STOCHASTIC INTEGRAL REPRESENTATION OF THE L

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in PROBABILITY

STOCHASTIC INTEGRAL REPRESENTATION OF THE L

²

MODU- LUS OF BROWNIAN LOCAL TIME AND A CENTRAL LIMIT THEO- REM

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]

SubmittedAugust 19, 2009, accepted in final formNovember 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 theL²-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 L²-modulus of the Brownian local time.

1 Introduction

LetB={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 L²modulus of continuity of the local time:

Theorem 1. For each fixed t>0, h⁻³²

Z

R

(L_t(x+h)−L_t(x))²d x−4th

−→L 8 Çαt

3η, (1.1)

as h tends to zero, where

αt= Z

R

(L_t(x))²d x, (1.2)

andηis a N(0, 1)random variable independent of B.

1D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0604207.

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We make use of the notation

G_t(h) = Z

R

(L_t(x+h)−L_t(x))²d x. (1.3)

It is proved in[3, Lemma 8.1]thatE G_t(h)

=4th+O(h²). Therefore, we can replace the term 4thin (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 variableG_t(h). Then, Section 3 is devoted to the proof of Theorem 1.

2 Stochastic integral representation of the L

²

-modulus of con- tinuity

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 thatB is defined on a complete probability space(Ω,F,P)such thatF is generated byB. Consider the setS of smooth random variables of the form

F= f

B_t₁, . . . ,B_t_n

, (2.4)

where t₁, . . . ,t_n≥0, f ∈ C_b^∞(Rⁿ)(the space of bounded functions which have bounded deriva- tives of all orders) andn∈N. The derivative operatorDon a smooth random variable of the form (2.4) is defined by

D_tF=

n

X

i=1

∂f

∂x_i

B_t

1, . . . ,B_t

n

I_[0,t

i](t),

which is an element of L²(Ω×[0,∞)). We denote by D^1,2 the completion ofS with respect to the normkFk1,2 given by

kFk²_1,2=E F²

+ E

Z _∞

0

D_tF2

d t

.

The classical Itô representation theorem asserts that any square integrable random variable can be expressed as

F=E[F]+

Z∞

0

u_td B_t,

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whereu={u_t,t≥0}is a unique adapted process such thatE R∞

0 u²_td t

<∞. If F belongs to D^1,2, thenu_t =E[DtF|Ft], where{Ft,t≥0}is the filtration generated byB, and we obtain the Clark-Ocone formula (see[5])

F =E[F]+

Z∞

0

E[DtF|Ft]d B_t. (2.5)

The random variable G_t(h) =R

R(L_t(x+h)−L_t(x))²d x can be expressed in terms of the self- intersection local time of Brownian motion. In fact,

G_t(h) = Z

R

Z t

0

δ(B_u+x+h)du− Zt

0

δ(B_u+x)du

2

d x

= −2

Z t

0

Zt

0

δ(B_v−B_u+h)dud v− Z t

0

Zt

0

δ(B_v−B_u)dud v

= −2 Zt

0

Z v

0

δ(B_v−B_u+h) +δ(B_v−B_u−h)−2δ(B_v−B_u) dud v.

The rigorous justification of above argument can be made easily by approximating the Dirac delta function by the heat kernel p_"(x) = ^p_2π"¹ e^−x²^/2" as"tends to zero. That is, G_t(h)is the limit in L²(Ω)as"tends to zero of

G^"_t(h) =−2

Z t

0

Z v

0

p_"(B_v−B_u+h) +p_"(B_v−B_u−h)−2p_"(B_v−B_u)

dud v. (2.6) 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 G_t(h) =E(G_t(h)) +

Z t

0

u_t,h(r)d B_r, where

u_t,h(r) = 4 Z r

0

Zh

0

p_t₋_r(B_r−B_u−η)−p_t₋_r(B_r−B_u+η) dηdu +4

Zr

0

I_[0,h](B_u−B_r)−I_[0,h](B_r−B_u)

du. (2.7)

Proof For anyu<vand anyh∈Rwe can write

D_rp_"(B_v−B_u+h) =p⁰_"(B_v−B_u+h)I_[_u,v_](r),

and for anyu<r<v E

D_rp⁰_"(B_v−B_u+h)|Fr

= Ep⁰_"(p

v−rη+B_r−B_u+h)

= p⁰_v₋_r_+"(B_r−B_u+h),

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whereηdenotes aN(0, 1)random variable independent ofB. Therefore, from Clark-Ocone formula (2.5) and Equation (2.6) we obtain

G_t^"(h) =E(G_t^"(h)) +

Zt

0

u^"_t,h(r)d B_r, where

u^"_t_,h(r) = −2

Zt

r

Z r

0

p⁰_v₋_r_+"(B_r−B_u+h) +p⁰_v₋_r_+"(B_r−B_u−h) +2p⁰_v₋_r_+"(B_r−B_u)

dud v.

This expression can be written as

u^"_t,h(r) =−2

Z t

r

Zr

0

Zh

0

(p⁰⁰_v−r_+"(B_r−B_u+η)−p⁰⁰_v−r_+"(B_r−B_u−η)

dηdud v.

Using the fact thatp⁰⁰_t(x) =2^∂_∂^p^t

t (x)we obtain

u^"_t,h(r) = −4

Zr

0

Zh

0

(p_t−r+"(B_r−B_u+η)−p_t−r+"(B_r−B_u−η)dηdu

− Z r

0

Zh

0

(p_"(B_r−B_u+η)−p_"(B_r−B_u−η)

dηdu

! .

Letting"tend to zero we get thatu^"_t,h(r)converges inL²(Ω×[0,t])tou_t_,h(r)ashtends to zero, which implies the desired result.

From Proposition 2 we can make the following decomposition u_t,h(r) = ˆu_t,h(r) +4Ψh(r), where

ˆ

u_t,h(r) = −4 Z r

0

Zh

0

p_t−r(B_r−B_u+η)−p_t−r(B_r−B_u−η)dηdu

= −4 Z r

0

Zh

0

Z η

−η

p⁰_t₋_r(B_r−B_u+ξ)dξdηdu (2.8) and

Ψh(r) =− Zr

0

I_[0,h_](B_r−B_u)−I_[0,h_](B_u−B_r)

du. (2.9)

As a consequence, we finally obtain G_t(h)−E(G_t(h)) =

Zt

0

ˆ

u_t,h(r)d B_r+4 Zt

0

Ψh(r)d B_r. (2.10)

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3 Proof of Theorem 1

The proof will be done in several steps. Along the proof we will denote byC a generic constant, which may be different form line to line.

Step 1We claim that the stochastic integralRt

0 uˆ_t,h(r)d B_rmakes no contribution to the limit (1.1).

That is,

h^−3/2 Z t

0

ˆ

u_t,h(r)d B_r

converges inL²(Ω)to zero ashtends to zero. This is a consequence of the next proposition.

Proposition 3. There is a constant C>0such that E

Z t

0

|ˆu_t,h(r)|²d r

≤Ch⁴, for all h>0.

Proof From (2.8) we can write E

|ˆu_t,h(r)|²

= Z r

0

Z r

0

Zh

0

Zh

0

Z η1

−η1

Zη2

−η2

E(p⁰_t₋_r(B_r−B_u

1+ξ1)

×p⁰_t−r(B_r−B_u₂+ξ2))dξ1dξ2dη1dη2du₁du₂. By a symmetry argument, it suffices to integrate in the region 0<u₁<u₂<r. Set

Φ(u₁,u₂,ξ1,ξ2) =E

p⁰_t₋_r(B_r−B_u

1+ξ1)p⁰_t₋_r(B_r−B_u

2+ξ2) . Then,

Φ(u₁,u₂,ξ1,ξ2) = E

p⁰_t_−r(B_r−B_u₂+B_u₂−B_u₁+ξ1)p⁰_t−r(B_r−B_u₂+ξ2)

= E

p⁰_t₋_r₊_u

2−u₁(B_r−B_u

2+ξ1)p⁰_t₋_r(B_r−B_u

2+ξ2)

= Z

R

p_r₋_u₂(z)p⁰_t−r+u

2−u1(z+ξ1)p⁰_t−r(z+ξ2)dz

≤ kp_r₋_u₂kp₁kp⁰_t−r+u

2−u1kp₂kp⁰_t−rkp₃, where _p¹

1+ _p¹

2+_p¹

3 =1. It is easy to see that

kp_r₋_u₂kp₁ ≤ C(r−u₂)⁻¹²⁺^2p¹¹, kp⁰_t−r+u

2−u1kp₂ ≤ C(t−r+u₂−u₁)⁻¹⁺^2p¹² ≤C(u₂−u₁)⁻¹⁺^2p¹², kp⁰_t−rkp₃ ≤ C(t−r)⁻¹⁺^2p¹³,

for some constantC>0. Thus E

|ˆu_t,h(r)|²

≤ C Z r

0

Zu₂

0

Zh

0

Zh

0

Zη1

−η1

Zη2

−η2

(r−u₂)⁻¹²⁺^2p¹¹

×(u₂−u₁)⁻¹⁺^2p¹²(t−r)⁻¹⁺^2p¹³dξ1dξ2dη1dη2du₁du₂

≤ C h⁴. This proves the proposition.

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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:

h⁻³² Zt

0

Ψh(r)d B_r→^L 2η Çαt

3 ,

whereη is a standard normal random variable independent ofB,αt has been defined in (1.2), andΨh(r)is given by (2.9). Notice that

M_t^h=h⁻³² Z t

0

Ψh(r)d B_r

is a martingale with quadratic variation

¬M^h¶

t=h⁻³ Z t

0

Ψ²h(r)d r.

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.

h⁻³ Z t

0

Ψ²_h(r)d r→4

3αt, (3.11)

and

¬M^h,B¶

t=h^−3/2 Z t

0

Ψh(r)d r→0, (3.12)

as htends to zero, where the convergence (3.12) is uniform in compact sets. In fact, let B^hbe the Brownian motion such that M_t^h= B^h

〈M^h〉t. Then, from Theorem (2.3) pag. 524 in[7], and the convergences (3.11) and (3.12), we deduce that(B,B^h,〈M^h〉t)converges in distribution to (B,β,⁴

3αt), where β is a Brownian motion independent of B. This implies that M_t^h = B^h

〈M^h〉t

converges in distribution toβ⁴

3αt, which yields the desired result.

Before proving (3.11) and (3.12) we will expressΨh(r)using Tanaka’s formula. By the occupation formula for the Brownian motion we can write

Ψh(r) = − Z

R

I_[0,h_](B_r−x)−I_[0,h_](x−B_r)

L_r(x)d x

= Z 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−B_s, 0≤s≤r}:

L_r(B_r+y)−L_r(B_r−y) = y+ (B_r−y)⁺−(B_r+y)⁺

− Z r

0

I_B_r_−B_s₊_y_>0−I_B_r_−B_s_−y_>0

dBb_s,

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wheredbB_s denote the backward stochastic Itô integral and y>0. Integrating in the variable y yields

Ψh(r) = h² 2 −

Zh

0

(B_r+y)⁺−(B_r−y)⁺ d y

− Zh

0

Z r

0

I_y_>|_B

r−B_s|dbB_s

d y. (3.13)

By stochastic Fubini’s theorem Zh

0

Z r

0

I_y>|B

r−Bs|dBb_s

d y=

Zr

0

(h− |B_r−B_s|)⁺dbB_s. (3.14) Hence,

Ψh(r) = h² 2 −

Zh

0

(B_r+y)⁺−(B_r−y)⁺d y

− Z r

0

(h− |B_r−B_s|)⁺dBb_s. (3.15) The convergences (3.11) and (3.12) will be proved in the next two steps.

Step 3The convergence (3.12) follows from the following lemma.

Lemma 4. For any t≥0,¬ M^h,B¶

t converges to zero in L²(Ω)uniformly in compact sets as h tends to zero.

Proof In view of (3.13) it suffices to show that sup

0≤t≤t₁

h^−3/2 Z t

0

Z r

0

(h− |B_r−B_s|)⁺dBb_s

d r

converges to zero inL²(Ω)ashtends to zero, for anyt₁>0. For anyp≥2 and any 0≤s<twe can write by Fubini’s theorem and Burkholder’s inequality

E

Z t

s

Z r

0

(h− |B_r−B_v|)⁺dbB_v

d r

p!

≤2^p⁻¹

¨ E

Z s

0

Z t

s

(h− |B_r−B_v|)⁺d r

dBb_v

p!

+E

Zt

s

Z t

v

(h− |B_r−B_v|)⁺d r

dbB_v

p! «

≤c_p

¨ E







Zs

0

Z t

s

(h− |B_r−B_v|)⁺d r

2

d v

p/2





+E







Z t

s

Z t

v

(h− |B_r−B_v|)⁺d r

2

d v

p/2





«

=c_p(I₁+I₂).

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The termI₁can be expressed using occupation formula as follows

I₁ = E







Zs

0

Z

R

(h− |x−B_v|)⁺(L_t(x)−L_s(x))d x

2

d v

p/2





≤ s^p^/2h^2pE

sup

x |L_t(x)−L_s(x)|^p

.

By the inequalities for local time proved, for instance, in[1]we obtain I₁≤c_ph^2p|t−s|^p^/2.

Similarly,

I₂ = E







Zt

s

Z

R

(h− |x−B_v|)⁺(L_t(x)−L_v(x))d x

2

d v

p/2





≤ h^2p|t−s|^p^/2 sup

s≤v≤tE

sup

x |L_t(x)−L_v(x)|^p

≤ c_ph^2p|t−s|^p.

Finally, a standard application of the Garsia-Rudemich-Rumsey lemma allows us to conclude.

Step 4 We are going to show that h⁻³

Z t

0

Ψh(r)²d r^L

2(Ω)

→ 4

3αt, (3.16)

ashtends to zero. Notice that

αt=2 Z t

0

Zv

0

δ0(B_v−B_u)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 h⁻³

Zt

0

Z r

0

(h− |B_r−B_s|)⁺dbB_s

2

d r^L

2(Ω)

→ 4

3αt, (3.17)

ashtends to zero. By Itô’s formula we can write

Z r

0

(h− |B_r−B_s|)⁺dbB_s

2

=2 Zr

0

Z r

s

(h− |B_r−B_u|)⁺dbB_u

×(h− |B_r−B_s|)⁺dbB_s+ Z r

0

(h− |B_r−B_s|)⁺2

ds. (3.18)

Finally, (3.17) follows form (3.18) and the next two lemmas.

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Lemma 5. We have

Zt

0

Z r

0

(h− |B_r−B_s|)⁺2

h³ dsd r^L

2(Ω)

→ 4 3αt, as h tends to zero.

Proof Notice that αt =

Z

R

L_t(x)²d x = Z t

0

Z

R

L_r(x)L_{d r}(x)d x= Z t

0

L_r(B_r)d r, and

Zt

0

Z r

0

(h− |B_r−B_s|)⁺2

h³ dsd r=

Z t

0

Z

R

(h− |B_r−x|)⁺2

h³ L_r(x)d x. As a consequence, taking into account that

Z

R

(h− |B_r−x|)⁺2

h³ d x=

Z

R

[(h− |x|)⁺]² h³ d x= 4

3, we obtain

Zt

0

Z r

0

(h− |B_r−B_s|)⁺2

h³ dsd r−4 3αt

≤ Z t

0

Z

R

(h− |B_r−x|)⁺2

h³

L_r(x)−L_r(B_r) d x d r

≤ 4 3

Zt

0

sup

|x−y|<h

L_r(x)−L_r(y) d r,

which clearly converges to zero in L² by the properties of the Brownian local time (see, for instance,[2]).

Lemma 6. We have 1 h⁶E





Z t

0

Z r

0

Z r

s

(h− |B_r−B_u|)⁺dBb_u

d r

2

→0, as h tends to zero.

Proof By the isometry property of the backward Itô integral we can write B_h : = 1

h⁶E





Z t

0

Z r

0

Z r

s

d r

2



= 1

h⁶E



 Z t

0

Z t

s

(h− |B_r−B_s|)⁺

Z r

s

d r

2

ds





≤ E

B¹_hB²_h ,

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where

B¹_h= Zt

0

Z t

s

(h− |B_r−B_s|)⁺

h² d r

2

ds and

B_h²= sup

0<s<r<t

Z r

s

(h− |B_r−B_u|)⁺ h dBb_u

2

. As in Lemma 5, we can show thatB_h¹converges to ⁹

4

Rt

0 L_t(B_s)−L_s(B_s)2

ds, and the convergence holds inL^p(Ω)for anyp≥2. Here we use

Z

R

(h− |x|)⁺ h² d x=3

2. On the other hand,Rr

s

(h−|B_r−B_u|)⁺

h dBb_ucan be expressed using again Tanaka’s formula:

1 h

Z r

s

(h− |B_r−B_u|)⁺dbB_u = Z h

0

Z r

s

I_y>|B

r−Bu|dbB_u

d y

= 1 h

Zh

0

L_r(B_r−y)−L_r(B_r+y)d y

−1 h

Zh

0

L_s(B_r−y)−L_s(B_r+y)d y+h 2 + 1

h Zh

0

(B_r−B_s+y)⁺−(B_r−B_s−y)⁺ d y.

Therefore, 1 h

Z r

s

≤ sup

s<r<t

sup

|x−y|≤2h

L_r(x)−L_r(y) +O(h), which also converges to zero in L^p(Ω)for anyp≥2.

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