Functional limit theorem for the self-intersection local time of the fractional Brownian motion

0.57 MB·

by  Arturo Jaramillo

FUNCTIONAL LIMIT THEOREM FOR THE SELF-INTERSECTION
LOCAL TIME OF THE FRACTIONAL BROWNIAN MOTION
ARTURO JARAMILLO AND DAVID NUALART
7
1 Abstract. Let Bt t≥0 bead-dimensionalfractionalBrownianmotionwithHurstparam-
{ }
0 eter 0<H <1, where d 2. Consider the approximationofthe self-intersectionlocaltime
≥
2 of B, defined as

n T t
ε
a IT = pε(Bt Bs)dsdt,
J 0 0 −
Z Z
9 where pε(x) is the heat kernel. We prove that the process ITε E[ITε] T≥0, rescaled by
{ − }
1 a suitable normalization, converges in law to a constant multiple of a standard Brownian
R] m3 o<tioHn f<or12,3din<thHe s≤pa43ceanCd[0t,o a),meunldtiopwleedofwaithsutmheoftoipnodleopgeyndoefnutnHifoerrmmitceonpvreorcgeesnsecsefoonr
4 ∞
P compacts.
.
h
t
a
m
[
1. Introduction

1
v Let B = B be a d-dimensional fractional Brownian motion of Hurst parameter
9 { t}t≥0
H (0,1). Fix T > 0. The self-intersection local time of B in the interval [0,T] is formally
8
∈
2 defined by
5
0 T t
1. I := δ(Bt −Bs)dsdt,
0 Z0 Z0
7 where δ denotes the Dirac delta function. A rigorous definition of this random variable may
1
be obtained by approximating the delta function by the heat kernel
:
v
i 1
X pε(x) := (2πε)−d2 exp x 2 , x Rd.
r −2ε k k ∈
a (cid:26) (cid:27)
In the case H = 1, B is a classical Brownian motion, and its self-intersection local time has
2
been studied by many authors (see Albeverio (1995), Hu (1996), Imkeller, Pérez-Abreu and
Vives (1995), Varadhan (1969), Yor (1985) and the references therein). In the case H = 1,
6 2
the self-intersection local time for B was first studied by Rosen in [13] in the planar case
and it was further investigated using techniques from Malliavin calculus by Hu and Nualart
in [5]. In particular, it was proved that the approximation of the self-intersection local time
Date: January 20, 2017.
2010 Mathematics Subject Classification. 60G05; 60H07; 60G15; 60F17.
Keywords andphrases. FractionalBrownianmotion,self-intersectionlocaltime,Wienerchaosexpansion,
central limit theorem.
D. Nualart was supported by the NSF grant DMS1512891.
1
2 ARTUROJARAMILLOAND DAVIDNUALART
of B in [0,T], defined by
T t
Iε := p (B B )dsdt, (1.1)
T ε t − s
Z0 Z0
converges in L2(Ω) when H < 1. Furthermore, it was shown that when 1 H < 3 ,
d d ≤ 2d
Iε E[Iε] to converges in L2(Ω), and for the case 3 < H < 3, the following limit theorem
T − T 2d 4
holds (see [5, Theorem 2]).
Theorem 1.1. If 23d < H < 43, then εd2−43H(ITε − E[ITε]) converges in law to a centered
Gaussian distribution with variance σ2T, as ε 0, where the constant σ2 is given by (3.3).
→
The case H = 3 was addressed as well in [5], where it was shown that the sequence
2d
(log(1/ε))−1(Iε E[Iε]) converges in law to a centered Gaussian distribution with variance
2 T − T
σ2 , as ε 0, where σ2 is the constant given by [5, Equation (42)].
log → log
The aim of this paper is to prove a functional version of Theorem 1.1, and extend it to
the case 3 H < 1. Our main results are Theorems 1.2, 1.3 and 1.4.
4 ≤
Theorem 1.2. Let 3 < H < 3, d 2 be fixed. Then,
2d 4 ≥
{εd2−43H(ITε −E[ITε])}T≥0 L→aw {σWT}T≥0, (1.2)
in the space C[0, ), endowed with the topology of uniform convergence on compact sets,
∞
where W is a standard Brownian motion, and the constant σ2 is given by (3.3).
Webriefly outlinetheproofof (1.2). Theproofoftheconvergence ofthefinite-dimensional
distributions, is based on the application of a multivariate central limit theorem established
by Peccati and Tudor in [12] (see Section 2.3), and follows ideas similar to those presented
in [5]. On the other hand, proving the tightness property for the process
Iε := ε2d−43H(Iε E[Iε]),
T T − T
presents a great technical difficulty. In fact, by the Billingsley criterion (see [2, Theo-
e
rem 12.3]), the tightness property can be obtained by showing that there exists p > 2,
such that for every 0 T T ,
1 2
≤ ≤
E ITε2 −ITε1 p ≤ C|T2 −T1|p2 , (1.3)
h(cid:12) (cid:12) i
for some constant C > 0 indepen(cid:12)dent of T(cid:12),T and ε. The problem of finding a bound like
(cid:12)e e (cid:12)1 2
(1.3) comes from the fact that the smallest even integer such that p > 2 is p = 4, and a direct
computation of the moment of order four E Iε Iε 4 is too complicated to be handled.
| T2 − T1|
To overcome this difficulty, in this paper weh introduce ia new approach to prove tightness
based on the techniques of Malliavin calculus.eLet ues describe the main ingredients of this
approach.
First, we write the centered random variable Z := Iε Iε as
T2 − T1
Z = δDL−1Z,
e e
−
FUNCTIONAL CLT FOR THE SELF-INTERSECTION LOCAL TIME OF THE FBM 3
where δ, D and L are the basic operators in Malliavin calculus. Then, taking into consider-
ation that E[DL−1Z] = 0 we apply Meyer’s inequalities to obtain a bound of the type
Z c D2L−1Z , (1.4)
Lp(Ω) p Lp(Ω;(Hd)⊗2)
k k ≤ k k
for any p > 1, where the Hilbert space H is defined in Section 2.1. Notice that
Z = εd2−43H (pε(Bt Bs) E[pε(Bt Bs)])dsdt.
− − −
Z0≤s≤t,T1≤t≤T2
Applying Minkowski’s inequality and (1.4), we obtain
Z Lp(Ω) cpεd2−43H D2L−1pε(Bt Bs) pdsdt.
k k ≤ k − k
Z0≤s≤t,T1≤t≤T2
Then, we get the desired estimate by choosing p > 2 close to 2, using the self-similarity of
the fractional Brownian motion, the expression of the operator L−1 in terms of the Ornstein-
Uhlenbeck semigroup, Mehler’s formula and Gaussian computations. In this way, we reduce
the problem to showing the finiteness of an integral (see Lemma 5.3), similar to the integral
appearing in the proof of the convergence of the variances. It is worth mentioning that this
approach for proving tightness has not been used before, and has its own interest.
In the case H > 34, the process εd2−23H+1(ITε −E[ITε]) also converges in law, in the topology
of C[0, ), but the limit is no longer a multiple of a Brownian motion, but a multiple of
∞
a sum of independent Hermite processes of order two. More precisely, if Xj denotes
{ T}T≥0
(j)
the second order Hermite process, with respect to B , defined in Section 2.1, then
{ t }t≥0
Iε satisfies the following limit theorem
ε∈(0,1)
{ }
Theorem 1.3. Let H > 3, and d 2 be fixed. Then, for every T > 0,
e 4 ≥
d
εd2−23H+1(Iε E[Iε]) L2(Ω) Λ Xj, (1.5)
T − T → − T
j=1
X
where the constant Λ is defined by
(2π)−d ∞
Λ := 2 (1+u2H)−d2−1u2du. (1.6)
2
Z0
In addition,
d
{εd2−23H+1(ITε −E[ITε])}T≥0 L→aw {−Λ XTj}T≥0, (1.7)
j=1
X
in the space C[0, ), endowed with the topology of uniform convergence on compact sets.
∞
We briefly outline the proof of Theorem 1.3. The convergence (1.5) is obtained from the
chaotic decomposition of Iε. It turns out that the chaos of order two completely determines
T
the asymptotic behavior of εd2−23H+1(ITε − E[ITε]), and consequently, (1.5) can be obtained
by the characterization of the Hermite processes presented in [8], applied to the second
chaotic component of Iε. Similarly to the case 3 < H < 3, we show that the sequence
T 2d 4
εd2−23H+1(ITε −E[ITε]) is tight, which proves the convergence in law (1.7).
4 ARTUROJARAMILLOAND DAVIDNUALART
The technique we use to prove tightness doesn’t work for the case Hd 3, so the conver-
≤ 2
gence in law of log(1/ε)−1(Iε E[Iε]) to a scalar multiple of a Brownian motion for
{ 2 T − T }T≥0
the case Hd = 3 still remains open. Nevertheless, for the critical case H = 3 and d 3, the
2 4 ≥
technique does work, and we prove the following limit theorem
Theorem 1.4. Suppose H = 3 and d 3. Then,
4 ≥
εd−1
2 (Iε E[Iε]) Law ρW , (1.8)
{ log(1/ε) T − T }T≥0 → { T}T≥0
in the space C[0, ), enpdowed with the topology of uniform convergence on compact sets,
∞
where W is a standard Brownian motion, and the constant ρ is defined by (3.52).
Remark
We impose the stronger condition d 3 instead of d 2, since the choice H = 3, d = 2
≥ ≥ 4
gives Hd = 3, and as mentioned before, it is not clear how to prove tightness for this case.
2
We briefly outline the proof of Theorem 1.4. The proof of the tightness property is
analogous to the case 3 < H < 3. On the other hand, the proof of the convergence of the
2d 4
finite dimensional distributions requires a new approach. First we show that, as in the case
H > 3, the chaos of order two determines the asymptotic behavior of Iε . Then we
4 { T}T≥0
describe the behavior of the second chaotic component of Iε, which we denote by J (Iε),
T 2 T
and is given by
J (Iε) = (2π)−2dε32−d2 d T ε−32(T−s) u23 H Bs(j+)ε32u −Bs(j) duds, (1.9)
2 T − 2 j=1 Z0 Z0 (1+u32)2d+1 2 √εu34 
X
 
where H denotes the Hermite polynomial of order 2. Then we show that we can replace
2
the domain of integration of u by [0, ), and this integral can be approximated by Riemann
∞
sums of the type
1 M2M u(k)23 T Bs(j+)ε2Mu(k) −Bs(j)
H ds, (1.10)
−2M k=2 (1+u(k)23)d2+1 Z0 2 √εu(k)43 
X
 
where u(k) = k , and M is some fixed positive number. By [3, Equation (1.4)], we have
2M
that, for k fixed, the random variable
(j) (j)
B B
ξε(T) := ε−31 T H s+ε32u(k) − s ds
k log(1/ε) 2 √εu(k)3 
Z0 4
converges in law to a Gaussianpdistribution as ε 0. Hence, aftera suitable analysis of the
→
covariances of the process ξε(T) 2 k M2M, and T 0 and an application of the
{ k | ≤ ≤ ≥ }
Peccati-Tudor criterion (see [12]), we obtain that the process (1.10) multiplied by the factor
(2π)−d2ε−31 converges to a constant multiple of a Brownian motion ρ W, for some ρ > 0.
2√log(1/ε) M M
The result then follows by proving that the approximations (1.10) to the integrals in the
FUNCTIONAL CLT FOR THE SELF-INTERSECTION LOCAL TIME OF THE FBM 5
right-hand side of (1.9) are uniform over ε (0,1/e) as M , and that ρ ρ as
M
∈ → ∞ →
M .
→ ∞
The paper is organized as follows. In Section 2 we present some preliminary results on the
fractional Brownian motion and the chaotic decomposition of Iε. In Section 3, we compute
T
the asymptotic behavior of the variances of the chaotic components of Iε as ε 0. The
T →
proofs of the main results are presented in Section 4. Finally, in Section 5 we prove some
technical lemmas.
2. Preliminaries and main results
2.1. Some elements of Malliavin calculus for the fractional Brownian motion.
(1) (d)
Throughout the paper, B = (B ,...,B ) will denote a d-dimensional fractional
Brownian motion with Hurst pa{ramteter H t(0,}1t≥),0defined on a probability space (Ω, ,P).
That is, B is a centered, Rd-valued Gaussia∈n process with covariance function F
δ
E B(i)B(j) = i,j(t2H +s2H t s 2H).
t s 2 −| − |
h i
We will denote by H the Hilbert space obtained by taking the completion of the space of
step functions on [0, ), endowed with the inner product
∞
, := E B(1) B(1) B(1) B(1) , for 0 a b, and 0 c d.
1[a,b] 1[c,d] H b − a d − c ≤ ≤ ≤ ≤
h(cid:16) (cid:17)(cid:16) (cid:17)i
(cid:10) (cid:11)
(j)
For every 1 j d fixed, the mapping B can be extended to linear isometry
≤ ≤ 1[0,t] 7→ t
between H and the Gaussian subspace of L2(Ω) generated by the process B(j). We will
denote this isometry by B(j)(f), for f H. If f Hd is of the form f = (f ,...,f ), with
1 d
∈ ∈
f H, we set B(f) := d B(j)(f ). Then f B(f) is a linear isometry between Hd and
j ∈ j=1 j 7→
the Gaussian subspace of L2(Ω) generated by B.
P
For any integer q 1, we denote by (Hd)⊗q and (Hd)⊙q the qth tensor product of Hd,
≥
and the qth symmetric tensor product of Hd, respectively. The qth Wiener chaos of L2(Ω),
denoted by , is the closed subspace of L2(Ω) generated by the variables
q
H
d d
H (B(j)(f )) q = q, and f ,...,f H, f = 1 ,
qj j | j 1 d ∈ k jkH
( )
j=1 j=1
Y X
where H is the qth Hermite polynomal, defined by
q
Hq(x) := (−1)qex22 ddxqqe−x22.
For q N, with q 1, and f Hd of the form f = (f ,...,f ), with f = 1, we can
∈ ≥ ∈ 1 d k jkH
write
d
f⊗q = f f .
i1 ⊗···⊗ iq
i1,.X..,iq=1
6 ARTUROJARAMILLOAND DAVIDNUALART
For such f, we define the mapping
d d
I (f⊗q) := H (B(j)(f )),
q qj(i1,...,iq) j
i1,.X..,iq=1Yj=1
where q (i ,...,i ) denotes the number of indices in (i ,...,i ) equal to j. The range of I is
j 1 q 1 q q
contained in . Furthermore, this mapping can be extended to a linear isometry between
q
H
H⊙q (equipped with the norm √q! ) and (equipped with the L2(Ω)-norm).
k·k(Hd)⊗q Hq
Denote by the σ-algebra generated by B. It is well known that every square integrable
G
random variable -measurable, has a chaos decomposition of the type
G
∞
F = E[F]+ I (f ),
q q
q=1
X
for some f (Hd)⊙q. In what follows, we will denote by J (F), for q 1, the projection of
q q
∈ ≥
F over the qth Wiener chaos , and by J (F) the expectation of F.
q 0
H
Let denote the set of all cylindrical random variables of the form
S
F = g(B(h ),...,B(h )),
1 n
where g : Rn R is an infinitely differentiable function with compact support, and h Hd.
j
→ ∈
The Malliavin derivative of F with respect to B, is the element of L2(Ω;Hd), defined by
n
∂g
DF = (B(h ),...,B(h ))h .
1 n i
∂x
i
i=1
X
By iteration, one can define the rth derivative Dr for every r 2, which is an element of
≥
L2(Ω;(Hd)⊗r).
For p 1 and r 1, the space Dr,p denotes the closure of with respect to the norm
≥ ≥ S
, defined by
Dr,p
k·k
1
r p
F := E[ F p]+ E DiF p .
k kDr,p | | (Hd)⊗i
!
Xi=1 h(cid:13) (cid:13) i
The operator Dr can be consistently extended to the(cid:13)space(cid:13)Dr,p. We denote by δ the adjoint
of the operator D, also called the divergence operator. A random element u L2(Ω;Hd)
∈
belongs to the domain of δ, denoted by Domδ, if and only if satisfies
1
E DF,u C E F2 2 , for every F D1,2,
h iHd ≤ u ∈
where C is a constan(cid:12)t o(cid:2)nly depen(cid:3)d(cid:12)ing on u(cid:2). If(cid:3)u Domδ, then the random variable δ(u) is
u
(cid:12) (cid:12) ∈
defined by the duality relationship
E[Fδ(u)] = E DF,u ,
h iHd
which holds for every F D1,2. The operator L(cid:2) is defined(cid:3)on the Wiener chaos by
∈
∞
LF := qJ F, for F L2(Ω),
q
− ∈
q=1
X
FUNCTIONAL CLT FOR THE SELF-INTERSECTION LOCAL TIME OF THE FBM 7
and coincides with the infinitesimal generator of the Ornstein-Uhlenbeck semigroup P ,
θ θ≥0
{ }
which is defined by
∞
P := e−qθJ .
θ q
q=0
X
A random variable F belongs to the domain of L if and only if F D1,2, and DF Domδ,
∈ ∈
in which case
δDF = LF.
−
We also define the operator L−1 as
∞
1
L−1F = J F, for F L2(Ω).
q
−q ∈
q=1
X
Notice that L−1 is a bounded operator and satisfies LL−1F = F E[F] for every F L2(Ω),
− ∈
sothatL−1 actsasapseudo-inverse ofL. TheoperatorL−1 satisfies thefollowingcontraction
property for every F L2(Ω) with E[F] = 0,
∈
E DL−1F 2 E F2 .
Hd ≤
h i
(cid:13) (cid:13) (cid:2) (cid:3)
In addition, by Meyer’s inequalities (see [10, Proposition 1.5.8]), for every p > 1, there exists
(cid:13) (cid:13)
a constant c > 0 such that the following relation holds for every F D2,p, with E[F] = 0
p
∈
δ(DL−1F) c ( D2L−1F + E DL−1F ). (2.1)
Lp(Ω) ≤ p Lp(Ω;(Hd)⊗2) (H)d
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:2) (cid:3)(cid:13)
Assume that(cid:13)B is an ind(cid:13)ependent co(cid:13)py of B, a(cid:13)nd such that B(cid:13),B are defin(cid:13)ed in the product
space (Ω Ω, ,P P). Given a random variable F L2(Ω), measurable with respect to
× F⊗F ⊗ ∈
the σ-algebra egenerated by B, we can write F = Ψ (B), where Ψe is a measurable mapping
F F
from RHd toeR, deteermineed P-a.s. Then, for every θ 0 we have the Mehler formula
≥
P F = E Ψ (e−θB + 1 e−2θB) , (2.2)
θ F
−
h p i
where E denotes the expectation weith respect to P. The opeerator L−1 can be expressed in
terms of P , as follows
θ
e ∞ e
L−1F = P Fdθ, for F such that E[F] = 0. (2.3)
θ
Z0
2.2. Hermite process. When H > 1, the inner product in the space H can be written, for
2
every step functions ϕ,ϑ on [0, ), as
∞
ϕ,ϑ = H(2H 1) ϕ(ξ)ϑ(ν) ξ ν 2H−2dξdν. (2.4)
h iH − | − |
R2
Z +
Following [8], we introduce the Hermite process Xj of order 2, associated to the jth
{ T}T≥0
(j)
component of B, B , and describe some of its properties. The family of kernels
{ t }t≥0
8 ARTUROJARAMILLOAND DAVIDNUALART
ϕε T 0,ε (0,1) (Hd)⊗2, defined, for every multi-index i = (i ,i ), 1 i ,i d,
{ j,T | ≥ ∈ } ⊂ 1 2 ≤ 1 2 ≤
by
T
ϕε (i,x ,x ) := ε−2 δ δ (x ) (x )ds, (2.5)
j,T 1 2 j,i1 j,i21[s,s+ε] 1 1[s,s+ε] 2
Z0
satisfies the following relation for every H > 3, and T 0
4 ≥
lim ϕε ,ϕη = H2(2H 1)2 s s 4H−4ds ds = c T4H−2, (2.6)
ε,η→0 j,T j,T (Hd)⊗2 − Z[0,T]2| 1 − 2| 1 2 H
(cid:10) (cid:11)
where c := H2(2H−1). This implies that ϕε converges, as ε 0, to an element of (Hd)⊗2,
H 4H−3 j,T →
denoted by πj. In particular, for every K > 0, ϕε is bounded by some constant
T j,K (Hd)⊗2
C , only depending on K and H. On the other hand, by (2.4) and (2.5), we deduce that
K,H (cid:13) (cid:13)
for every T [0,K], it holds ϕε ϕε (cid:13) (cid:13), and hence
∈ j,T (Hd)⊗2 ≤ j,K (Hd)⊗2
sup ϕε ,(cid:13)(cid:13)ϕη (cid:13)(cid:13) (cid:13)(cid:13) sup(cid:13)(cid:13) ϕε ϕη
j,T1 j,T2 (Hd)⊗2 ≤ j,T1 (Hd)⊗2 j,T2 (Hd)⊗2
T1,T2∈(0,K](cid:12) (cid:12) T1,T2∈(0,K]
ε,η∈(0,1) (cid:10) (cid:11) ε,η∈(0,1) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:12) (cid:12) sup ϕε 2 C . (2.7)
≤ j,K (Hd)⊗2 ≤ K,H
ε∈(0,1)
(cid:13) (cid:13)
The element πj, can be characterized as follows(cid:13). For(cid:13)any vector of step functions with
T
compact support f = (f(1),...,f(d)) Hd, i = 1,2, we have
i i i ∈
πj,f f = lim ϕε ,f f
t 1 ⊗ 2 (Hd)⊗2 ε→0 j,t 1 ⊗ 2 (Hd)⊗2
(cid:10) (cid:11) = lim(cid:10)ε−2H2(2H (cid:11)1)2
ε→0 −
T s+ε T
ξ η 2H−2f(j)(η)dηdξds
× | − | i
Z0 i=1,2Zs Z0
Y
and hence
T T
πj,f f = H2(2H 1)2 s η 2H−2f(j)(η)dηds. (2.8)
t 1 ⊗ 2 (Hd)⊗2 − | − | i
Z0 i=1,2Z0
(cid:10) (cid:11) Y
We define the second order Hermite process Xj , with respect to B(j) , as Xj :=
{ T}T≥0 { t }t≥0 T
I (πj).
2 T
2.3. A multivariate central limit theorem. In the seminal paper [11], Nualart and Pec-
cati established a central limit theorem for sequences of multiple stochastic integrals of a
fixed order. In this context, assuming that the variances converge, convergence in distri-
bution to a centered Gaussian law is actually equivalent to convergence of just the fourth
moment. Shortly afterwards, in [12], Peccati and Tudor gave a multidimensional version of
this characterization. More recent developments on these type of results have been addressed
by using Stein’s method and Malliavin techniques (see the monograph by Nourdin and Pec-
cati [9] and the references therein). In the sequel, we will use the following multivariate
central limit theorem obtained by Peccati and Tudor in [12] (see also Theorems 6.2.3 and
6.3.1 in [9]).
FUNCTIONAL CLT FOR THE SELF-INTERSECTION LOCAL TIME OF THE FBM 9
Theorem 2.1. For r N fixed, consider a sequence F of random vectors of the form
n n≥1
∈ { }
F = (F(1),...,F(r)). Suppose that for i = 1,...,r and n N, the random variables F(i)
n n n n
∈
belong to L2(Ω), and have chaos decomposition
∞
F(i) = I (f ),
n q q,i,n
q=1
X
for some f (Hd)⊗q. Suppose, in addition, that for every q 1, there is a real symmetric
q,i,n
∈ ≥
non negative definite matrix C = Ci,j 1 i,j r , such that the following conditions
q { q | ≤ ≤ }
hold:
(i) For every fixed q 1, and 1 i,j r, we have q! f ,f Ci,j as
≥ ≤ ≤ h q,i,n q,j,ni(Hd)⊗q → q
n .
→ ∞
(ii) There exists a real symmetric nonnegative definite matrix C = Ci,j 1 i,j r ,
{ | ≤ ≤ }
such that Ci,j = lim Q Ci,j.
Q→∞ q=1 q
(iii) For all q 1 and i = 1,...,r, the sequence I (f ) converges in law to a
q q,i,n n≥1
≥ P { }
centered Gaussian distribution as n .
(iv) lim sup ∞ q! f 2 →= 0∞, for all i = 1,...,r.
Q→∞ n≥1 q=Q k q,i,nk(Hd)⊗q
Then, F converges inPlaw as n , to a centered Gaussian vector with covariance matrix
n
→ ∞
C.
2.4. Chaos decomposition for the self-intersection local time. In this section we
describe the chaos decomposition of the variable Iε defined by (1.1). Let ε (0,1), and
T ∈
T 0 be fixed. Define the set
≥
:= (s,t) R2 s t 1 .
R { ∈ + | ≤ ≤ }
For every γ > 0, we will denote by γ the set γ := γv v . First we write
R R { | ∈ R}
Iε = (s,t)p (B B )dsdt. (2.9)
T R2 1TR ε t − s
Z +
We can determine the chaos decomposition of the random variable p (B B ) appearing in
ε t s
(2.9) as follows. Given a multi-index i = (i ,...,i ), n N, 1 i d,−we set
n 1 n j
∈ ≤ ≤
α(i ) := E[X X ],
n i1··· in
where the X are independent standard Gaussian random variables. Notice that
i
(2q )! (2q )!
α(i ) = 1 ··· d , (2.10)
2q (q )! (q )!2q
1 d
···
if n = 2q is even and for each k = 1,...,d, the number of components of i equal to k,
2q
denoted by 2q , is also even, and α(i ) = 0 otherwise. Proceeding as in [5, Lemma 7], we
k n
can prove that
∞
p (B B ) = E[p (B B )]+ I fε , (2.11)
ε t − s ε t − s 2q 2q,s,t
q=1
X (cid:0) (cid:1)
10 ARTUROJARAMILLOAND DAVIDNUALART
where fε is the element of (Hd)⊗2q, given by
2q,s,t
(2π)−dα(i ) 2q
f2εq,s,t(i2q,x1,...,x2q) := (−1)q (22q)! 2q (ε+(t−s)2H)−d2−q 1[s,t](xj), (2.12)
j=1
Y
and
E[pε(Bt Bs)] = (2π)−d2(ε+(t s)2H)−d2. (2.13)
− −
By (2.9), (2.11) and (2.13), it follows that the random variable Iε has the chaos decomposi-
T
tion
∞
Iε = E[Iε]+ I (hε ), (2.14)
T T 2q 2q,T
q=1
X
where
hε (i ,x ,...,x ) := (s,t)fε (i ,x ,...,x )dsdt, (2.15)
2q,T 2q 1 2q R2 1TR 2q,s,t 2q 1 2q
Z +
and
E[ITε] = (2π)−d2 R2 1TR(s,t)(ε+(t−s)2H)−2ddsdt. (2.16)
Z +
InSection3,wewilldescribethebehaviorasε 0ofthecovariancefunctionoftheprocesses
→
Iε and I (hε ) . In order to address this problem, we will first introduce some
{ T}T≥0 { 2q 2q,T }T≥0
notation that will help us to describe the covariance function of the variables p (B B )
ε t s
−
and its chaotic components, which ultimately will lead to an expresion for the covariance
function of Iε.
T
First we describe the inner product fε ,fε . From (2.12), we can prove
2q,s1,t1 2q,s2,t2 (Hd)⊗2q
that for every 0 s t and 0 s t ,
≤ 1 ≤ 1 ≤ 2 ≤(cid:10)2 (cid:11)
(2π)−dα(i )2
f2εq,s1,t1,f2εq,s2,t2 (Hd)⊗2q = (2q1,...,2qd)! ((2q)!)22q (ε+(t1 −s1)2H)−d2−q
(cid:10) (cid:11) q1+·X··+qd=q
×(ε+(t2 −s2)2H)−d2−q 1⊗[s12,qt1],1⊗[s22,qt2] H⊗2q , (2.17)
D E
where (2q ,...,2q )! denotes the multinomial coefficient (2q ,...,2q )! = (2q)! . To
1 d 1 d (2q1)!···(2qd)!
computetheterm ⊗2q , ⊗2q appearinginthepreviousexpression, wewillintroduce
1[s1,t1] 1[s2,t2] H⊗2q
the following notaDtion. For everyEx,u ,u > 0, define
1 2
µ(x,u ,u ) := E B(1) B(1) B(1) . (2.18)
1 2 u1 x+u2 − x
h (cid:16) (cid:17)i
Define as well µ(x,u ,u ), for x < 0, by µ(x,u ,u ) := µ( x,u ,u ). Using the property of
1 2 1 2 2 1
−
stationary increments of B, we can check that for every s ,s ,t ,t 0, such that s t
1 2 1 2 1 1
≥ ≤
and s t , it holds
2 2
≤
E B(1) B(1) B(1) B(1) = µ(s s ,t s ,t s ). (2.19)
t1 − s1 t2 − s2 2 − 1 1 − 1 2 − 2
h(cid:16) (cid:17)(cid:16) (cid:17)i