Parameter Estimation of Complex Fractional Ornstein-Uhlenbeck Processes with Fractional Noise

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by  Yong Chen

PARAMETER ESTIMATION OF COMPLEX FRACTIONAL
ORNSTEIN-UHLENBECK PROCESSES WITH FRACTIONAL NOISE
YONGCHEN,YAOZHONGHU,ANDZHIWANG
Abstract. We obtain strong consistency and asymptotic normality of a least squares esti-
mator of the drift coefficient for complex-valued Ornstein-Uhlenbeck processes disturbed by
7 fractional noise, extending the result of Y. Hu and D. Nualart, [Statist. Probab. Lett., 80
1 (2010),1030-1038]toaspecial2-dimensions. Thestrategy istoexploittheGarsia-Rodemich-
0 Rumseyinequalityandcomplexfourthmomenttheorems. Themainingredientsofthispaper
2
arethesamplepathregularityofamultipleWiener-Itôintegralandtwoequivalentconditions

n ofcomplexfourthmomenttheoremsintermsofthecontractions ofintegralkernelsandcom-
a
plexMalliavinderivatives.
J
Keywords : Complex Wiener-Itô multiple integral; fractional Brownian motion; fractional
6
Ornstein-Uhlenbeck process; fourth moment theorems; strong consistency; asymptotic nor-
2
mality.

] MSC 2000:60H07;60F05;62M09.
R
P
.
h
t
a 1. Introduction
m
To model the Chandler wobble, or variation of latitude conerning with the rotation of the
[

earth, M. Arató, A.N. Kolmogorov and Ya.G. Sinai [2] (see also [1]) proposed the following
1
v stochastic linear equation
8
dZ = γZ dt+√adζ , t 0, (1.1)
6 t t t
− ≥
5
7 where Zt = X1(t)+iX2(t) is a complex-valued process, γ = λ iω, λ > 0, a > 0 and ζt is a
−
0 complex Brownian motion. It is also suggested in [1] that the Brownian motion in (1.1) may
.
1 be replaced by other processes. In this paper we consider the statistical estimator of γ when
0
the complex Brownian motion ζ in (1.1) is replaced by a complex fractional Brownian motion
7
1 ζ = Bt1+iBt2, where (B1,B2) is a two dimensional fractional Brownian motion with H [1,3).
: t √2 t t ∈ 2 4
v [We shall fix the Hurst parameter and then omit the explicit dependence of the process on the
i
X Hurst parameter.] From now on we assume that ζ is a complex fractional Brownian motion of
r Hurst parameter H (1/2,3/4).
a ∈
To compare with the work in [9], we write (1.1) as
dX (t) λ ω X (t) a dB1
1 = − − 1 dt+ t . (1.2)
"dX2(t)# " ω −λ#"X2(t)# r2 "dBt2#
Thus (1.1) can be considered as a particular two dimensional Langevin equation driven by frac-
tional Brownian motions. However, we find it is more convenient to use the complex valued
equation (1.1).
1
2 Y.CHEN,Y.HU,ANDZ.WANG
Motivatedby the workof Hu andNualart[9] we alsoconsider a leastsquaresestimatorfor γ.
To this end, we intuitively rewrite (1.1) as
Z˙ +γZ =√aζ˙ , 0 t T .
t t t
≤ ≤
2
We minimize T Z˙ +γZ dt to obtain a least squares estimator of γ as follows.
0 t t
(cid:12) (cid:12)
R (cid:12) (cid:12) T Z¯ dZ T Z¯ dζ
(cid:12) (cid:12)γˆ = 0 t t =γ √a 0 t t . (1.3)
T − R0T |Zt|2dt − R0T |Zt|2dt
ThemainresultsofthepresentpaRperarethestrongconRsistencyandtheasymptoticnormality
of the estimator γˆ which we state as follows.
T
Theorem 1.1. Let H [1,3).
∈ 2 4
(i) The least squares estimator γˆ is strongly consistent. Namely, γˆ converges to γ almost
T T
surely as T .
→∞
(ii) √T (γˆ γ) is asymptotically normal. Namely,
T
−
1
√T[γˆ γ]law (0, C) as T , (1.4)
T − → N 2d2a →∞
σ2+c b
where C= with
" b σ2 c#
−
2 (xy)1 2H
σ2 = dxdy −
Γ(2 2H)2 (x+y)(x+γ¯)(y+γ)
− Z[0,∞)2
Γ2(2H 1) 2 1 1
+ − + + (1.5)
2λ γ 4H−2 γ4H−2 γ¯4H−2
| |
(cid:0) (cid:1)
2 (xy)1 2H 1 1
−
c+ib = + dxdy. (1.6)
Γ(2 2H)2 (y+γ)2 x+y x+γ
− Z[0,∞)2 h i
Γ(2H 1) 1 1
d = − + . (1.7)
2λ γ2H 1 γ¯2H 1
− −
(cid:16) (cid:17)
In the special case when H = 1, we have
2
λ
√T[γˆ γ]law (0, Id ), (1.8)
T 2
− → N 4a
where Id is a 2 2 identity matrix.
2
×
Remark 1.2. An important new feature for the case of fractional Ornstein-Uhlenbeck process
(H (1/2,3/4)) is that the limiting distribution is no longer independent Gaussian as in the
∈
case of Brownian motion case (H = 1/2). We will discuss exclusively the case H = 1/2 since
6
the case H =1/2 is easy.
A minor difference between the case of one dimensional fractional Orstein-Uhlenbeck process
considered in [9] and our complex case is that in our least squares estimator γˆ defined by (1.3),
we have T Z¯ dZ in the numerator, while in [9] it is T X dX . However, this minor difference
0 t t 0 t t
causes a big unpleasant trouble. By using Itô formula the later is expressed as X2 plus another
R R T
manageable term. This is critical in the proof of the strong consistency of the estimator since it
COMPLEX FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 3
allows us to use a famous theorem of Pickands in [9]. However, we cannot no longer apply Itô
formulato T Z¯ dZ toobtainsimilaridentity. Togetaroundthis difficultyweshalluseanother
0 t t
famous result, the Garsia-Rodemich-Rumsey inequality [8, Theorem 2.1].
R
To show the asymptotic normality, we may use a multi-dimensional fourth moment theorem.
However, we develop a complex version of the fourth moment theorem which is easier to use in
our case. To state the theorem we denote αH = H(2H 1) and φ(s,t) = αH s t2H−2 and
− | − |
define the Hilbert space
H:=L2 = f f :R R, f 2 := ∞ ∞f(s)f(t)φ(s,t)dsdt< . (1.9)
φ { | + → | |φ ∞}
Z0 Z0
Now the theorem is stated as follows.
Theorem 1.3 (Fourth Moment Theorems). Let Fk =Im,n(fk) with fk H⊙Cm H⊙Cn be a
{ } ∈ ⊗
sequenceof(m,n)-thcomplexWiener-Itômultipleintegrals(seethenextsectionforadiscussion),
with m and n fixed and m+n 2. Suppose that as k , E[F 2] σ2 and E[F2] c+ib,
≥ →∞ | k| → k →
where is the absolutevalue (or modulus)of a complex numberand c,b R. Then thefollowing
|·| ∈
statements are equivalent:
(i) The sequence (ReF , ImF ) converges in law to a bivariate normal distribution with
k k
σ2+c b
covariance matrix C= 1 ,
2" b σ2 c#
−
(ii) E[F 4] c2+b2+2σ4.
k
| | →
(iii) f f 0 and f h 0 for any 0 < i+j l 1
k k⊗i,j kkH⊗(2(l−i−j)) → k k⊗i,j kkH⊗(2(l−i−j)) → ≤ −
where l =m+n and h is the kernel of F¯ , i.e., F¯ =I (h ).
k k k n,m k
(iv) kDFkk2H, DF¯k 2H and hDFk, DF¯kiH converge to a constant in L2(Ω) as k tends to
infinity,whereD isthecomplexMalliavin derivatives. Thatistosay,Var( DF 2) 0,
Var( DF¯k(cid:13)(cid:13)2H)→(cid:13)(cid:13)0 and Var(hDFk, DF¯kiH)→0 as k tends to infinity. k kkH →
Remark 1.4(cid:13)(cid:13). (cid:13)(cid:13)1) If m = n and E[Fk2] = 0 or if m 6= n, then C = σ22Id2. That is to say,
the limit is a complex Gaussian variable (0,σ2) in this case. Theorem 7 of [11] is
CN
concerning multi-dimensional fourth moment theorems, but it requires C = σ2Id. Thus,
2
our findings are partly more general.
2) We give a different and simpler proof of the theorem. The equivalence (i) (ii) is shown
⇔
by an indirect method in [6] and by stein’s method in [4]. In this paper, we show that
(i) (ii) (iii) (iv) (i) directly. We make use of (iii) to show the asymptotic
⇒ ⇒ ⇒ ⇒
normality which is simpler than to use (iv) as in previous work [9].
2. Preliminaries: complex multiple Wiener-Itô integrals
Denote by (B , t 0) a fBm of Hurst parameter H (1/2,3/4). Then Gaussian isonormal
t
≥ ∈
processassociatedwith H is givenby Wiener integralswith respecttofBm forany deterministic
kernel H (where H is defined by (1.9)):
∈
B(f)= ∞f(s)dB , f H. (2.1)
s
∀ ∈
Z0
4 Y.CHEN,Y.HU,ANDZ.WANG
Let B˜() be an independent copy ofthe fractionalBrownianmotion B(). Followingthe same
· ·
idea of [6], we define complex Gaussian isonormal process and complex multiple Wiener-Itô
integrals with respect to fBm as follows. For any f =f +if with f , f H, define that
1 2 1 2
∈
HC :={f1+if2 : f1,f2 ∈H}, hf1+if2,f1+if2iHC =hf1,f1iH+hf2,f2iH, (2.2)
1
B(f)=B(f )+iB(f ), ζ(f)= [B(f)+iB˜(f)]. (2.3)
1 2
√2
Then ζ is called a complex isonormal Gaussian process over HC, which is a centered complex
Gaussian family satisfying
E[ζ(h)2]=0, E[ζ(g)ζ(h)]=hg,hiHC, ∀g,h∈HC.
From now on, without ambiguity, we still denote HC by H.
Definition 2.1 (Complex multiple Wiener-Itô integrals). For a fixed (p,q), suppose that g
∈
H p H q, we call I (g) the complex multiple Wiener-Itô integral of g with respect to ζ (see
⊙ ⊙ p,q
⊗
[6]). And if f H (p+q) then we define
⊗
∈
I (f)=I (f˜), (2.4)
p,q p,q
where f˜is the symmetrization of f in the sense of Itô [10]:
1
f˜(t ,...,t )= f(t ,...,t ,t ,...,t ), (2.5)
1 p+q p!q! π(1) π(p) σ(1) σ(q)
π σ
XX
where π and σ run over all permutations of (1,...,p) and (p+1,...,p+q) respectively.
It is easy to see that I (f)=I (f¯) and
p,q q,p
E[Ip,q(f)Ip′,q′(g)]=δp,p′δq,q′p!q! f˜,g˜ , (Itô’s isometry) (2.6)
h i
where the Kronecker delta δp,p′ is 1 when p′ is equal to p, and is 0 otherwise, and , is the
h· ·i
inner product on H (p+q). As a consequence,
⊗
2
E[I (f)2]=p!q! f˜ p!q! f 2, (Itô’s isometry). (2.7)
p,q
| | ≤ k k
(cid:13) (cid:13)
The proof of Theorem 1.3 proceeds thro(cid:13)ug(cid:13)h several propositions and lemmas. Firstly, we define
(cid:13) (cid:13)
the contraction of (i,j) indices of two symmetric functions.
Definition 2.2. For two symmetric functions f H⊙p1 H⊙q1, g H⊙p2 H⊙q2 and i
∈ ⊗ ∈ ⊗ ≤
p q , j q p , the contraction of (i,j) indix is defined as (see [5])
1 2 1 2
∧ ≤ ∧
f g(t ,...,t ;s ,...,s )
⊗i,j 1 p1+p2−i−j 1 q1+q2−i−j
= d~udu~φ(u ,u )...φ(u ,u )f(t ,...,t ,u ,...,u ;s ...,s ,v ...,v )
ZR2+l ′ 1 ′1 i ′i 1 p1−i 1 i 1 q1−j 1 j
×g(tp1−i+1,...,tp−l,v1′,...,vj′;sq1−j+1...,sq−l,u′1,...,u′i)φ(v1,v1′)...φ(vj,vj′)d~vdv~′,
where l = i + j, p = p + p , q = q + q , ~u = (u , ,u ), u~ = (u , ,u ) and ~v =
1 2 1 2 1 ··· i ′ ′1 ··· ′i
(v , ,v ), v~ =(v , ,v ).
1 ··· j ′ 1′ ··· j′
COMPLEX FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 5
By convention, f g = f g denotes the tensor product of f and g. We write f˜ g for
0,0 p,q
⊗ ⊗ ⊗
the symmetrizationof f g. In what follows,we use the conventionf g =0 if i>p q
p,q i,j 1 2
⊗ ⊗ ∧
or j >q p .
1 2
∧
Our next result is a technical lemma.
Lemma 2.3. Suppose that F =I (f) with f H m H n and that F¯ =I (h). Then
m,n ⊙ ⊙ n,m
∈ ⊗
E[F 4] 2 E[F 2] 2 E[F2] 2
| | − | | −
= (cid:0)m 2 (cid:1)n 2((cid:12)(cid:12)m!n!)2(cid:12)(cid:12) f f 2 + l−1((l r)!)2 ψ 2 (2.8)
i j k ⊗i,j kH⊗(2(l−i−j)) − k rkH⊗(2(l−r))
0<i+j<l(cid:18) (cid:19) (cid:18) (cid:19) r=1
X X
m n n m
= (m!n!)2 f h 2 (2.9)
i i j j k ⊗i,j kH⊗(2(l−i+j))
0<i+j<l(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
X
l 1
+ − (2m r)!(2n r)! ϕ 2 ,
− − k rkH⊗2(l−r)
r=1
X
where l=m+n and
2 2
m n
ψ = i!j! f˜ h, (2.10)
r i,j
i j ⊗
i+j=r (cid:18) (cid:19) (cid:18) (cid:19)
X
m n n m
ϕ = i!j! f˜ f. (2.11)
r i,j
i i j j ⊗
i+j=r (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
X
Proof. It follows from Lemma 4.1 of [5] that
E[F 4] E[F 2] 2 (2.12)
| | − | |
= m(cid:0)2 n 2(cid:1)(m!n!)2 f f 2 + l (l r)! 2 ψ 2 .
i j k ⊗i,j kH⊗(2(l−i−j)) − k rkH⊗(2(l−r))
i+j>0(cid:18) (cid:19) (cid:18) (cid:19) r=1
X X(cid:0) (cid:1)
We calculate the term ψ =f h:
l m,n
⊗
f h = d~udu~d~vdv~φ(u ,u )...φ(u ,u )φ(v ,v )...φ(v ,v )
⊗m,n Rm+n ′ ′ 1 ′1 m ′m 1 1′ n n′
Z +
×f(u1,...,um;v1,...,vn)h(v1′,...,vn′;u′1,...,u′m)
1
= f 2 = E[F 2], (2.13)
k kH⊗(m+n) m!n! | |
where the last equality is from Ito’s isometry (2.7). Next, we calculate the term f f in
m,n
⊗
Eq.(2.12) according to whether m = n or not. We consider the case m = n first. Without loss
6
generalitywecantakem>n. ByDefinition2.2wehavethatifi>norj >nthenf f =0.
i,j
⊗
Therefore, if m=n then
6
f f =0=E[F2], (2.14)
⊗m,n k
where the lastequalityis Itô’s isometry (2.6). Ifm=n, similarlyto show (2.13), we obtainthat
1
f f = f, h = E[F2]. (2.15)
⊗m,m h iH⊗(m+n) (m!)2
6 Y.CHEN,Y.HU,ANDZ.WANG
Substituting (2.15) or (2.14) according to whether m = n or not, and (2.13), into (2.12), we
obtain (2.8). Similarly, we can show (2.9). (cid:3)
Notation 1. Suppose that f(~tm,~sn) H m H n. Denote
⊙ ⊙
∈ ⊗
f (~tm 1,~sn)=f(~tm 1,u,~sn), fv(~tm,~sn 1)=f(~tm,~sn 1,v). (2.16)
u − − − −
Clearly, f (~tm 1,~sn) H m 1 H n and fv(~tm,~sn 1) H m H n 1.
u − ⊙ − ⊙ − ⊙ ⊙ −
∈ ⊗ ∈ ⊗
Definition 2.4 (Complex Malliavin Derivatives). Let denote the set of all random variables
S
of the form
f ζH(ϕ ), ,ζH(ϕ ) , (2.17)
1 m
···
where f C∞(Cm) and ϕi H,i =(cid:0)1,2, ,m. Let F (cid:1) be given by (2.17). The complex
∈ ↑ ∈ ··· ∈ S
Malliavin derivative of F is the element of L2(Ω,H) defined by:
m
DF = ∂ f(ζH(ϕ ),...,ζH(ϕ ))ϕ , (2.18)
i 1 m i
i=1
X
m
D¯F = ∂¯f(ζH(ϕ ),...,ζH(ϕ ))ϕ¯ , (2.19)
i 1 m i
i=1
X
where ∂ f = ∂ f(z ,...,z ), ∂¯ f = ∂ f(z ,...,z ) are the Wirtinger derivatives [4].
j ∂zj 1 m j ∂z¯j 1 m
Proposition 2.5. Suppose that l =m+n and
m 2 n 2
η = i i!j!f˜ h, (2.20)
r i,j
i j ⊗
i+j=r (cid:18) (cid:19) (cid:18) (cid:19)
X
m 2 n 2
ξ = j i!j!h˜ f, (2.21)
r j,i
i j ⊗
i+j=r (cid:18) (cid:19) (cid:18) (cid:19)
X
m n n m
ν = i i!j!f˜ f, (2.22)
r i,j
i i j j ⊗
i+j=r (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
X
then we have that
l 1
Var( DI (f) 2)= − [(l r)!]2 η 2 , (2.23)
k m,n kH − k rkH⊗(2(l−r))
r=1
X
l 1
Var( D¯I (f) 2)= − [(l r)!]2 ξ 2 , (2.24)
m,n H − k rkH⊗(2(l−r))
r=1
(cid:13) (cid:13) X
(cid:13) (cid:13) l 1
Var(hDIm,n(f), DIm,n(f)iH)= − (2m−r)!(2n−r)!kνrk2H⊗(2(l−r)). (2.25)
r=1
X
Proof. We need only to show (2.23) since the other two are similar. Denote l =m+n 1.
′
−
Step 1: Using product formula. By Theorem 12(D) of [10] and the product formula of complex
Wiener-Itô multiple integrals (Theorem 3.2 of [5]), we have that
1
D(I (f)) 2
m2 k · m,n kH
= I f (~tm 1,~sn) 2
m 1,n u − H
−
(cid:13) (cid:0) (cid:1)(cid:13)
(cid:13) (cid:13)
COMPLEX FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 7
= dudvφ(u,v)I f (~tm 1,~sn) I f (~tm 1,~sn)
m 1,n u − m 1,n v −
Z[0,∞)2 − (cid:0) (cid:1) − (cid:0) (cid:1)
m 1 n 2 2
− m 1 n
= − i!j! dudvφ(u,v)Il′ i j,l′ i j fu(~tm−1,~sn) i,j hv(~tn,~sm−1)
Xi=0 Xj=0 (cid:18) i (cid:19) (cid:18)j(cid:19) Z[0,∞)2 −− −− (cid:0) ⊗ (cid:1)
where hv(~tn,~sm 1)=f¯(~sm 1,~tn)=f¯(~sm 1,v,~tn) and
− v − −
f (~tm 1,~sn) hv(~tn,~sm 1) (t¯l′ i j,u,s¯l′ i j,v) (2.26)
u − i,j − −− −−
⊗
(cid:16) (cid:17)
= d~xidx~′iφ(x1,x′1)...φ(xi,x′i)fu(~tm−1−i,~xi,~sn−j,~yj)
Z[0,∞)i+j
×f¯v(sn−j+1,...,sl′−j−i,x~′i,tm−i,...,tl′−i−j,y~′j)φ(y1,y1′)...φ(yj,yj′)d~yjdy~′j.
Then we obtain that
l′
1
m2 kD·(Im,n(f))k2H =Xr=0Z[0,∞)2dudvφ(u,v)Il′−r,l′−r(gr(u,v)). (2.27)
where
2 2
m 1 n
gk(u,v)= − i!j!fu(~tm−1,~sn)˜i,jhv(~tn,~sm−1). (2.28)
i j ⊗
i+j=k(cid:18) (cid:19) (cid:18) (cid:19)
X
Taking expectation to Eq.(2.27), we have that
1
m2E[kD·(Im,n(f))k2H]=Z[0,∞)2dudvφ(u,v)gl′(u,v)
=(m 1)!n! dudvφ(u,v)f hv
u m 1,n
− Z[0,∞)2 ⊗ −
=(m 1)!n! f 2 . (2.29)
− k kH⊗(m+n)
Step 2: Calculating variance. It follows from Fubini’s theorem and Itô’s isometry that we have:
1 E[ D I (f) 4] (2.30)
m4 · m,n H
l′ (cid:13) (cid:0) (cid:1)(cid:13)
(cid:13) (cid:13)
= dudvdu′dv′φ(u,v)φ(u′,v′)[(l′−r)!]2hgr(u,v),gr(u′,v′)iH⊗2(l′−r).
Xr=0Z[0,∞)4
It is easy to check that
dudvdu′dv′φ(u,v)φ(u′,v′)hfu⊗˜i,jhv,fu′⊗˜i,jhv′iH⊗2(l′−k)
Z[0,∞)4
=hf⊗˜i+1,jh,f⊗˜i+1,jhiH⊗2(l′−k) = f⊗˜i+1,jh 2H⊗2(l′−k),
which implies that (cid:13) (cid:13)
(cid:13) (cid:13)
l′
1
m4E[ D· Im,n(f) 4H]= [(l′−r)!]2hGr, GriH⊗2(l′−r),
r=0
(cid:13) (cid:0) (cid:1)(cid:13) X
where (cid:13) (cid:13)
m 1 2 n 2
G = − i!j!f(~tm,~sn)˜ h(~tn,~sm). (2.31)
k i+1,j
i j ⊗
i+j=k(cid:18) (cid:19) (cid:18) (cid:19)
X
8 Y.CHEN,Y.HU,ANDZ.WANG
Especially, for the term with k=l , we have that
′
Gl′ 2 =[(m 1)!n!]2 f m,nh2
| | − | ⊗ |
1 2
=[(m 1)!n!]2 f 4 = E[ D(I (f)) 2] .
− k kH⊗(m+n) m2 k · m,n kH
(cid:16) (cid:17)
Substituting the above equality displayed into (2.30), we have that
l′ 1
Var(kDIm,n(f)k2H)=m4 − [(l′−r′)!]2hGr′, Gr′iH⊗2(l′−r′)
r′=0
X
l 1
= − [(l r)!]2 η 2 ( let l=l +1, r =r +1),
− k rkH⊗(2(l−r)) ′ ′
r=1
X
where ηr =m2Gr′, which implies the desired expressions (2.20)-(2.23).
(cid:3)
Proof of Theorem 1.3. Since (i) (ii) is wellknown, we needonly to showthe followingimplica-
⇒
tions:
(ii) (iii) (iv) (i).
⇒ ⇒ ⇒
[(ii) (iii) ] Condition (ii) implies that as k ,
⇒ →∞
E[F 4] 2 E[F 2] 2 E[F 2] 2 0, (2.32)
k k k
| | − | | − →
which implies that Condition (iii) holds(cid:0)by (2.8)-(cid:1)(2.9)(cid:12), (see L(cid:12)emma 2.3).
(cid:12) (cid:12)
[(iii) (iv)] Suppose that Condition (iii) holds. The inequality (5.2) of [10] implies that as
⇒
k ,
→∞
f ˜ f 0, f ˜ h = h ˜ f 0.
k⊗i,j k H⊗(2(l−i−j)) → k⊗i,j k H⊗(2(l−i−j)) k⊗i,j k H⊗(2(l−i−j)) →
Theref(cid:13)ore, it foll(cid:13)ows from Minkowski’s(cid:13)inequalit(cid:13)y and Proposit(cid:13)ion 2.5 tha(cid:13)t as k ,
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) →∞
ηk 0, ξk 0, νk 0, r =1,...,l 1,
r → r → r → −
whereηk, ξk, νk aregivenasEquations(2.20)-(2.22). By(2.23)-(2.25),weobtainthatCondition
r r r
(iv) holds.
[(iv) (i)]Wefollowtheideaof[11,Theorem4],i.e. wecombineMalliavincalculusandpartial
⇒
differential equations. Let
ϕ (z)=E ei(z¯Fk+zF¯k)/2 .
k
Then we have that (cid:2) (cid:3)
∂∂ϕzk = 2iE F¯k×ei(z¯Fk+zF¯k)/2 , (2.33)
( ∂∂ϕz¯k = 2iE(cid:2)Fk×ei(z¯Fk+zF¯k)/2(cid:3).
By the assumption E[F 2] σ2, F a(cid:2)re tight. Now sup(cid:3)pose that the subsequence F
| k| → { k} { nk}
converges to G in law. Without ambiguity, we still denote F by F . By the hypercon-
{ nk} { k}
tractivity inequality ofcomplex multiple Wiener-Itô integrals[5], F r is uniformly integrable
k
{| | }
and thus E[Gr]=lim E[F r] for all r 1 [3, Theorem 5.4]. Therefore, the characteristic
k k
function ϕ(z|)=| E[e2i(z¯G→+z∞G¯)] |has|continuous≥partial derivatives of any order.
COMPLEX FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES 9
It is not difficult to see that
1
E F¯ ei(z¯Fk+zF¯k)/2 = E (δD)F ei(z¯Fk+zF¯k)/2
k k
× m ×
(cid:2) (cid:3)= 1 E(cid:2) D(ei(z¯Fk+zF¯k)/2), DFk(cid:3)H
m h i
= 1 E(cid:2) ei(z¯Fk+zF¯k)/2(z¯DFk+zD(cid:3)F¯k), DFk H
m h i
Clearly, for any z C, ei(z¯Fk+zF¯k)/2 ei(z¯(cid:2)G+zG¯)/2 in L2(Ω). Thus, Condition (iv(cid:3)) implies that
∈ →
as k ,
→∞
E ei(z¯Fk+zF¯k)/2(z¯DFk+zDF¯k), DFk H
h i
→(cid:2) (z¯klim E[kDFkk2H]+zklim E[hDF¯kD(cid:3)FkiH])ϕ(z), ∀z ∈C,
→∞ →∞
sincethescalarproductintheHilbertspaceL2(Ω)dependscontinuouslyonitsfactors. Itfollows
from (2.29) and
1
E DIm,n(f), DIm,n(f) H =δm,nm!(m 1)!f m,mf,
mn h i − ⊗
that (cid:2) (cid:3)
lim E[ DF 2]=m lim E[ F 2]=mσ2,
k k kkH k k kkH
→∞ →∞
klim E[hDF¯kDFkiH]=mδm,nklim E[Fk2]
→∞ →∞
=m(c ib)δ .
m,n
−
Therefore, it follows from (2.33) that for any z C,
∈
∂ϕ
=[z¯σ2+z (c ib)δ ]ϕ(z). (2.34)
m,n
∂z · −
In the same way,
∂ϕ
=[z¯ (c+ib)δ +zσ2]ϕ(z). (2.35)
m,n
∂z¯ ·
Clearly, ϕ(0) = 1. Therefore, G is a bivariate normal distribution with covariance matrix
σ2+c b
C= 1 . Prokhorov’stheoremimpliesthat F convergestoabivariatenormal
2" b σ2 c# { k}
distribution with th−e desired covariance matrix C. ✷
3. Asymptotic consistency and normality
We need several propositions and lemmas before the proof of Theorem 1.1. The following
lemma’s proof is easy.
Lemma 3.1. For any H (1,1), we have that
∈ 2
e−γu1−γ¯u2 u1 u2 2H−2du1du2 =d, (3.1)
| − |
Z[0,∞)2
where d is defined by (1.7).
10 Y.CHEN,Y.HU,ANDZ.WANG
Proposition 3.2. Let Z be the solution to (1.1). As T , we have that
→∞
1 T
Z 2dt aα d, a.s. (3.2)
t H
T | | →
Z0
Proof. Denote Y = √a t e γ(t s)dζ . It is easy to see that Y is centered complex Gaussian
t − − s
process. Itô’s isometry im−p∞lies that for any t R,s 0,
R ∈ ≥
E[Yt+sY¯t]=αHa ∞dv1 ∞dv2e−γ(v1+s)e−γ¯v2 v1 v2 2H−2 =E[YsY¯0]. (3.3)
| − |
Z−s Z0
Thus Y is stationary. It is easy to check that as s , E[Y Y¯ ] 0 with the same order as
t s 0
→ ∞ →
s2H−2, which implies that Yt is ergodic [7, p78].
| | { }
Then we have that
t
Z =e γtZ +√a e γ(t s)dζ
t − 0 − − s
Z0
t 0
=e γtZ +√a e γ(t s)dζ e γt√a eγsdζ
− 0 − − s − s
−
Z−∞ Z−∞
=Y +e γt(Z Y ).
t − 0 0
−
The ergodicity and Cauchy-Schwarzinequality imply that as T ,
→∞
1 T 1 T
Z 2dt= Y 2 2 (e γt(Z Y )Y¯)+e 2λt Z Y 2 dt
t t − 0 0 t − 0 0
T | | T | | − ℜ − | − |
Z0 Z0 h i
lim E[Y 2]=aα d, a.s.
T H
→T | |
→∞
where the last equality is from Eq.(3.3) and Lemma 3.1. (cid:3)
Denote
ψt(r,s)=e−γ¯(r−s)1 0 s r t and Xt =I1,1(ψt(r,s)). (3.4)
{ ≤ ≤ ≤ }
Lemma 3.3. As n , the sequence ξ := 1X converges to zero almost sure.
→∞ n n n
Proof. Denote F = 1 X . Lemma 3.(cid:8)5 implies th(cid:9)at sup E[F 2] < . From the hypercon-
T √T T n | n| ∞
tractivity of multiple Wiener-Itô integrals [5, Proposition 2.4], we see that sup E[F 4] < .
n | n| ∞
For any fixed ε>0, it follows from Chebyshev’s inequality that
1 34
P ξ >ε =P F >√nε E[F 4] E[F 2]2.
| n| | n| ≤ n2ε4 | n| ≤ n2ε4 | n|
The Borel-Cantell(cid:0)i lemma(cid:1)implie(cid:0)s that ξ c(cid:1)onverges to zero almost surely. (cid:3)
n
{ }
Proposition 3.4. For any real number p 2 and integer n 1,
≥ ≥
n+1 n+1 X X p
t s
B := | − | dsdt (3.5)
n Zn Zn |t−s|2pH
is finite. Moreover, for any real numbers p>2, q >1 and integer n 1,
≥
X X R nq/p, t ,t [n,n+1], (3.6)
| t2 − t1|≤ p,q ∀ 1 2 ∈
where R is a random constant independent of n.
p,q