Representation of self-similar Gaussian processes

0.15 MB·

by  Adil Yazigi

REPRESENTATION OF SELF-SIMILAR GAUSSIAN PROCESSES
ADILYAZIGI
Abstract. We develop the canonical Volterra representation for a self-similar
Gaussian process by using the Lamperti transformation of the corresponding
stationary Gaussian process, where this latter one admits a canonical integral
representationundertheassumptionofpurenon-determinism. Weapplytherep-
resentationobtainedtotheequivalenceinlawforself-similar Gaussianprocesses.
4
1
0 Mathematics Subject Classification (2010): 60G15,60G18, 60G22.
2
Keywords: Self-similar processes; Gaussian processes; canonical Volterra representation;

l
u Lamperti transformation; stationary Gaussian process; equivalence in law; homogeneous
J kernels.

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2

1. Introduction and preliminaries
]
R
In this paper, we will construct the canonical Volterra representation for a given
P
. self-similar centered Gaussian processes. The role of the canonical Volterra repre-
h
t sentation which was first introduced by Levy in [15] and [16], and later developed
a
by Hida in [9], is to provide an integral representation for a Gaussian process X in
m
terms of a Brownian motion W and a non-random Volterra kernel k such that the
[

expression
2
v t
6 Xt = k(t,s)dWs
3 Z0
2 holds for all t and the Gaussian processes X and W generate the same filtration. It
3
. is known, see [3] and [15], that if the kernel k satisfies the homogeneity property for
1
somedegreeα,i.e. k(at,as) = aαk(t,s),a > 0,theGaussianprocessX isself-similar
0
4 with index α+ 1. Thus, the main goal of this paper is to give, under some suitable
2
1
conditions, a general construction of the canonical Volterra representation for self-
:
v
similar Gaussian processes, and which also guaranties the homogeneity property
i
X of the kernel. In section 2, the linear Lamperti transform that defines the one-
r one correspondence between stationary processes and self-similar processes, will be
a
used to express the explicit form of the canonical Volterra representation for self-
similarGaussianprocessesinthelightoftheclassical canonicalrepresentation ofthe
stationary processes given by Karhunen in [12]. In section 3, we give an application
of the representation obtained to a Gaussian process equivalent in law to the self-
similar Gaussian process.
In our mathematical settings, we take T > 1 to be a fixed time horizon, and
on a complete probability space (Ω,F,P) we consider a centered Gaussian process
Date: July 24, 2014.
The author wishes to thankTommi Sottinen for discussions and helpfulcomments. The author
alsothankstheFinnishDoctoralProgrammeinStochasticsandStatisticsandtheFinnishCultural
Foundation for financial support.
1
2 YAZIGI,A.
X = (X ;t ∈ [0,T]) that enjoys the self-similarity property for some β > 0, i.e.
t
(X ) =d (aβX ) , for all a > 0,
at 0≤t≤T/a t 0≤t≤T
d
where = denotes equality in distributions, or equivalently ,
t s
r(t,s)= E(X X ) = T2βr , , 0 ≤ t,s ≤ T. (1.1)
t s
T T
(cid:18) (cid:19)
In particular, we have r(t,t) = t2βE(X2), which is finite and continuous function
1
at every (t,t) in [0,T]2, and therefore, is continuous at every (t,s) ∈ [0,T]2, see
[17]. A consequence of the continuity of the covariance function r is that X is
mean-continuous.
WedenotebyH (t)theclosedlinearsubspaceofL2([0,T])generatedbyGaussian
X
random variables X for s ≤ t, and by (FX) , where FX := σ(X ,s ≤ t), the
s t t∈[0T] t s
completed natural filtration of X. We call the Volterra representation of X the
integral representation of the form
t
X = k(t,s)dW , t ∈ [0,T], (1.2)
t s
Z0
where W = (W ;t ∈ [0,T]) is a standard Brownian motion and the kernel k(t,s)
t
is a Volterra kernel, i.e. a measurable function on [0,T] × [0,T] that satisfies
T tk(t,s)2dsdt < ∞, and k(t,s) = 0 for s > t. The Gaussian process X with
0 0
such representation is called a Gaussian Volterra process, provided with k and W.
R R
Moreover, the Volterra representation is said to be canonical if the canonical
property
FX = FW
t t
holds for all t, or equivalently
H (t) = H (t), for all t. (1.3)
X W
Remark 1.1. (i) An equivalent to the canonical property is that if there ex-
ists a random variable η = T φ(s)dW , φ ∈ L2([0,T]), such that it is
0 s
t
independent of X for all 0 ≤ t ≤ T, i.e. k(t,s)φ(s)ds = 0 , one
t R 0
has φ ≡ 0. This means that the family {k(t,·),0 ≤ t ≤ T} is free and
R
spans a vector space that is dense in L2([0,T]). If we associate with the
canonical kernel k a Volterra integral operator K defined on L2([0,T]) by
K φ(t) = tk(t,s)φ(s)ds, it follows from thecanonical property (1.3)that
0
K is injective and K (L2([0,T])) is dense in L2([0,T]). The covariance in-
R
tegral operator R associated with the kernel r(t,s) has the decomposition
R = K K ∗, where K ∗ is the adjoint operator of K . In this case, the
covariance r is factorable, i.e.
t∧s
r(t,s) = k(t,u)k(s,u)du.
Z0
(ii) A special property for a Volterra integral operator is that it has no eigen-
values, see [7].
REPRESENTATION OF SELF-SIMILAR GAUSSIAN PROCESSES 3
2. The Canonical Volterra representation and self-similarity
The Gaussian process X is β–self-similar, and according to Lamperti [14], it can
be transformed into a stationary Gaussian process Y defined by:
Y(t):= e−βtX(et), t ∈ (−∞,logT]. (2.1)
Conversely, X can be recovered from Y by the inverse Lamperti transformation
X(t) = tβY(logt), t ∈[0,T]. (2.2)
It is obvious that the mean-continuity of the process Y follows from the fact that
E(Y −Y )2 = 2 r(1,1)−e−(t−s)βr(et−s,1)
t s
(cid:16) (cid:17)
converges to zero when t approaches s. As was shown by Hida & Hitsuda (§3, [10]),
whichisawell-knownclassicalresultthathasbeenfirstestablishedbyKarhunen(§3,
Satz 5, [12]), the stationary Gaussian process Y admits the canonical representation
t
Y = G (t−s)dW∗, (2.3)
t T s
−∞
Z
where G is a measurable function that belongs to L2(R,du) such that G (u) =
T T
0 when u < 0, and W∗ is a standard Brownian motion satisfying the canonical
property, i.e., HY(t) = HW∗(t), t ∈ [0,T]. A necessary and sufficient condition
for the existence of the representation (2.3) is that Y is purely non-deterministic.
Following Cramer [4], a process Z is purely non-deterministic if and only if the
condition
H (t) = {0}, (C)
Z
t
\
is fulfilled, where {0} is the L2–subspace spanned by the constants. The condition
(C) means that the remote past is trivial, i.e. FZ is trivial; see also [8], [10] and
0+
[12].
Next, we shall extend the property of pure non-determinism to the self-similar
centered Gaussian process X.
Theorem 2.1. The self-similar centered Gaussian process X = (X ;t ∈ [0,T])
t
satisfies the condition (C) if and only if there exist a standard Brownian motion W
and a Volterra kernel k such that X has the representation
t
X = k(t,s)dW , (2.4)
t s
Z0
where the Volterra kernel k is defined by
k(t,s) = tβ−21 F s , s < t, (2.5)
t
(cid:16) (cid:17)
for some function F ∈ L2(R ,du) independent of β, with F(u) = 0 for 1< u.
+
Moreover, H (t)= H (t) holds for each t.
X W
Remark 2.2. Inthecase wheretheprocessX istrivial self-similar, i.e. X = tβW ,
t 1
0 ≤ t ≤ T, the condition (C) is not satisfied since H (t) = H (1). Thus,
t∈(0,T) X W
X has no Volterra representation in this case.
T
4 YAZIGI,A.
Proof. The fact that X is purely non-deterministic is equivalent to that Y is purely
non-deterministic since
H (t) = H (logt)= H (t).
X Y Y
t∈(0,T) t∈(0,T) t∈(−∞,logT)
\ \ \
Thus Y admits the representation (2.3) for some square integrable kernel G and a
T
standard Brownian motion W∗. By the inverse Lamperti transformation, we obtain
logt t t
X(t) = tβGT(logt−s)dWs∗ = tβs−21GT log s dWs,
Z−∞ Z0 (cid:18) (cid:19)
where dWs = s12dWl∗ogs. We take the Volterra kernel k to be defined as k(t,s) =
tβ−21 F st , where F(u) = u−21GT(logu−1) ∈ L2(R+,du) vanishing when u < 1
since G (u) = 0 when u < 0, i.e. for t < s, we have F(s) = 0, and then, k(t,s) = 0.
(cid:0)T (cid:1) t
Indeed,
∞ ∞ du ∞
F(u)2du= G (logu−1)2 = G (v)2dv < ∞,
T T
u
Z0 Z0 Z−∞
and
T t s 2 T 1
F dsdt = tdt F(u)2du
t
Z0 Z0 (cid:16) (cid:17) Z0 Z0
T ∞
= tdt G (v)2dv < ∞.
T
Z0 Z0
Thus,
T t s 2 T 1
t2β−1F dsdt = t2β F (u)2 du dt< ∞
t
Z0 Z0 (cid:16) (cid:17) (cid:18)Z0 (cid:19)(cid:18)Z0 (cid:19)
Considering the closed linear subspace H (t) of L2([0,T]) that is generated by
dW
W −W for all u ≤ s ≤ t, we have H (t) = H (t) since W = 0, and therefore,
s u dW W 0
the canonical property follows from the equalities
HX(t) = HY(logt) = HdW∗(logt)= HdW(t) = HW(t).
(cid:3)
Example2.3(FractionalBrownianmotion). ThefractionalBrownianmotion(fBm)
on [0,T] with index H ∈ (0,1) is a centered Gaussian process BH = (B ;0 ≤ t ≤ T)
t
with the covariance function R (t,s) = 1(s2H+t2H−|t−s|2H). ThefBm is H-self–
H 2
similar, and following [1] and [5], it admits the canonical Volterra representation
with the canonical kernel
kH(t,s) = cHs21−H t(u−s)H−32 uH−21 du, for H > 1,
2
Zs
kH(t,s) = dH t H−12 (t−s)H−12
s

(cid:18) (cid:19)
− H − 1 s12−H tuH−23 (u−s)H−21 du , for H < 1,
2 2
(cid:18) (cid:19) Zs !
REPRESENTATION OF SELF-SIMILAR GAUSSIAN PROCESSES 5
1 1
where c = H(2H−1) 2, d = 2H 2, here B denotes the
H B(2−2H,H−1) H (1−2H)B(1−2H,H+1)
2 2
Beta function(cid:16). So, the funct(cid:17)ion F that(cid:16)corresponds to the c(cid:17)anonical Volterra repre-
sentation of fBm has the expressions:
F(u) = cH u21−H 1(z−u)H−32 zH−21 dz , for H > 1,
2
(cid:18) Zu (cid:19)
and
F(u) = dH 1 −1 H−21 − H − 1 (u)12−H 1zH−32 (z−u)H−12 dz ,
u 2
(cid:18) (cid:19) (cid:18) (cid:19) Zu !
for H < 1.
2
A function f(t,s) is said to be homogeneous with degree α if the equality
f(at,as) = aαf(t,s), a > 0,
holds for all t,s in [0,T]. From the expression (2.5) of the canonical kernel, it is easy
to see that k is homogeneous with degree β − 1, i.e. k(t,s) = Tβ−12k(t, s), for all
2 T T
s < t ∈ [0,T].
Given X with the canonical Volterra representation (2.4), let U to be a bounded
unitary endomorphism on L2([0,T]) with adjoint U∗ = U−1, and define the pro-
cess B = (B) := (U∗(W)) for each t ∈ [0,T]. Indeed, B is a standard Brownian
t t
motion since the Gaussian measure is preserved under the unitary transformations.
With the notation k (·) := k(t,·), the Gaussian process associated with the ker-
t
nel (Uk )(s) and the standard Brownian motion B has same law as X. For the
t
covariance operator, we write
R = K K ∗ = K U∗UK ∗ = (K U∗)(K U∗)∗,
where the operator K U∗ is defined by
t T
(K U∗)φ(t) = k(t,s)(U∗φ)(s)ds = (Uk )(s)φ(s)ds, φ ∈ L2([0,T]).
t
Z0 Z0
TheassociatedGaussianprocesshasthentheintegralrepresentation T(Uk )(s)dB
0 t s
for all t ∈ [0,T].
R
Corollary 2.4. For any bounded unitary endomorphism U on L2([0,T]), the ho-
mogeneity of k is preserved under U.
Proof. Let U bea boundedunitary endomorphism on L2([0,T]), and let the scaling
operator Sf(t) = T21f(Tt) with adjoint S∗f(t) = T−12f(t) to be defined for all
T
f ∈ L2([0,T]). The homogeneity of k means that
kt(s)= Tβ(S∗kt)(s),
T
then we have
s
Ukt(s) = Tβ(US∗kt)(s) = Tβ−21(SUS∗kt )( ).
T T T
6 YAZIGI,A.
To show the equality SUS∗kt = Ukt, we will use the Mellin transform
T T
∞ ∞
(SUS∗kt)(s)sp−1ds = (US∗kt)(s)(S∗sp−1)ds
T T
Z0 Z0
∞
= T12−p (US∗kt)(s)sp−1ds
T
Z0
∞
= T21−p (S∗kt)(s)(U∗sp−1)ds
T
Z0
∞ s
= T−p kt( )(U ∗sp−1)ds
T T
Z0
∞ ∞
= kt(u)(U∗up−1)du= Ukt(u)up−1du,
T T
Z0 Z0
and the uniqueness property of the Mellin transform implies that
SUS∗kt = Ukt.
T T
(cid:3)
Remark 2.5. Thefact that the β-self–similar Gaussian process X satisfies the con-
dition (C), guaranties the existence of the canonical kernel k which is homogeneous
with degree β − 1, and its homogeneity is preserved under unitary transformation.
2
If we consider again the example in Remark 2.2, one has the representation
T
X = tβ1 (s)dW ,0 ≤ t ≤ T,
t [0,1] s
Z0
where1 (s)istheindicator function. Inthiscase, weseethatthekerneltβ1 (s)
[0,1] [0,1]
does not satisfy the homogeneity property of any degree.
3. Application to the equivalence in law
In this section, we shall emphasize the self-similarity property under the equiv-
alence of laws of Gaussian processes. First, We recall the results shown by Hida-
Hitsuda in the case of Brownian motion, see [10] and [11]. Following Hitsuda’s
representation theorem, a centered Gaussian process W = (W ;t ∈ [0,T]) is equiv-
t
alent in law to a standard Brownian motion W = (W ;t ∈ [0,T]) if and only if W
t
can be represented in a unique way by f f
f
t s
W = W − l(s,u)dW ds, (3.1)
t t u
Z0 Z0
where l(s,u) is a Volterrafkernel, i.e.
T t
l(t,s)2dsdt< ∞, l(t,s) = 0 for t < s, (3.2)
Z0 Z0
and such that the equality HWf(t) = HW(t) holds for each t. We note here that
the uniqueness of the canonical decomposition (3.1) is in the sense that if l′ is a
Volterra kernel and W′ = (W′;t ∈ [0,T]) is a standard Brownian motion such that
t
for 0 ≤ t ≤ T
t s t s
W′− l′(s,u)dW′ ds= W − l(s,u)dW ds,
t u t u
Z0 Z0 Z0 Z0
then l = l′ and W = W′.
REPRESENTATION OF SELF-SIMILAR GAUSSIAN PROCESSES 7
If we denote by P and P the laws of W and W respectively, these two processes
are equivalent in law if P and P are equivalent, and the Radon-Nikodym density is
given by e f
e
dP T s 1 T s 2
= exp l(s,u)dW dW − l(s,u)dW ds .
dP u s 2 s
(Z0 Z0 Z0 (cid:18)Z0 (cid:19) )
e
The centered Gaussian process W is a standard Brownian motion under P with
E(W W )= E(W W ), hence, it is self-similar with index 1 under P. It follows from
t s t s 2
(3.1) that the covariance of W unfder P has the form of e
e f f e
t∧s s t∧s t
E(W W ) = t∧sf− l(v,u)dvdu− l(v,u)dvdu
t s
Z0 Zu Z0 Zu
t s v1∧v2
ff
+ l(v ,u)l(v ,u)dudv dv .
1 2 1 2
Z0 Z0 Z0
TheHitsudarepresentationcanbeextendedtotheclassofthecanonicalGaussian
Volterra processes,see[2]and[20]. Acentered GaussianprocessX = (X ;t ∈[0,T])
t
is equivalent in law to a Gaussian Volterra process X if and only if there exits a
unique centered Gaussian process, namely W, satisfying (3.1) aend (3.e2), and such
that
t ft s
X = k(t,s)dW = X − k(t,s) l(s,u)dW ds, (3.3)
t s t u
Z0 Z0 Z0
where the kernel k(t,s) and the standard Brownian motion stand for (1.2), the
e f
canonical Volterra representation of X. Moreover, we have HXe(t) = HX(t) for all t.
Under the condition (C), the kernel k is (β − 1)-homogeneous, and the centered
2
GaussianprocessX isβ–self-similarunderPsinceW isastandardBrownianmotion.
It is obvious that if X has same law as X, it is β–self-similar under P, and this
condition is also enecessary, see [18]. Hoewever, fin the next proposition, we will
use the homogeneity peroperty of the Volterra kernel l as a necessary and sufficient
condition for the self–similarity for the process X, and equivalently for W, under
the law P.
e f
Proposition 3.1. Let X = (X ;t ∈ [0,T]) be a centered β-self–similar Gaussian
t
process satisfying the condition (C), then
(i) a centered Gaussian process X = (X ;t ∈ [0,T]) is equivalent in law to X
t
if and only if X admits a representation of the form of
e e
t
Xt =e Xt−tβ−21 z(t,s)dWs, 0 ≤ t ≤ T, (3.4)
Z0
where W isea standard Brownian motion on [0,T], and the kernel z(t,s) is
independent of β and expressed by
t u
z(t,s) = F l(u,s)du, s< t,
t
Zs (cid:16) (cid:17)
for a Volterra kernel l and some function F ∈ L2(R ,du) vanishing on
+
(1,∞].
(ii) In addition, X is β–self-similar if and only if l ≡ 0.
For the proof, we need the following lemma.
e
8 YAZIGI,A.
Lemma 3.2. If a Volterra kernel on [0,T]×[0.T] is homogeneous with degree (−1),
then it vanishes on [0,T]×[0.T].
Proof. Let a Volterra kernel h be (−1)-homogeneous. Combining the square inte-
grability and the homogeneity property h(t,s) = 1 h(t, s), a > 0, 0 ≤ s < t ≤ T,
a a a
yields
T t Ta at t s 2 1 Ta t′
h(t,s)2dsdt= h , dsdt= h(t′,s′)2ds′dt′
a a a2
Z0 Z0 Z0 Z0 (cid:18) (cid:19) Z0 Z0
which is finite for all a > 0. This implies that h vanishes on [0,T]×[0.T]. (cid:3)
Proof. (i) X satisfies the condition (C), and by Theorem (2.1), it admits a canonical
Volterra representation with a standard Brownian motion W and a kernel of the
form of k(t,s) = tβ−21F st , F ∈ L2(R+,du) vanishing on (1,∞]. By using Fubini
theorem, (3.3) gives
(cid:0) (cid:1)
t t
X = X − k(t,u)l(u,s)dudW , 0 ≤ t ≤ T,
t t s
Z0 Zs
which proves theeclaim.
ii) Suppose that X is β-self–similar. From (i), X has the representation
t
Xet = k(t,s)−tβ−12z(t,s) edWs, 0≤ t ≤ T,
Z0 (cid:16) (cid:17)
which is a canoneical Volterra representation. Indeed, if L denotes the Volterra
integral operator associated with the Volterra kernel l(t,s), the integral operator
K −K L = K (I −L)thatcorrespondstotheVolterra kernel k(t,s)−tβ−21z(t,s)
is also a Volterra integral operator, [7]. Here, I denotes the Identity operator. In
particular, if we let f ∈L2([0,T]) be such that K (I −L)f = 0. By (i) in Remark
1.1, the operator K is injective, hence, (I −L)f = 0, i.e., Lf = f. Therefore,
the Volterra integral operator L admits an eigenvalue, which is a contradiction by
(ii) in Remark 1.1. So, f ≡ 0.
Now, using the fact that HXe(t) = HX(t) for all t, X satisfies also the condition
(C), and by Theorem (2.1), the canonical kernel k(t,s) − tβ−21z(t,s) is (β − 1)-
2
e
homogeneous. For a > 0, we write
t s t s
k(t,s)−tβ−21z(t,s) = aβ−21 k , −tβ−12z , ,
a a a a
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
which implies that z(t,s) =z(t, s), and by the change of variable , we have
a a
t
t u a u s t v 1 v s
F l(u,s)du= F l u, du= F l , dv, s < t,
t s t a t a a a
Zs (cid:16) (cid:17) Za (cid:18)a(cid:19) (cid:16) (cid:17) Zs (cid:16) (cid:17) (cid:16) (cid:17)
which equivalent to
t u t u 1 u s
F l(u,s)du = F l , dv, s < u < t.
t t a a a
Z0 (cid:16) (cid:17) Z0 (cid:16) (cid:17) (cid:16) (cid:17)
Taking derivatives with respect to t on both sides, and since F u 6= 0, we obtain
t
1 u s
(cid:0) (cid:1)
l(u,s)= l , , s < u,
a a a
(cid:16) (cid:17)
which means that l is homogeneous with degree (−1). By applying the Lemma 3.2,
we get l ≡ 0.
REPRESENTATION OF SELF-SIMILAR GAUSSIAN PROCESSES 9
If l ≡ 0, we have E(X X ) = E(X X ) which means that X =d X. Therefore, X is
t s t s
β–self-similar. (cid:3)
e e e e
Remark 3.3. The importance of the condition (C) in Proposition (3.1) can been
seen in the case of the fBm with index H = 1, i.e. BH = tBH, 0 ≤ t ≤ T.
t 1
Here the condition (C) fails. Since fBm is Gaussian, each process is determined
by its covariance E(BHBH) = tsE((BH)2). However, the laws of processes that
t s 1
correspond to different values of E((BH)2) are equivalent, on the other hand, these
1
laws are different.
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10 YAZIGI,A.
AdilYazigi,DepartmentofMathematicsandStatistics,UniversityofVaasaP.O.Box
700, FIN-65101 Vaasa, Finland
E-mail address: [email protected]