getdoccc7e. 138KB Jun 04 2011 12:04:59 AM
Elect. Comm. in Probab. 14 (2009), 529–539
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
STOCHASTIC INTEGRAL REPRESENTATION OF THE L 2 MODULUS OF BROWNIAN LOCAL TIME AND A CENTRAL LIMIT THEOREM
YAOZHONG HU
Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045
email: [email protected]
DAVID NUALART1
Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045
email: [email protected]
Submitted August 19, 2009, accepted in final form November 11, 2009
AMS 2000 Subject classification: 60H07, 60F05, 60J55, 60J65
Keywords: Malliavin calculus, Clark-Ocone formula, Brownian local time, Knight theorem, central
limit theorem, Tanaka formula.
Abstract
The purpose of this note is to prove a central limit theorem for the L 2 -modulus of continuity of the
Brownian local time obtained in [3], using techniques of stochastic analysis. The main ingredients
of the proof are an asymptotic version of Knight’s theorem and the Clark-Ocone formula for the
L 2 -modulus of the Brownian local time.
1
Introduction
Let B = {B t , t ≥ 0} be a standard Brownian motion, and denote by {L t (x), t ≥ 0, x ∈ R} its local
time. In [3] the authors have proved the following central limit theorem for the L 2 modulus of
continuity of the local time:
Theorem 1. For each fixed t > 0,
Z
− 23
2
h
R
(L t (x + h) − L t (x)) d x − 4th
as h tends to zero, where
−→ 8
αt
3
η,
(1.1)
Z
(L t (x))2 d x,
αt =
R
and η is a N (0, 1) random variable independent of B.
1
Ç
L
D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0604207.
529
(1.2)
530
Electronic Communications in Probability
We make use of the notation
Z
G t (h) =
R
(L t (x + h) − L t (x))2 d x.
(1.3)
It is proved in [3, Lemma 8.1] that E G t (h) = 4th + O(h2 ). Therefore, we can replace the term
4th in (1.1) by E G t (h) .
The proof of Theorem 1 is done in [3] by the method of moments. The purpose of this paper is
to provide a simple proof of this result. Our method is based on an asymptotic version of Knight’s
theorem (see Revuz and Yor [7], Theorem (2.3), page 524) combined with the techniques of
stochastic analysis and Malliavin calculus. The main idea is to
apply the Clark-Ocone stochastic
integral representation formula to express G t (h) − E G t (h) as a stochastic integral. Then, by
means of simple estimates using Hölder’s inequality, it is proved that the leading term is a martingale, to which we can apply an asymptotic version of Knight’ theorem. An important ingredient
is to show the convergence of the quadratic variation of this martingale, which will be derived by
using Tanaka’s formula and backward Itô stochastic integrals.
The paper is organized as follows. In the next section we recall some preliminaries on Malliavin
calculus and we establish a stochastic integral representation for the random variable G t (h). Then,
Section 3 is devoted to the proof of Theorem 1.
2
Stochastic integral representation of the L 2 -modulus of continuity
Let us introduce some basic facts on the Malliavin calculus with respect the the Brownian motion
B = {B t , t ≥ 0}. We refer to [4] for a complete presentation of these notions. We assume that B
is defined on a complete probability space (Ω, F , P) such that F is generated by B. Consider the
set S of smooth random variables of the form
F = f Bt1 , . . . , Bt n ,
(2.4)
where t 1 , . . . , t n ≥ 0, f ∈ C b∞ (Rn ) (the space of bounded functions which have bounded derivatives of all orders) and n ∈ N. The derivative operator D on a smooth random variable of the form
(2.4) is defined by
n
X
∂f
B t 1 , . . . , B t n I[0,t i ] (t),
Dt F =
∂ xi
i=1
which is an element of L 2 (Ω × [0, ∞)). We denote by D1,2 the completion of S with respect to
the norm kF k1,2 given by
kF k21,2
=E F
2
Z
+ E
∞
Dt F
2
dt
.
0
The classical Itô representation theorem asserts that any square integrable random variable can
be expressed as
Z∞
F = E[F ]+
ut d Bt ,
0
Brownian local time and a central limit theorem
531
where u = {u t , t ≥ 0} is a unique adapted process such that E
R ∞
0
u2t d t
1,2
< ∞. If F belongs to
D , then u t = E[D t F |F t ], where {F t , t ≥ 0} is the filtration generated by B, and we obtain the
Clark-Ocone formula (see [5])
Z∞
F = E[F ]+
0
E[D t F |F t ]d B t .
(2.5)
R
The random variable G t (h) = R (L t (x + h) − L t (x))2 d x can be expressed in terms of the selfintersection local time of Brownian motion. In fact,
Z Z
Z
t
Z
=
−2
0
−2
0
v
0
dx
0
tZ t
Z
δ(B v − Bu + h)dud v −
0
Z tZ
=
0
R
tZ t
δ(Bu + x)du
δ(Bu + x + h)du −
G t (h) =
2
t
0
0
δ(B v − Bu )dud v
δ(B v − Bu + h) + δ(B v − Bu − h) − 2δ(B v − Bu ) dud v .
The rigorous justification of above argument can be made easily by approximating the Dirac delta
2
1
function by the heat kernel pǫ (x) = p2πǫ
e−x /2ǫ as ǫ tends to zero. That is, G t (h) is the limit in
L 2 (Ω) as ǫ tends to zero of
Z tZ v
ǫ
G t (h) = −2
pǫ (B v − Bu + h) + pǫ (B v − Bu − h) − 2pǫ (B v − Bu ) dud v.
(2.6)
0
0
Applying Clark-Ocone formula we can derive the following stochastic integral representation for
G t (h).
Proposition 2. The random variable G t (h) defined in (1.3) can be expressed as
Z t
G t (h) = E(G t (h)) +
u t,h (r)d B r ,
0
where
Z rZ
u t,h (r)
=
0
0
Z
r
+4
0
Proof
h
4
p t−r (B r − Bu − η) − p t−r (B r − Bu + η) dηdu
I[0,h] (Bu − B r ) − I[0,h] (B r − Bu ) du.
For any u < v and any h ∈ R we can write
D r pǫ (B v − Bu + h) = pǫ′ (B v − Bu + h)I[u,v] (r),
and for any u < r < v
E D r pǫ′ (B v − Bu + h)|F r
p
=
Epǫ′ (
=
p′v−r+ǫ (B r
v − rη + B r − Bu + h)
− Bu + h),
(2.7)
532
Electronic Communications in Probability
where η denotes a N (0, 1) random variable independent of B. Therefore, from Clark-Ocone formula (2.5) and Equation (2.6) we obtain
Z
G tǫ (h)
=
t
uǫt,h (r)d B r ,
E(G tǫ (h)) +
0
where
Z tZ r
uǫt,h (r)
=
−2
p′v−r+ǫ (B r − Bu + h) + p′v−r+ǫ (B r − Bu − h)
0
r
+2p′v−r+ǫ (B r
− Bu ) dud v.
This expression can be written as
Z tZ rZ
uǫt,h (r)
= −2
0
r
h
0
Using the fact that p′′t (x) = 2
∂ pt
∂t
(x) we obtain
Z rZ
uǫt,h (r)
=
−4
0
Z rZ
−
0
(p′′v−r+ǫ (B r − Bu + η) − p′′v−r+ǫ (B r − Bu − η) dηdud v.
h
(p t−r+ǫ (B r − Bu + η) − p t−r+ǫ (B r − Bu − η) dηdu
0
h
!
(pǫ (B r − Bu + η) − pǫ (B r − Bu − η) dηdu
0
.
Letting ǫ tend to zero we get that uǫt,h (r) converges in L 2 (Ω × [0, t]) to u t,h (r) as h tends to zero,
which implies the desired result.
From Proposition 2 we can make the following decomposition
u t,h (r) = û t,h (r) + 4Ψh (r),
where
Z rZ
û t,h (r)
=
−4
0
h
0
p t−r (B r − Bu + η) − p t−r (B r − Bu − η) dηdu
Z r Z hZ
=
−4
0
and
Ψh (r) = −
−η
0
Z
r
0
η
p′t−r (B r − Bu + ξ)dξdηdu
I[0,h] (B r − Bu ) − I[0,h] (Bu − B r ) du.
(2.8)
(2.9)
As a consequence, we finally obtain
Z
G t (h) − E(G t (h)) =
Z
t
t
Ψh (r)d B r .
û t,h (r)d B r + 4
0
0
(2.10)
Brownian local time and a central limit theorem
3
533
Proof of Theorem 1
The proof will be done in several steps. Along the proof we will denote by C a generic constant,
which may be different form line to line.
Rt
Step 1 We claim that the stochastic integral 0 û t,h (r)d B r makes no contribution to the limit (1.1).
That is,
Z
t
h−3/2
û t,h (r)d B r
0
converges in L 2 (Ω) to zero as h tends to zero. This is a consequence of the next proposition.
Proposition 3. There is a constant C > 0 such that
Z t
2
≤ Ch4 ,
|û t,h (r)| d r
E
0
for all h > 0.
Proof
From (2.8) we can write
E |û t,h (r)|
2
Z r Z r Z hZ hZ
η1
Z
η2
=
0
0
0
−η2
−η1
0
E(p′t−r (B r − Bu1 + ξ1 )
×p′t−r (B r − Bu2 + ξ2 ))dξ1 dξ2 dη1 dη2 du1 du2 .
By a symmetry argument, it suffices to integrate in the region 0 < u1 < u2 < r. Set
Φ(u1 , u2 , ξ1 , ξ2 ) = E p′t−r (B r − Bu1 + ξ1 )p′t−r (B r − Bu2 + ξ2 ) .
Then,
Φ(u1 , u2 , ξ1 , ξ2 )
=
=
E p′t−r (B r − Bu2 + Bu2 − Bu1 + ξ1 )p′t−r (B r − Bu2 + ξ2 )
E p′t−r+u2 −u1 (B r − Bu2 + ξ1 )p′t−r (B r − Bu2 + ξ2 )
Z
=
R
≤
where
1
p1
+
1
p2
+
1
p3
kp r−u2 k p1 kp′t−r+u2 −u1 k p2 kp′t−r k p3 ,
= 1. It is easy to see that
− 12 + 2p1
kp r−u2 k p1
≤
C(r − u2 )
kp′t−r+u2 −u1 k p2
≤
C(t − r + u2 − u1 )
kp′t−r k p3
≤
C(t − r)
for some constant C > 0. Thus
p r−u2 (z)p′t−r+u2 −u1 (z + ξ1 )p′t−r (z + ξ2 )dz
E |û t,h (r)|
2
≤
≤
This proves the proposition.
Z rZ
u2
1
−1+ 2p1
3
Z hZ hZ
,
−1+ 2p1
0
×(u2 − u1 )
4
Ch .
0
0
−1+ 2p1
2
−1+ 2p1
2
,
,
η1
Z
η2
C
0
≤ C(u2 − u1 )
2
−η1
(t − r)
−η2
(r − u2 )
−1+ 2p1
3
− 12 + 2p1
1
dξ1 dξ2 dη1 dη2 du1 du2
534
Electronic Communications in Probability
Step 2 Taking into account Proposition 3 and Equation (2.10), in order to show Theorem 1 it
suffices to show the following convergence in law:
Z
Ç
t
L
− 32
Ψh (r)d B r → 2η
h
0
αt
3
,
where η is a standard normal random variable independent of B, α t has been defined in (1.2),
and Ψh (r) is given by (2.9). Notice that
Z
t
3
M th = h− 2
Ψh (r)d B r
0
is a martingale with quadratic variation
¬
M
h
Z
¶
t
t
−3
Ψ2h (r)d r.
=h
0
From the asymptotic version of Knight’s theorem (see Revuz and Yor [7], Theorem (2.3) page.
524) it suffices to show the following convergences in probability.
Z
t
−3
h
0
Ψ2h (r)d r →
and
¬
h
M ,B
Z
¶
t
4
αt ,
(3.11)
Ψh (r)d r → 0,
(3.12)
3
t
−3/2
=h
0
as h tends to zero, where the convergence (3.12) is uniform in compact sets. In fact, let B h be
h
the Brownian motion such that M th = B〈M
h 〉 . Then, from Theorem (2.3) pag. 524 in [7], and
t
the convergences (3.11) and (3.12), we deduce that (B, B h , 〈M h 〉 t ) converges in distribution to
h
(B, β, 43 α t ), where β is a Brownian motion independent of B. This implies that M th = B〈M
h〉
t
converges in distribution to β 4 α t , which yields the desired result.
3
Before proving (3.11) and (3.12) we will express Ψh (r) using Tanaka’s formula. By the occupation
formula for the Brownian motion we can write
Z
Ψh (r) = −
I[0,h] (B r − x) − I[0,h] (x − B r ) L r (x)d x
Z
R
h
=
0
L r (B r + y) − L r (B r − y) d y.
We can express the difference L r (B r − y) − L r (B r + y) by means of Tanaka’s formula for the
Brownian motion {B r − Bs , 0 ≤ s ≤ r}:
L r (B r + y) − L r (B r − y)
=
y + (B r − y)+ − (B r + y)+
Z r
bs ,
I Br −Bs + y>0 − I Br −Bs − y>0 d B
−
0
Brownian local time and a central limit theorem
535
bs denote the backward stochastic Itô integral and y > 0. Integrating in the variable y
where d B
yields
Zh
h2
Ψh (r) =
−
(B r + y)+ − (B r − y)+ d y
2
0
Z h Z r
bs d y.
−
I y>|Br −Bs | d B
(3.13)
0
0
By stochastic Fubini’s theorem
Z h Z r
0
0
bs
I y>|Br −Bs | d B
Z
r
dy =
0
bs .
(h − |B r − Bs |)+ d B
(3.14)
Hence,
Ψh (r)
=
h2
2
Z
−
Z
−
h
0
(B r + y)+ − (B r − y)+ d y
r
0
bs .
(h − |B r − Bs |)+ d B
(3.15)
The convergences (3.11) and (3.12) will be proved in the next two steps.
Step 3 The convergence (3.12) follows from the following lemma.
¬
¶
Lemma 4. For any t ≥ 0, M h , B t converges to zero in L 2 (Ω) uniformly in compact sets as h tends
to zero.
Proof
In view of (3.13) it suffices to show that
Z t Z r
−3/2
+ b
(h − |B r − Bs |) d Bs d r
sup h
0≤t≤t 1
0
0
converges to zero in L 2 (Ω) as h tends to zero, for any t 1 > 0. For any p ≥ 2 and any 0 ≤ s < t we
can write by Fubini’s theorem and Burkholder’s inequality
Z Z
p !
r
t
+ b
(h − |B r − B v |) d B v d r
E
s
0
p !
Z Z
¨
t
s
+
p−1
bv
(h − |B r − B v |) d r d B
E
≤2
0
s
p ! «
Z Z
t
t
bv
(h − |B r − B v |)+ d r d B
+E
s
v
2 p/2
¨
Z s Z t
(h − |B r − B v |)+ d r
d v
≤ c p E
0
s
2 p/2 «
Z t Z t
+E
(h − |B r − B v |)+ d r
d v
s
v
= c p (I1 + I2 ).
536
Electronic Communications in Probability
The term I1 can be expressed using occupation formula as follows
2 p/2
Z s Z
I 1 = E
(h − |x − B v |)+ (L t (x) − Ls (x))d x
d v
0
R
≤ s p/2 h2p E sup |L t (x) − Ls (x)| p .
x
By the inequalities for local time proved, for instance, in [1] we obtain
I1 ≤ c p h2p |t − s| p/2 .
Similarly,
I2
=
≤
≤
2 p/2
Z t Z
d v
(h − |x − B v |)+ (L t (x) − L v (x))d x
E
s
R
h2p |t − s| p/2 sup E sup |L t (x) − L v (x)| p
x
s≤v≤t
2p
p
c p h |t − s| .
Finally, a standard application of the Garsia-Rudemich-Rumsey lemma allows us to conclude.
Step 4 We are going to show that
Z
t
−3
L 2 (Ω)
Ψh (r)2 d r →
h
0
4
3
αt ,
(3.16)
as h tends to zero. Notice that
Z tZ
v
αt = 2
0
0
δ0 (B v − Bu )dud v
is the self-intersection local time of B, and Equation (3.16) provides an approximation for this
self-intersection local time which has its own interest.
Taking into account (3.13) and (3.14), the convergence (3.16) will follow from
Z t Z
2
r
bs
(h − |B r − Bs |) d B
+
−3
h
0
0
L 2 (Ω)
dr →
4
3
αt ,
(3.17)
as h tends to zero. By Itô’s formula we can write
Z
bs
(h − |B r − Bs |) d B
+
0
Z r Z
2
r
bs +
×(h − |B r − Bs |) d B
0
Z
r
0
bu
(h − |B r − Bu |) d B
+
=2
+
r
s
(h − |B r − Bs |)+
Finally, (3.17) follows form (3.18) and the next two lemmas.
2
ds.
(3.18)
Brownian local time and a central limit theorem
Lemma 5. We have
Z tZ
0
r
537
(h − |B r − Bs |)+
2
h3
0
L 2 (Ω)
dsd r →
4
3
αt ,
as h tends to zero.
Proof
Notice that
2
0
R
Z tZ
0
r
(h − |B r − Bs |)+
h3
0
t
L r (B r )d r,
L r (x)L d r (x)d x =
L t (x) d x =
αt =
and
Z
Z tZ
Z
0
R
Z tZ
2
dsd r =
0
(h − |B r − x|)+
2
L r (x)d x.
h3
R
As a consequence, taking into account that
Z
R
(h − |B r − x|)+
h3
Z
2
[(h − |x|)+ ]2
dx =
R
h3
dx =
4
3
,
we obtain
≤
≤
Z Z
t r (h − |B − B |)+ 2
4
r
s
dsd r − α t
0 0
3
h3
Z tZ
2
(h − |B r − x|)+
L (x) − L (B ) d x d r
r
r
r
3
h
0
R
Z t
4
sup L r (x) − L r ( y) d r,
3 0 |x− y|
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
STOCHASTIC INTEGRAL REPRESENTATION OF THE L 2 MODULUS OF BROWNIAN LOCAL TIME AND A CENTRAL LIMIT THEOREM
YAOZHONG HU
Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045
email: [email protected]
DAVID NUALART1
Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045
email: [email protected]
Submitted August 19, 2009, accepted in final form November 11, 2009
AMS 2000 Subject classification: 60H07, 60F05, 60J55, 60J65
Keywords: Malliavin calculus, Clark-Ocone formula, Brownian local time, Knight theorem, central
limit theorem, Tanaka formula.
Abstract
The purpose of this note is to prove a central limit theorem for the L 2 -modulus of continuity of the
Brownian local time obtained in [3], using techniques of stochastic analysis. The main ingredients
of the proof are an asymptotic version of Knight’s theorem and the Clark-Ocone formula for the
L 2 -modulus of the Brownian local time.
1
Introduction
Let B = {B t , t ≥ 0} be a standard Brownian motion, and denote by {L t (x), t ≥ 0, x ∈ R} its local
time. In [3] the authors have proved the following central limit theorem for the L 2 modulus of
continuity of the local time:
Theorem 1. For each fixed t > 0,
Z
− 23
2
h
R
(L t (x + h) − L t (x)) d x − 4th
as h tends to zero, where
−→ 8
αt
3
η,
(1.1)
Z
(L t (x))2 d x,
αt =
R
and η is a N (0, 1) random variable independent of B.
1
Ç
L
D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0604207.
529
(1.2)
530
Electronic Communications in Probability
We make use of the notation
Z
G t (h) =
R
(L t (x + h) − L t (x))2 d x.
(1.3)
It is proved in [3, Lemma 8.1] that E G t (h) = 4th + O(h2 ). Therefore, we can replace the term
4th in (1.1) by E G t (h) .
The proof of Theorem 1 is done in [3] by the method of moments. The purpose of this paper is
to provide a simple proof of this result. Our method is based on an asymptotic version of Knight’s
theorem (see Revuz and Yor [7], Theorem (2.3), page 524) combined with the techniques of
stochastic analysis and Malliavin calculus. The main idea is to
apply the Clark-Ocone stochastic
integral representation formula to express G t (h) − E G t (h) as a stochastic integral. Then, by
means of simple estimates using Hölder’s inequality, it is proved that the leading term is a martingale, to which we can apply an asymptotic version of Knight’ theorem. An important ingredient
is to show the convergence of the quadratic variation of this martingale, which will be derived by
using Tanaka’s formula and backward Itô stochastic integrals.
The paper is organized as follows. In the next section we recall some preliminaries on Malliavin
calculus and we establish a stochastic integral representation for the random variable G t (h). Then,
Section 3 is devoted to the proof of Theorem 1.
2
Stochastic integral representation of the L 2 -modulus of continuity
Let us introduce some basic facts on the Malliavin calculus with respect the the Brownian motion
B = {B t , t ≥ 0}. We refer to [4] for a complete presentation of these notions. We assume that B
is defined on a complete probability space (Ω, F , P) such that F is generated by B. Consider the
set S of smooth random variables of the form
F = f Bt1 , . . . , Bt n ,
(2.4)
where t 1 , . . . , t n ≥ 0, f ∈ C b∞ (Rn ) (the space of bounded functions which have bounded derivatives of all orders) and n ∈ N. The derivative operator D on a smooth random variable of the form
(2.4) is defined by
n
X
∂f
B t 1 , . . . , B t n I[0,t i ] (t),
Dt F =
∂ xi
i=1
which is an element of L 2 (Ω × [0, ∞)). We denote by D1,2 the completion of S with respect to
the norm kF k1,2 given by
kF k21,2
=E F
2
Z
+ E
∞
Dt F
2
dt
.
0
The classical Itô representation theorem asserts that any square integrable random variable can
be expressed as
Z∞
F = E[F ]+
ut d Bt ,
0
Brownian local time and a central limit theorem
531
where u = {u t , t ≥ 0} is a unique adapted process such that E
R ∞
0
u2t d t
1,2
< ∞. If F belongs to
D , then u t = E[D t F |F t ], where {F t , t ≥ 0} is the filtration generated by B, and we obtain the
Clark-Ocone formula (see [5])
Z∞
F = E[F ]+
0
E[D t F |F t ]d B t .
(2.5)
R
The random variable G t (h) = R (L t (x + h) − L t (x))2 d x can be expressed in terms of the selfintersection local time of Brownian motion. In fact,
Z Z
Z
t
Z
=
−2
0
−2
0
v
0
dx
0
tZ t
Z
δ(B v − Bu + h)dud v −
0
Z tZ
=
0
R
tZ t
δ(Bu + x)du
δ(Bu + x + h)du −
G t (h) =
2
t
0
0
δ(B v − Bu )dud v
δ(B v − Bu + h) + δ(B v − Bu − h) − 2δ(B v − Bu ) dud v .
The rigorous justification of above argument can be made easily by approximating the Dirac delta
2
1
function by the heat kernel pǫ (x) = p2πǫ
e−x /2ǫ as ǫ tends to zero. That is, G t (h) is the limit in
L 2 (Ω) as ǫ tends to zero of
Z tZ v
ǫ
G t (h) = −2
pǫ (B v − Bu + h) + pǫ (B v − Bu − h) − 2pǫ (B v − Bu ) dud v.
(2.6)
0
0
Applying Clark-Ocone formula we can derive the following stochastic integral representation for
G t (h).
Proposition 2. The random variable G t (h) defined in (1.3) can be expressed as
Z t
G t (h) = E(G t (h)) +
u t,h (r)d B r ,
0
where
Z rZ
u t,h (r)
=
0
0
Z
r
+4
0
Proof
h
4
p t−r (B r − Bu − η) − p t−r (B r − Bu + η) dηdu
I[0,h] (Bu − B r ) − I[0,h] (B r − Bu ) du.
For any u < v and any h ∈ R we can write
D r pǫ (B v − Bu + h) = pǫ′ (B v − Bu + h)I[u,v] (r),
and for any u < r < v
E D r pǫ′ (B v − Bu + h)|F r
p
=
Epǫ′ (
=
p′v−r+ǫ (B r
v − rη + B r − Bu + h)
− Bu + h),
(2.7)
532
Electronic Communications in Probability
where η denotes a N (0, 1) random variable independent of B. Therefore, from Clark-Ocone formula (2.5) and Equation (2.6) we obtain
Z
G tǫ (h)
=
t
uǫt,h (r)d B r ,
E(G tǫ (h)) +
0
where
Z tZ r
uǫt,h (r)
=
−2
p′v−r+ǫ (B r − Bu + h) + p′v−r+ǫ (B r − Bu − h)
0
r
+2p′v−r+ǫ (B r
− Bu ) dud v.
This expression can be written as
Z tZ rZ
uǫt,h (r)
= −2
0
r
h
0
Using the fact that p′′t (x) = 2
∂ pt
∂t
(x) we obtain
Z rZ
uǫt,h (r)
=
−4
0
Z rZ
−
0
(p′′v−r+ǫ (B r − Bu + η) − p′′v−r+ǫ (B r − Bu − η) dηdud v.
h
(p t−r+ǫ (B r − Bu + η) − p t−r+ǫ (B r − Bu − η) dηdu
0
h
!
(pǫ (B r − Bu + η) − pǫ (B r − Bu − η) dηdu
0
.
Letting ǫ tend to zero we get that uǫt,h (r) converges in L 2 (Ω × [0, t]) to u t,h (r) as h tends to zero,
which implies the desired result.
From Proposition 2 we can make the following decomposition
u t,h (r) = û t,h (r) + 4Ψh (r),
where
Z rZ
û t,h (r)
=
−4
0
h
0
p t−r (B r − Bu + η) − p t−r (B r − Bu − η) dηdu
Z r Z hZ
=
−4
0
and
Ψh (r) = −
−η
0
Z
r
0
η
p′t−r (B r − Bu + ξ)dξdηdu
I[0,h] (B r − Bu ) − I[0,h] (Bu − B r ) du.
(2.8)
(2.9)
As a consequence, we finally obtain
Z
G t (h) − E(G t (h)) =
Z
t
t
Ψh (r)d B r .
û t,h (r)d B r + 4
0
0
(2.10)
Brownian local time and a central limit theorem
3
533
Proof of Theorem 1
The proof will be done in several steps. Along the proof we will denote by C a generic constant,
which may be different form line to line.
Rt
Step 1 We claim that the stochastic integral 0 û t,h (r)d B r makes no contribution to the limit (1.1).
That is,
Z
t
h−3/2
û t,h (r)d B r
0
converges in L 2 (Ω) to zero as h tends to zero. This is a consequence of the next proposition.
Proposition 3. There is a constant C > 0 such that
Z t
2
≤ Ch4 ,
|û t,h (r)| d r
E
0
for all h > 0.
Proof
From (2.8) we can write
E |û t,h (r)|
2
Z r Z r Z hZ hZ
η1
Z
η2
=
0
0
0
−η2
−η1
0
E(p′t−r (B r − Bu1 + ξ1 )
×p′t−r (B r − Bu2 + ξ2 ))dξ1 dξ2 dη1 dη2 du1 du2 .
By a symmetry argument, it suffices to integrate in the region 0 < u1 < u2 < r. Set
Φ(u1 , u2 , ξ1 , ξ2 ) = E p′t−r (B r − Bu1 + ξ1 )p′t−r (B r − Bu2 + ξ2 ) .
Then,
Φ(u1 , u2 , ξ1 , ξ2 )
=
=
E p′t−r (B r − Bu2 + Bu2 − Bu1 + ξ1 )p′t−r (B r − Bu2 + ξ2 )
E p′t−r+u2 −u1 (B r − Bu2 + ξ1 )p′t−r (B r − Bu2 + ξ2 )
Z
=
R
≤
where
1
p1
+
1
p2
+
1
p3
kp r−u2 k p1 kp′t−r+u2 −u1 k p2 kp′t−r k p3 ,
= 1. It is easy to see that
− 12 + 2p1
kp r−u2 k p1
≤
C(r − u2 )
kp′t−r+u2 −u1 k p2
≤
C(t − r + u2 − u1 )
kp′t−r k p3
≤
C(t − r)
for some constant C > 0. Thus
p r−u2 (z)p′t−r+u2 −u1 (z + ξ1 )p′t−r (z + ξ2 )dz
E |û t,h (r)|
2
≤
≤
This proves the proposition.
Z rZ
u2
1
−1+ 2p1
3
Z hZ hZ
,
−1+ 2p1
0
×(u2 − u1 )
4
Ch .
0
0
−1+ 2p1
2
−1+ 2p1
2
,
,
η1
Z
η2
C
0
≤ C(u2 − u1 )
2
−η1
(t − r)
−η2
(r − u2 )
−1+ 2p1
3
− 12 + 2p1
1
dξ1 dξ2 dη1 dη2 du1 du2
534
Electronic Communications in Probability
Step 2 Taking into account Proposition 3 and Equation (2.10), in order to show Theorem 1 it
suffices to show the following convergence in law:
Z
Ç
t
L
− 32
Ψh (r)d B r → 2η
h
0
αt
3
,
where η is a standard normal random variable independent of B, α t has been defined in (1.2),
and Ψh (r) is given by (2.9). Notice that
Z
t
3
M th = h− 2
Ψh (r)d B r
0
is a martingale with quadratic variation
¬
M
h
Z
¶
t
t
−3
Ψ2h (r)d r.
=h
0
From the asymptotic version of Knight’s theorem (see Revuz and Yor [7], Theorem (2.3) page.
524) it suffices to show the following convergences in probability.
Z
t
−3
h
0
Ψ2h (r)d r →
and
¬
h
M ,B
Z
¶
t
4
αt ,
(3.11)
Ψh (r)d r → 0,
(3.12)
3
t
−3/2
=h
0
as h tends to zero, where the convergence (3.12) is uniform in compact sets. In fact, let B h be
h
the Brownian motion such that M th = B〈M
h 〉 . Then, from Theorem (2.3) pag. 524 in [7], and
t
the convergences (3.11) and (3.12), we deduce that (B, B h , 〈M h 〉 t ) converges in distribution to
h
(B, β, 43 α t ), where β is a Brownian motion independent of B. This implies that M th = B〈M
h〉
t
converges in distribution to β 4 α t , which yields the desired result.
3
Before proving (3.11) and (3.12) we will express Ψh (r) using Tanaka’s formula. By the occupation
formula for the Brownian motion we can write
Z
Ψh (r) = −
I[0,h] (B r − x) − I[0,h] (x − B r ) L r (x)d x
Z
R
h
=
0
L r (B r + y) − L r (B r − y) d y.
We can express the difference L r (B r − y) − L r (B r + y) by means of Tanaka’s formula for the
Brownian motion {B r − Bs , 0 ≤ s ≤ r}:
L r (B r + y) − L r (B r − y)
=
y + (B r − y)+ − (B r + y)+
Z r
bs ,
I Br −Bs + y>0 − I Br −Bs − y>0 d B
−
0
Brownian local time and a central limit theorem
535
bs denote the backward stochastic Itô integral and y > 0. Integrating in the variable y
where d B
yields
Zh
h2
Ψh (r) =
−
(B r + y)+ − (B r − y)+ d y
2
0
Z h Z r
bs d y.
−
I y>|Br −Bs | d B
(3.13)
0
0
By stochastic Fubini’s theorem
Z h Z r
0
0
bs
I y>|Br −Bs | d B
Z
r
dy =
0
bs .
(h − |B r − Bs |)+ d B
(3.14)
Hence,
Ψh (r)
=
h2
2
Z
−
Z
−
h
0
(B r + y)+ − (B r − y)+ d y
r
0
bs .
(h − |B r − Bs |)+ d B
(3.15)
The convergences (3.11) and (3.12) will be proved in the next two steps.
Step 3 The convergence (3.12) follows from the following lemma.
¬
¶
Lemma 4. For any t ≥ 0, M h , B t converges to zero in L 2 (Ω) uniformly in compact sets as h tends
to zero.
Proof
In view of (3.13) it suffices to show that
Z t Z r
−3/2
+ b
(h − |B r − Bs |) d Bs d r
sup h
0≤t≤t 1
0
0
converges to zero in L 2 (Ω) as h tends to zero, for any t 1 > 0. For any p ≥ 2 and any 0 ≤ s < t we
can write by Fubini’s theorem and Burkholder’s inequality
Z Z
p !
r
t
+ b
(h − |B r − B v |) d B v d r
E
s
0
p !
Z Z
¨
t
s
+
p−1
bv
(h − |B r − B v |) d r d B
E
≤2
0
s
p ! «
Z Z
t
t
bv
(h − |B r − B v |)+ d r d B
+E
s
v
2 p/2
¨
Z s Z t
(h − |B r − B v |)+ d r
d v
≤ c p E
0
s
2 p/2 «
Z t Z t
+E
(h − |B r − B v |)+ d r
d v
s
v
= c p (I1 + I2 ).
536
Electronic Communications in Probability
The term I1 can be expressed using occupation formula as follows
2 p/2
Z s Z
I 1 = E
(h − |x − B v |)+ (L t (x) − Ls (x))d x
d v
0
R
≤ s p/2 h2p E sup |L t (x) − Ls (x)| p .
x
By the inequalities for local time proved, for instance, in [1] we obtain
I1 ≤ c p h2p |t − s| p/2 .
Similarly,
I2
=
≤
≤
2 p/2
Z t Z
d v
(h − |x − B v |)+ (L t (x) − L v (x))d x
E
s
R
h2p |t − s| p/2 sup E sup |L t (x) − L v (x)| p
x
s≤v≤t
2p
p
c p h |t − s| .
Finally, a standard application of the Garsia-Rudemich-Rumsey lemma allows us to conclude.
Step 4 We are going to show that
Z
t
−3
L 2 (Ω)
Ψh (r)2 d r →
h
0
4
3
αt ,
(3.16)
as h tends to zero. Notice that
Z tZ
v
αt = 2
0
0
δ0 (B v − Bu )dud v
is the self-intersection local time of B, and Equation (3.16) provides an approximation for this
self-intersection local time which has its own interest.
Taking into account (3.13) and (3.14), the convergence (3.16) will follow from
Z t Z
2
r
bs
(h − |B r − Bs |) d B
+
−3
h
0
0
L 2 (Ω)
dr →
4
3
αt ,
(3.17)
as h tends to zero. By Itô’s formula we can write
Z
bs
(h − |B r − Bs |) d B
+
0
Z r Z
2
r
bs +
×(h − |B r − Bs |) d B
0
Z
r
0
bu
(h − |B r − Bu |) d B
+
=2
+
r
s
(h − |B r − Bs |)+
Finally, (3.17) follows form (3.18) and the next two lemmas.
2
ds.
(3.18)
Brownian local time and a central limit theorem
Lemma 5. We have
Z tZ
0
r
537
(h − |B r − Bs |)+
2
h3
0
L 2 (Ω)
dsd r →
4
3
αt ,
as h tends to zero.
Proof
Notice that
2
0
R
Z tZ
0
r
(h − |B r − Bs |)+
h3
0
t
L r (B r )d r,
L r (x)L d r (x)d x =
L t (x) d x =
αt =
and
Z
Z tZ
Z
0
R
Z tZ
2
dsd r =
0
(h − |B r − x|)+
2
L r (x)d x.
h3
R
As a consequence, taking into account that
Z
R
(h − |B r − x|)+
h3
Z
2
[(h − |x|)+ ]2
dx =
R
h3
dx =
4
3
,
we obtain
≤
≤
Z Z
t r (h − |B − B |)+ 2
4
r
s
dsd r − α t
0 0
3
h3
Z tZ
2
(h − |B r − x|)+
L (x) − L (B ) d x d r
r
r
r
3
h
0
R
Z t
4
sup L r (x) − L r ( y) d r,
3 0 |x− y|