of semimartingales, and the mathematical theory has been studied in a number
ofpapers[6,17,24,27]whilevariousapplicationsarethemainsubjectsin[14–16,25,33].
Multipowervariationturnsoutto be usefulfor analysingpropertiesofpartsofaprocess
that are not directly observable. In this section we present the definition of realised
multipower variation and recall its asymptotic properties for some classes of processes.
Letusconsideracontinuous-timeprocessX,definedonsomefilteredprobabilityspace
(Ω, ,( ) ,P), that is observed at equidistant time points t =i/n, i=0,...,[nt]. A
t t 0 i
F F ≥
realised multipower variation of the process X is an object of the type
[nt] k+1 k
−
∆n X pj, ∆nX=X X , p ,...,p 0, (2.1)
| i+j−1 | i i/n− (i−1)/n 1 k≥
i=1 j=1
X Y
for some fixed number k 1. We now present an overview of the asymptotic theory for
≥
quantities of the form (2.1) for various types of processes X.
We start with the case
BSM
t t
X =X + a ds+ σ dW , (2.2)
t 0 s s s
Z0 Z0
where W is a Brownianmotion, a is a locally boundedand predictable drift processand
σ is an adapted and cadlag volatility process. As was established in [6], the convergence
in probability
np+/2−1[nti]−=k1+1j=k1|∆ni+j−1X|pj −u→cp µp1···µpkZ0t|σs|p+ds (2.3)
X Y
holds, where p = k p and µ =E[up], u N(0,1) and we write Zn ucp Z when
+ j=1 j p | | ∼ −→
supt∈[0,T]|Ztn−Zt|P−P→0 for any T >0. Under a further condition on the volatility pro-
cess, one obtains the associated stable central limit theorem:
[nt]−k+1 k t
√n np+/2−1 i=1 j=1|∆ni+j−1X|pj −µp1···µpkZ0 |σs|p+ds!
X Y
(2.4)
t
st √A σ p+dB ,
s s
−→ | |
Z0
1162 O.E. Barndorff-Nielsen, J.M. Corcuera and M. Podolskij
where B is another Brownian motion, defined on an extension of the probability space
(Ω, ,( ) ,P) and independent of , and the constant A is given by
t t 0
F F ≥ F
k k k 1 m k k m
− −
A= µ (2k 1) µ2 +2 µ µ µ .
2pl− − pl pl pl pl+pl+m
l=1 l=1 m=1l=1 l=k m+1 l=1
Y Y XY Y− Y
Recall that the stable convergence of processes is defined as follows. A sequence of
processes Zn convergesstably in law towardsthe process Z (written Zn st Z), that is,
−→
definedontheextensionoftheoriginalprobabilityspace(Ω, ,( ) ,P),ifandonlyif
t t 0
F F ≥
foranybounded andcontinuousreal-valuedfunctional f andany -measurablerandom
F
variable V it holds that
lim E[f(Zn)V]=E[f(Z)V],
n
→∞
we use the notation Zn st Z.
−→
Acrucialpropertyoftherealisedmultipowervariationisitsrobustnesstojumpswhen
max (p )<2 [17, 26]. Assume for a moment that X is a general Itoˆ semimartingale
i i
withcontinuouspartXc satisfying(2.2).Then,by (2.3)andthe robustnessproperty,we
obtain the convergence
µ−12 [in=t1]−1|∆niX||∆ni+1X| P [Xc],
[nt] ∆nX 2 −→ [X]
P i=1| i |
where [X] denotes the quadraticPvariation of the semimartingale X and the limit is less
than or equal to 1. The latter result, together with the stable convergence in (2.4), can
be usedto constructaformaltestforjumps (see[13]). Onthe otherhand,weknow that
if the limit of the left-hand side is greater than 1 (which is the case for some typical
turbulence data), the process X cannot be an Itoˆ semimartingale.
In another direction, a study [5] was made of the asymptotic behaviour of bipower
variation for processes of the type
t
X =X + σ dG , t 0, (2.5)
t 0 s s
≥
Z0
where G is a continuous Gaussian process with centered and stationary increments (the
latterintegralisdefinedasaRiemann–Stieltjes integral).The processdefinedin(2.5)is,
ingeneral,alsonotasemimartingale,andthe theoryin[6]doesnotapply.Inparticular,
a different normalisationis required. Define the (normalised) multipower variation by
[nt] k+1 k
1 −
V(X,p ,...,p )n= ∆n X pj, p ,...,p 0,
1 k t nτnp+ | i+j−1 | 1 k≥
i=1 j=1
X Y
where τ >0 is given by
n
τ2=R¯[(1/n)], (2.6)
n
Asymptotic results 1163
with
R¯(t)=E[(G G )2]. (2.7)
s+t s
−
Under some assumptions on R¯ and the volatility process σ it was shown that
t
V(X,p ,...,p )n ucp ρ σ p+ds
1 k t −→ p1,...,pk | s|
Z0
for a certain constant ρ that depends on the behaviour of R¯ near 0. Furthermore,
p1,...,pk
anassociated(stable)centrallimittheorem,ofaformsimilarto(2.4),wasderived.Note,
however, that in general there are essential differences between the characters of
BSS
processes and processes of type (2.5). In the latter case, the process σ has only a local
influence in the value of X whereas, in the case, the process is also affected by the
BSS
past of σ.
3. Multipower variation of Gaussian triangular arrays
In this section we derive some asymptotic results for functionals of arrays of stationary
Gaussiansequences.Weconsideratriangulararray(X ) (t>0)ofrow-wise
i,n n 1,1 i [nt]
stationary Gaussian variables with mean 0 and variance 1.≥Let≤≤
r (j)=cor(X ,X ), j 0, (3.1)
n 1,n 1+j,n
≥
be the correlation function of (X ) . Assume that the array (X )
i,n 1 i [nt] i,n n 1,1 i [nt]
is “non-degenerate”,that is, the covar≤ian≤ce matrix of (X ,...,X ) is inv≥erti≤ble≤for
i,n i+k,n
any k 1 and n 1 (otherwise the results below do not hold).
≥ ≥
Now, define the multipower variation associatedwith the sequence (X ) :
i,n n 1,1 i [nt]
≥ ≤≤
[nt] k+1 k
1 −
V(p ,...,p )n= X pj, p ,...,p 0. (3.2)
1 k t n | i+j−1,n| 1 k≥
i=1 j=1
X Y
Our first result is the weak law of large numbers.
Theorem 1. Assume that there exists a sequence r(j) with
n 1
1 −
r2(j) r(j), r(j) 0 (3.3)
n ≤ n →
j=1
X
as n . Then it holds that
→∞
V(p ,...,p )n ρ(n) t ucp 0, (3.4)
1 k t − p1,...,pk −→
where
ρ(pn1),...,pk =E[|X1,n|p1···|Xk,n|pk]. (3.5)
1164 O.E. Barndorff-Nielsen, J.M. Corcuera and M. Podolskij
Proof. See Section 8. (cid:3)
Before we present the associated central limit theorem, we need to introduce another
Gaussian process. Suppose that r (j) ρ(j), j =1,...,k 1, for some numbers ρ(j).
n
→ −
Let (Q ) be a non-degenerate, stationary, centered (discrete-time) Gaussian process
i i 1
≥
with variance 1 and correlationfunction
ρ(j)=cor(Q ,Q ), j 1. (3.6)
1 1+j
≥
Define
[nt] k+1 k
1 −
V (p ,...,p )n= Q pj (3.7)
Q 1 k t n | i+j−1|
i=1 j=1
X Y
andletρp1,...,pk =E[|Q1|p1···|Qk|pk].Thenρp(n1,)...,pk →ρp1,...,pk andinthiscaseweobtain
the uniform convergence on compacts in probability (or ucp convergence):
V(p ,...,p )n ucp ρ t.
1 k t −→ p1,...,pk
Now we have the following central limit theorem for the family (V(pj,...,pj)n) .
1 k t 1 j d
≤ ≤
Theorem 2. Assume that
r (j) ρ(j), j 0, (3.8)
n
→ ≥
and that, for any j,n 1, there exists a sequence r(j) with
≥
∞
r2(j) r(j), r(j)< . (3.9)
n ≤ ∞
j=1
X
Then we have
√n(V(pj1,...,pjk)nt −ρp(nj1,)...,pjkt)1≤j≤d→L β1/2Bt, (3.10)
where B is a d-dimensional Brownian, β is a d d-dimensional matrix given by
×
β = lim ncov(V (pi,...,pi)n,V (pj,...,pj)n), 1 i,j d, (3.11)
ij n Q 1 k 1 Q 1 k 1 ≤ ≤
→∞
andtheweakconvergenceholdsinthespace ([0,T]d)equippedwiththeuniformtopology.
D
Proof. See Section 8. (cid:3)
Remark 1. Similarasymptoticresultscanbeobtainedforgeneralquantitiesoftheform
[nt] k+1
1 −
H(X ,...,X ) (3.12)
i,n i+k 1,n
n −
i=1
X
Asymptotic results 1165
for some function H:Rk R. Let m denote the Hermite index of H (notice that the
→
Hermite index of the power function used in (3.2) is 2). Replace condition (3.3) by
n 1
1 −
rm(j) r(j), r(j) 0
| n |≤ n →
j=1
X
and (3.9) by
∞
rm(j) r(j), r(j)< .
| n |≤ ∞
j=1
X
ThenTheorems1and2holdtrueforthefunctional(3.12)providedthatEH2(N (0,Σ))<
k
for any invertible Σ Rk k. We omit the details.
×
∞ ∈
Remark 2. Ho and Sun [23] have shown a non-functional version of Theorem 2 for
statistics of the type (3.12) when the correlation function r does not depend on n. To
n
the best of our knowledge, Theorem 2 is the first central limit theorem for (general)
multipower variation of a row-wise stationary Gaussian process.
4. Multipower variation for BSS processes
Armed with the general theorems proved in Section 3, we are now set to establish laws
of large numbers and central limit results for multipower variations in the framework of
the Browniansemistationary processes.The regularity conditions invokedare givenin a
first subsection, while the next states the theorems, the main parts of the proofs being
postponed to Section 8; the third subsection discusses the nature of the rather technical
regularityconditions anddescribes a setofsimpler assumptions that aremoreamenable
to checking.
4.1. Conditions
Weconsiderafilteredprobabilityspace(Ω, ,( ) ,P),assumingtheexistencethereon
t t 0
F F ≥
of a process, without drift term for the time being, that is,
BSS
t
Y = g(t s)σ W(ds), (4.1)
t s
−
Z−∞
where W is an ( )-Brownian measure on R, σ is an ( )-adapted and cadlag volatility
t t
process and g:RF R is a deterministic continuous mFemory function with g(t)=0 for
t 0andg L2((→0, )).Wealsorequire t g2(t s)σ2ds< a.s.toensurethatY <
≤ ∈ ∞ − s ∞ t
a.s. for all t 0. By an ( )-Brownian m−∞easure we understand a Gaussian stochastic
t
∞ ≥ F R
measure such that, for any Borelian set A with E[W(A)2]< ,
∞
W(A)∼N(0,m(A)),
1166 O.E. Barndorff-Nielsen, J.M. Corcuera and M. Podolskij
where m is the Lebesgue measure, and if A [t,+ ), then W(A) is independent of .
t
t ⊆ ∞ F
Note that B := W(ds),t a is a standard Brownian motion starting in a.
{ t a ≥ }
The process Y is assumed to be observed at time points t =i/n, i=1,...,[nt]. Now,
i
R
let G be the stationary Gaussian process defined as
t
G = g(t s)W(ds). (4.2)
t
−
Z−∞
Thisisanimportantauxiliaryobjectinthestudyof processes.NotethatGbelongs
BSS
to the type of processes occurring in (2.5), and that the autocorrelationfunction of G is
∞g(t+u)g(u)du
r(t)= 0 . (4.3)
R 0∞g2(u)du
R
We are interested in the asymptotic behaviour of the functionals
[nt] k+1 k
1 −
V(Y,p ,...,p )n= ∆n Y pj, p ,...,p 0,
1 k t nτnp+ | i+j−1 | 1 k≥
i=1 j=1
X Y
where∆nY =Y Y andτ2=R¯(1/n)withR¯(t)=E[G G 2], t 0.Inthe
i i/n− (i−1)/n n | s+t− s| ≥
following we assume that the function g is continuously differentiable on (0, ), g is
′
∞ | |
non-increasing on (b, ) for some b>0 and g L2((ε, )) for any ε>0. Moreover,we
′
∞ ∈ ∞
assume that for any t>0,
F = ∞(g (s))2σ2 ds< (4.4)
t Z1 ′ t−s ∞
almost surely.
Remark 3. Assumption(4.4)ensuresthattheprocessY hasthesame“smoothness”as
the process G (see Lemma 1 in Section 8). It is rather easy to check in practice,because
itisimpliedbytheconditionEF < fort>0.Furthermore,ifg hasboundedsupport,
t
∞
assumption (4.4) is trivially fulfilled since σ is cadlag.
Remark 4. Let us note again that the process Y is, in general, not a semimartingale.
In particular, this is the case when g /L2((0, )). For a closer discussion, see [12]. On
′
∈ ∞
the other hand, the process Y is not of the form(2.5). Thus,we requirenew methods to
prove the asymptotic results for V(Y,p ,...,p )n. Processes of the form (4.1) are used
1 k t
for modelling velocity of turbulent flows; see [8–10]. The function g, which is used in
suchmodels, behaves often as xδ near the origin.Hence, when δ ( 1/2,1/2) 0 ,Y is
∈ − \{ }
neither a differentiable process nor a semimartingale (because g /L2((0, ))). This is
′
∈ ∞
the primary case of our interest.
Asymptotic results 1167
We define the correlationfunction of the increments of G:
∆nG ∆n G R¯((j+1)/n)+R¯((j 1)/n) 2R¯(j/n)
r (j)=cov 1 , 1+j = − − , j 0.
n τ τ 2τ2 ≥
(cid:18) n n (cid:19) n
Next, we introduce a class of measuresthat is crucialfor our purposes.We define (recall
that g(x):=0 for x 0)
≤
(g(x 1/n) g(x))2dx
πn(A)= A − − , A (R). (4.5)
R0∞(g(x−1/n)−g(x))2dx ∈B
Note that πn is a probabilitRy measure on R .
+
For the weak law of large numbers we require the following assumptions:
(LLN) There exists a sequence r(j) with
n 1
1 −
r2(j) r(j), r(j) 0.
n ≤ n →
j=1
X
Moreover,it holds that
lim πn((ε, ))=0 (4.6)
n ∞
→∞
for any ε>0.
Remark 5.
(i) The first condition of (LLN) is adapted from Theorem 1. It guarantees the ucp
convergence of V(G,p ,...,p )n. The second condition of (LLN) says that the
1 k t
whole mass of the measure πn concentrates at 0. In particular, it is equivalent to
the weak convergence
πn δ ,
0
→
where δ is the Dirac measure at 0.
0
(ii) Condition (4.6) is absolutely crucial for the limit theorems given in the next sub-
section. When this condition is violated things become more complicated. In par-
ticular,it may leadto a different stochasticlimit of V(Y,p ,...,p )n (see the first
1 k t
example in Section 5). Intuitively, this can be explained by the observation that
theincrements∆nY containsubstantialinformationaboutthevolatility(far)out-
i
side of the interval [i 1, i] when condition (4.6) does not hold. Thus, in general,
−n n
we can not expect the limit described in Theorem 3 below.
Now, we introduce the assumptions for the central limit theorem:
(CLT) Assumption (LLN) holds, and
r (j) ρ(j), j 0,
n
→ ≥
1168 O.E. Barndorff-Nielsen, J.M. Corcuera and M. Podolskij
where ρ(j) is the correlation function of (Q ) , as introduced in (3.6). Furthermore,
i i 1
≥
there exists a sequence r(j) such that, for any j,n 1,
≥
∞
r2(j) r(j), r(j)< ,
n ≤ ∞
j=1
X
and for some γ (0,1] we have
∈
E[σ σ A] C t sAγ (4.7)
t s
| − | ≤ | − |
for any A>0. Finally, set p=min (pj). Assume that γ(p 1)> 1 and that
there exists a constant λ< 1 su1≤chi≤tkh,1a≤tjf≤odr aniy ε =O(n κ), κ ∧(0,1), w2e have
−p 1 n − ∈
∧
πn((εn, ))=O(nλ(1−κ)). (4.8)
∞
4.2. Limit theorems
In this section we present the limit laws of multipower variations of processes,
BSS
in part widening the scope slightly to allow more general drift terms. Recall that the
(realised) multipower variation of a process Y of the form (4.1) is defined as
[nt] k+1 k
1 −
V(Y,p ,...,p )n= ∆n Y pj, p ,...,p 0, (4.9)
1 k t nτnp+ | i+j−1 | 1 k≥
i=1 j=1
X Y
where τ2 =R¯(1) and p = k p . Our first result is the following probability limit
n n + j=1 j
theorem.
P
Theorem 3. Consider a process Z=Z +Z , where Z =Y is given by (4.1). Assume
1 2 2
that the condition (LLN) holds and that
[nt] k+1 k
1 − ∆n Z pj P 0, (4.10)
nτnp+ | i+j−1 ιj| −→
i=1 j=1
X Y
where ι ,...,ι 1,2 , for any t>0 and any (ι ,...,ι )=(2,...,2). Define
1 k 1 k
∈{ } 6
∆nG p1 ∆nG pk
ρ(n) =E 1 k .
p1,...,pk τ ··· τ
(cid:20)(cid:12) n (cid:12) (cid:12) n (cid:12) (cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)
Then we have (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
t
V(Z,p ,...,p )n ρ(n) σ p+ds ucp 0. (4.11)
1 k t − p1,...,pk | s| −→
Z0
Proof. See Section 8. (cid:3)