in PROBABILITY
CENTRAL LIMIT THEOREM FOR THE THIRD MOMENT IN SPACE
OF THE BROWNIAN LOCAL TIME INCREMENTS
YAOZHONG HU
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045 USA
email: hu@math.ku.edu
DAVID NUALART1
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045 USA
email: nualart@math.ku.edu
SubmittedFebruary 19, 2010, accepted in final formSeptember 13, 2010 AMS 2000 Subject classification: 60H07, 60H05, 60F05
Keywords: Brownian motion, local time, Clark-Ocone formula
Abstract
The purpose of this note is to prove a central limit theorem for the third integrated moment of the Brownian local time increments 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 third integrated moment of the Brownian local time increments.
1
Introduction
Let{Bt,t≥0}be a standard one-dimensional Brownian motion. We denote by{Ltx,t≥0,x∈R} a continuous version of its local time. The following central limit theorem for the L2modulus of continuity of the local time has been recently proved:
h−32
Z
R
(Lxt+h−Lxt)2d x−4th
L
−→p8
3
Z
R
(Ltx)2d x 1
2
η, (1)
whereηis a N(0, 1)random variable independent of B andL denotes the convergence in law. This result has been first proved in[3]by using the method of moments. In[4]we gave a simple proof based on Clark-Ocone formula and an asymptotic version of Knight’s theorem (see Revuz and Yor[9], Theorem (2.3), page 524). Another simple proof of this result with the techniques of stochastic analysis has been given in[11].
The following extension of this result to the case of the third integrated moment has been proved recently by Rosen in[12]using the method of moments.
1D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0904538.
Theorem 1. For each fixed t>0 1
h2 Z
R
(Lxt+h−Ltx)3d x−→L 8p3
Z
R
(Lxt)3d x 1
2
η
as h tends to zero, where ηis a normal random variable with mean zero and variance one that is independent of B.
The purpose of this paper is to provide a proof of Theorem 1 using the same ideas as in[4]. The main ingredient is to use Clark-Ocone stochastic integral representation formula which allows us to express the random variable
Fth=
Z
R
(Lxt+h−Lxt)3d x (2)
as a stochastic integral. In comparison with theL2modulus of continuity, the situation is here more
complicated and we require some new and different techniques. First, there are four different terms (instead of two) in the stochastic integral representation, and two of them are martingales. Surprisingly, some of the terms of this representation converge in L2(Ω)to the derivative of the self-intersection local time and the limits cancel out. Finally, there is a remaining martingale term to which we can apply the asymptotic version of Knight’s theorem. As in the proof of (1), to show the convergence of the quadratic variation of this martingale and other asymptotic results we make use of Tanaka’s formula for the time-reversed Brownian motion and backward Itô stochastic integrals.
We believe that a similar result could be established for the integrated moment of order p for an integer p≥4 using Clark-Ocone representation formula, but the proof would be much more involved.
These results are related to the behavior of the Brownian local time in the space variable. It was proved by Perkins[7]that for any fixed t >0, {Lx
t,x ∈R} is a semimartingale with quadratic variation〈Lt〉b− 〈Lt〉a=4Rb
a L x
td x. This property provides an heuristic explanation of the central limit theorems presented above. The proof of these theorems, however, requires more complicated tools.
The paper is organized as follows. In the next section we recall some preliminaries on Malliavin calculus. In Section 3 we establish a stochastic integral representation for the derivative of the self-intersection local time, which has its own interest, and for the random variable Fthdefined in (2). Section 4 is devoted to the proof of Theorem 1, and the Appendix contains two technical lemmas.
2
Preliminaries on Malliavin Calculus
Let us recall some basic facts on the Malliavin calculus with respect the Brownian motion B =
{Bt,t≥0}. We refer to[5]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 set
S of smooth random variables of the formF= f Bt1, . . . ,Btn
,, where t1, . . . ,tn≥0,n∈Nand
f is bounded and infinitely differentiable with bounded derivatives of all orders. The derivative operatorDon a smooth random variable of this is defined by
DtF= n
X
i=1
∂f
∂xi
Bt1, . . . ,Btn
We denote by D1,2the completion ofS with respect to the normkFk1,2 given by
kFk21,2=EF2+E Z∞
0
DtF2d t
.
The classical Itô representation theorem asserts that any square integrable random variable can be expressed as
F=E[F] +
Z∞
0
utd Bt,
whereu={ut,t≥0} is a unique adapted process such thatER∞ 0 u
2
td t
<∞. If F belongs to D1,2, thenu
t=E[DtF|Ft], where{Ft,t≥0}is the filtration generated byB, and we obtain the Clark-Ocone formula (see[6])
F=E[F]+
Z∞
0
E[DtF|Ft]d Bt. (3)
3
Stochastic integral representations
Consider the random variableγt defined rigorously as the limit inL2(Ω)
γt=lim
ǫ→0 Z t
0 Zu
0
pǫ′(Bu−Bs)dsdu, (4)
wherepǫ(x) = (2πǫ)−
1
2exp(−x2/2ǫ). The processγ
t coincides with the derivative−d yd αt(y)|y=0
of the self-intersection local time
αt(y) =
Z t
0 Z u
0
δy(Bu−Bs)dsdu.
The derivative of the self-intersection local time has been studied by Rogers and Walsh in[10]and by Rosen in[11]. We are going to use Clark-Ocone formula to show that the limit (4) exists and to provide an integral representation for this random variable.
Lemma 2. Setγεt =Rt
0 Ru
0 p
′
ǫ(Bu−Bs)dsdu. Then,γεt converges in L
2(Ω)asǫ tends to zero to the random variable
γt =2
Z t
0 Z r
0
pt−r(Br−Bs)ds−LBrr
d Br.
Proof. By Clark-Ocone formula applied toγε
t we obtain the integral representation
γεt =
Z 1
0
E(Drγεt|Fr)d Br,
where{Ft,t≥0}denotes the filtration generated by the Brownian motion. Then,
Drγεt =
Z t
0 Zu
0
and for anyr≤t
E(Drγεt|Fr) =
Z t
r
Zr
0
p′′ε+u−r(Br−Bs)dsdu=2
Z t
r
Zr
0
∂pε+u−r
∂u (Br−Bs)dsdu
= 2
Z r
0
(pε+t−r(Br−Bs)−pε(Br−Bs))ds.
Asǫtends to zero this expression converges in L2(Ω×[0,t])to
2
Z r
0
pt−r(Br−Bs)ds−LBr
r
,
which completes the proof.
Let us now obtain a stochastic integral representation for the third integrated moment Fth =
R R(L
x+h
t −L
x t)
3d x. Notice first that E(Fh
t) =0 because F h
t is an odd functional of the Brown-ian motion.
Proposition 3. We have Fth=Rt
0Φrd Br, whereΦr= P4
i=1Φ(ri), and Φ(1)
r = 6
Z
R
Lzr+h−Lzr21[0,h](Br−z)dz
Φ(2)
r = −6
Z
R Zh
0
Lzr+h−Lzr2pt−r(Br−z−y)d y dz
Φ(3)
r =
12h p
2π
Zr
0 Zh
−h
Z∞
h2
t−r
pt
−r−h2 z
(Br−Bs+y)z−
3 2(1−e−
z
2)dzd y ds
Φ(4)
r = −
12h p
2π
Z r
0
1[−h,h](Br−Bs)ds
Z∞
h2
t−r
z−32(1−e−
z
2)dz.
Proof. Let us write
Fth = lim
ǫ→0 Z
R Z t
0
pǫ(Bs−x−h)−pǫ(Bs−x)
ds
3 d x
= 6 lim
ǫ→0 Z
D
Z
R 3 Y
i=1
pǫ(Bsi−x−h)−pǫ(Bsi−x)
d x ds,
whereD={(s1,s2,s3)∈[0,t]3:s1<s2<s3}. We can pass to the limit asεtends to zero the first
factorpǫ(Bs1−x−h)−pǫ(Bs1−x), and we obtain
Fth = lim
ǫ→06 Z
D
n
pǫ(Bs2−Bs1)−pǫ(Bs2−Bs1+h) pǫ(Bs3−Bs1)−pǫ(Bs3−Bs1+h)
−pǫ(Bs2−Bs1−h)−pǫ(Bs2−Bs1)
pǫ(Bs3−Bs1−h)−pǫ(Bs3−Bs1)
o ds
= lim
ǫ→06 Z
D
We are going to apply the Clark-Ocone formula to the random variableR
DΦǫ(s)ds. Fixr∈[0,t].
We need to computeR
DE Dr
Φǫ(s)
|Fr
ds. To do this we decompose the regionD, up to a set of zero Lebesgue measure, asD=D0∪D1∪D2, where
D1 = {s: 0≤s1<s2<r<s3≤t}, D2 = {s: 0≤s1<r<s2<s3≤t},
andD0={s: 0≤r≤s1} ∪ {s:s3≤r≤t}. Notice that onD0,DrΦǫ(s)
=0.
Step 1For the regionD1we obtain
E Dr
Φǫ(s)|Fr = pǫ(Bs2−Bs1)−pǫ(Bs2−Bs1+h)
×hpǫ′+s
3−r(Br−Bs1)−p
′
ǫ+s3−r(Br−Bs1+h)
i
− pǫ(Bs2−Bs1−h)−pǫ(Bs2−Bs1)
×hpǫ′+s
3−r(Br−Bs1−h)−p
′
ǫ+s3−r(Br−Bs1)
i
.
We can write this in the following form
E DrΦǫ(s)
|Fr
=
Zh
0
pǫ(Bs2−Bs1+h)−pǫ(Bs2−Bs1)
p′′ǫ+s
3−r(Br−Bs1+y)d y
− Zh
0
pǫ(Bs2−Bs1)−pǫ(Bs2−Bs1−h)
pǫ′′+s
3−r(Br−Bs1−y)d y
= 2
Zh
0
pǫ(Bs2−Bs1+h)−pǫ(Bs2−Bs1)
∂pǫ+s3−r
∂s3
(Br−Bs1+y)d y
−2
Zh
0
pǫ(Bs2−Bs1)−pǫ(Bs2−Bs1−h)
∂pǫ+s3−r
∂s3 (Br−Bs1−y)d y.
Integrating with respect to the variables3yields Z
D
E DrΦǫ(s)
|Fr
ds = 2
Z
0≤s1<s2≤r
Zh
0
pǫ(Bs2−Bs1+h)−pǫ(Bs2−Bs1)
×pǫ+t−r(Br−Bs1+y)−pǫ(Br−Bs1+y)
d y ds1ds2 −2
Z
0≤s1<s2≤r
Zh
0
pǫ(Bs2−Bs1)−pǫ(Bs2−Bs1−h)
×pǫ+t−r(Br−Bs1−y)−pǫ(Br−Bs1−y)d y ds1ds2. This expression can be written in terms of the local time:
Z
D
E Dr
Φǫ(s)|Frds = 2
Z
R2
Zh
0 Z r
0
pǫ(x−z+h)−pǫ(x−z)
(Lrx−Lsx)Lzds ×pǫ+t−r(Br−z+y)−pǫ(Br−z+y)
d y d x dz −2
Z
R2
Zh
0 Z r
0
pǫ(x−z)−pǫ(x−z−h)
(Lxr−Lsx)Lzds ×pǫ+t−r(Br−z−y)−pǫ(Br−z−y)
We make the change of variables y→h−y andz→z+hin the last integral and we obtain
Z
D
E Dr
Φǫ(s)
|Fr
ds = 2
Z
R2
Zh
0 Z r
0
pǫ(x−z+h)−pǫ(x−z)
(Lrx−Lsx)Lzds−Lzds−h ×pǫ+t−r(Br−z+y)−pǫ(Br−z+y)
d y d x dz. Taking the limit asǫtends to zero in L2(Ω×[0,t])and integrating inxyields
lim
ε→0 Z
D
E DrΦǫ(s)|Frds = 2
Z
R Zh
0 Zr
0
Lzr−h−Lsz−h−Lzr+Lzs Lzds−Lzds−h ×pt−r(Br−z+y)−δ0(Br−z+y)d y dz
=
Z
R
Lzr+h−Lzr2 1[0,h](Br−z)dz
− Z
R Zh
0
Lzr+h−Lzr2pt−r(Br−z−y)d y dz:= Ψ(r1).
Step 2 For the regionD2, taking first the conditional expectation with respect toFs2which contains
Fr, and integrating with respect to the law ofBs2−Brwe obtain
E Dr
Φǫ(s)|Fr =
Z
R
d y ps2−r(y){
p′ǫ(Br−Bs1+y)−p
′
ǫ(Br−Bs1+y+h)
×pǫ+s3−s2(Br−Bs1+y)−pǫ+s3−s2(Br−Bs1+h+y)
+pǫ(Br−Bs1+y)−pǫ(Br−Bs1+y+h)
×hpǫ′+s
3−s2(Br−Bs1+y)−p
′
ǫ+s3−s2(Br−Bs1+h+y)
i
−p′ǫ(Br−Bs1+y−h)−p
′
ǫ(Br−Bs1+y)
×pǫ+s3−s2(Br−Bs1+y−h)−pǫ+s3−s2(Br−Bs1+y)
−pǫ(Br−Bs1+y−h)−pǫ(Br−Bs1+y)
× h
pǫ′+s
3−s2(Br−Bs1+y−h)−p
′
ǫ+s3−s2(Br−Bs1+y)
i }.
Integrating by parts yields
E DrΦǫ(s)
|Fr
= −
Z
R d y p′s
2−r(y){
pǫ(Br−Bs1+y)−pǫ(Br−Bs1+y+h)
×pǫ+s3−s2(Br−Bs1+y)−pǫ+s3−s2(Br−Bs1+h+y)
−pǫ(Br−Bs1+y−h)−pǫ(Br−Bs1+y)
×pǫ+s3−s2(Br−Bs1+y−h)−pǫ+s3−s2(Br−Bs1+y)}. Lettingǫtend to zero we obtain
lim
ε→0E Dr
Φǫ(s)
|Fr
= ps3−s2(0)−ps3−s2(h) hp′s
2−r(Br−Bs1+h)−p
′
s2−r(Br−Bs1−h)
i
= ps3−s2(0)−ps3−s2(h)
Zh
−h
ps′′
Hence,
lim
ε→0 Z
D2
E DrΦǫ(s)
|Fr
ds = 2
Z r
0 ds1
Zt
r
ds2 Z t
s2
ps3−s2(0)−ps3−s2(h)ds3 !
× Z h
−h ∂ps2−r
∂s2 (Br−Bs1+y)d y:= Ψ
(2)
r . (5)
We have Z
t
s2
ps3−s2(0)−ps3−s2(h)ds3=ph 2π
Z∞
h2
t−s2
z−32(1−e−
z
2)dz. (6)
Substituting (6) into (5) yields
Ψ(r2)= p2h 2π
Zh
−h
Z∞
h2 t−r
Z t−h2 z
r
Zr
0
∂pu−r
∂u (Br−Bs+y)×z
−32(1−e−
z
2)dsdudzd y.
Now we integrate in the variableuand we obtain
Ψ(r2) = p2h 2π
Zh
−h
Z∞
h2 t−r
Z r
0 p
t−r−h2
z
(Br−Bs+y)z−
3 2(1−e−
z
2)dsdzd y
−p2h
2π
Z r
0
1[−h,h](Br−Bs)ds
Z∞
h2 t−r
z−32(1−e−
z
2)dz.
Thus,
Mt =6
Z t
0
Ψ(1)
r d Br+6
Zt
0
Ψ(2)
r d Br, which completes the proof.
4
Proof of Theorem 1
The proof will be done in several steps. Along the proof we will denote by Ca generic constant, which may be different from line to line.
Step 1 Notice first that by Lemma 2 and the equation 1
p
2π
Z∞
0
z−32(1−e−
z
2)dz=1, (7)
we obtain thath−2Rt 0(Φ
(3)
r + Φ
(4)
r )d Br converges inL2(Ω)to 12γt.
Step 2 In order to handle the termh−2R0t(Φ(1)
r + Φ
(2)
r )d Brwe consider the function
φh(ξ) =
Zh
0
and we can write
Φ(1)
r + Φ
(2)
r =−6
Z
R
LBr−x+h
r −L
Br−x r
2
φh(x)d x.
Applying Tanaka’s formula to the time reversed Brownian motion{Br−Bs, 0≤s≤r}, we obtain 1
2
LrBr−x+h−LrBr−x = −(−x+h)++ (−x)++ (Br−x+h)+−(Br−x)+
− Zr
0
1[−h,0](Br−Bs−x)dbBs,
wheredbBsdenotes the backward Itô integral. Clearly, the only term that gives a nonzero contri-bution is
24
Z
R Z r
0
1[−h,0](Br−Bs−x)dBbs 2
φh(x)d x. We are going to use the following notation
δh
r,s(x) = 1[−h,0](Br−Bs−x), (8)
Ahσ,r(x) =
Z r
σ
δh
r,s(x)dBbs, (9)
where 0< σ <r. With this notation we want to find the limit in distribution of
Yh:=−24h−2
Z t
0 Z
R
(Ah0,r(x))2φ
h(x)d x
d Br,
ashtends to zero. By Itô’s formula,
(Ah0,r(x))2=2
Zr
0
Ahσ,r(x)δrh,σ(x)dBbσ+ Z r
0
δhr,s(x)ds.
Therefore,
Yh = −48h−2
Z t
0 Z
R Z r
0
Ahσ,r(x)δh
r,σ(x)dBbσ
φh(x)d x
d Br
−24h−2 Z t
0 Z
R Z r
0
δhr,s(x)φh(x)dsd x
d Br. (10)
Notice that, by Lemma 2
−24h−2 Z t
0 Z
R Z r
0
δhr,s(x)φh(x)dsd x
d Br
=−24h−2 Zt
0 Z
R Z r
0
1[−h,0](Br−Bs−x)
Z h
0
pt−r(x−y)d y−1[0,h](x)
! dsd x
! d Br
To handle the first summand in the right-hand side of (10) we make the decomposition
Z
R Z r
0 Ah
σ,r(x)δhr,σ(x)dbBσ
φh(x)d x= Γr,h−∆r,h, where
Γr,h=
Zh
0 Z
R Z r
0 Ah
σ,r(x)δhr,σ(x)dBbσ
pt−r(x−y)d x d y, (11) and
∆r,h=
Zh
0 Z r
0
Ahσ,r(x)δhr,σ(x)dBbσ
d x, (12)
Step 3 We claim that
lim h→0h
−4E(Γ2
r,h) =0, (13)
which implies thath−2Rt
0Γr,hd Br converges to 0 in L
2(Ω)ashtends to zero. Let us prove (13).
We can write
E(Γ2r,h) =
Zr
0 Zh
0 Z
R
Ahσ,r(x)δhr,σ(x)pt−r(x−y)d x d y
!2
dσ
≤ E
sup
0≤σ≤r≤t sup x∈R|
Ahσ,r(x)|2
Zr
0 Zh
0 Z
R
δhr,σ(x)pt−r(x−y)d x d y !2
dσ
.
Clearly, for any p≥1,
h−4 Z r
0 Z h
0 Z
R
δhr,σ(x)pt−r(x−y)d x d y !2
dσ
converges in Lp(Ω)toRr
0 pt−r(Br−Bs)
2ds, and, on the other hand, by Lemma 4 in the Appendix, ksup0≤σ≤r≤tsupx∈R|Ahσ,r(x)|kpconverges to zero ashtend to zero, for anyp≥2. This completes the proof of (13).
Step 4 Finally, we will discuss the limit of the martingale
Mht =48h−2 Z t
0
∆r,hd Br,
where∆r,h is defined in (12). Applying the asymptotic version of Knight’s theorem (see Revuz and Yor[9], Theorem 2.3 page 524) as in[4], it suffices to show the following convergences in probability ashtends to zero
〈Mh,B〉t→0, (14) uniformly in compact sets, and
〈Mh〉t→192
Z
R
(Lxt)3d x. (15)
Exchanging the order of integration we can write∆r,has
∆r,h=
Z r
0 Zh
0
Ahσ,r(x)δhr,σ(x)d x !
dbBσ=
Z r
0
where
Step 5 Let us prove (14). For anyp≥2 we have, by Burkholder’s inequality
E〈Mh,B〉t− 〈Mh,B〉s
Applying Cauchy-Schwarz inequality and lemmas 4 and 5 in the Appendix we obtain
B1≤Cp,t,ε(t−s)
p
2h2p+
p
2−ε.
A similar estimate can be deduced for the termB2. Finally, an application of the
Garsia-Rodemich-Rumsey lemma allows us to conclude that the convergence in (14) holds in probability, uniformly in compact sets.
Step 6 Let us prove (15). We have, by Itô’s formula and in view of (16) and (17)
We are going to see that only the first summand in the above expression gives a nonzero contribu-tion to the limit. Consider first the termR1t,h. We can express(Ψh
and, by Itô’s formula
Substituting the above equality in the expression of(Ψh r,σ)
2yields
R1t,h=
Z t
0 Zr
0 Zr
σ Zh
0 Zh
0
δhr,s(x)δhr,s(y)δhr,σ(x)δhr,σ(y)d x d y dsdσd r
+
Z t
0 Zr
0 Zh
0 Zh
0 Z r
σ
Ahs,r(x)δh r,s(y)dbBs
δh r,σ(x)δ
h
r,σ(y)d x d y dσd r
+
Z t
0 Zr
0 Zh
0 Zh
0 Z r
σ
Ahs,r(y)δhr,s(x)dbBs
δhr,σ(x)δhr,σ(y))d x d y dσd r
=
3 X
i=1 Ait,h.
Only the first term in the above sum will give a nonzero contribution to the limit. Let us consider first this term. We have
Zh
0
δhr,s(x)δhr,σ(x)d x=gh(Br−Bs,Br−Bσ),
where
gh(x,y) = (h− |x| − |y|)+1{x y<0}+ [(h− |x|)+∧(h− |y|)+]1{x y≥0}.
As a consequence,
482h−4A1,h
t = 48
2h−4 Z t
0 Z r
0 Zr
σ
gh(Br−Bs,Br−Bσ)2dsdσd r
= 1
248
2h−4 Zt
0 Z r
0 Z r
0
gh(Br−Bs,Br−Bσ)2dsdσd r
= 1
248
2h−4 Zt
0 Z
R2
gh(Br−x,Br−y)2LrxLryd x d y
d r.
Ashtends to zero this converges to 1
448 2Rt
0(L
Br
r )
2d r=192R R(L
x t)
3d x. This follows form the fact
that
Z
R2
gh(x,y)2d x d y= 1 2h
4.
Using Hölder’s inequality with 1
The termB1,thcan be estimated as follows
Zr
Furthermore, for the termB2,thwe can write
Using Lemma 4 in the Appendix with the exponentqyields
As a consequence,
h−4EA2,th≤Ch−4h
It only remains to show that the term h−4R2t,hin the right-hand side of (18) converges to zero. Using Fubini’s theorem we can write
R2t,h=
c =1. Using lemmas 4 and 5 in the Appendix we can show that the first two factors are bounded by a constant times h52−ε for some arbitrarily small ε >0. Using Doob’s maximal
inequality, the third factor can be estimated by
sup Finally, applying Lemma 4, we obtain
h−8E(R2t,h)2≤Ch1−δ
for some arbitrary smallδ. This completes the proof of Theorem 1.
5
Appendix
In this section we prove two technical results used in the paper.
Proof. By Tanaka’s formula applied to the time reversed Brownian motion we can write
Aσ,r(x) = −(Br−Bσ+h)++ (Br−Bσ)++ (Br−x+h)+−(Br−x)+ +1
2
LσBr−x+h−LBσr−x−LBrr−x+h+LBr−x
r
.
Therefore,
|Ar,σ(x)| ≤2h+sup
x∈R
sup
0≤r≤t|
Lxr+h−Lxr|. (19) Finally, the result follows from the inequalities for the local time proved by Barlow and Yor in
[1].
Lemma 5. Letδh
r,σ(x)be the random variable defined in (9). Then, for any p ≥ 2there exists a constant Ct,psuch that for all0≤s≤t
E sup
0≤σ≤s
Z t
s
Zh
0
δh
r,σ(x)d x d r
p
≤h2pCt,p(t−s)
p
2.
Proof. We can write
Z t
s
Zh
0
δh
r,σ(x)d x d r = Z t
s
(h− |Br−Bσ|)+d r=
Z
R
(Lxt −Lsx)(h− |x−Bσ|)+d x ≤h2sup
x
(Ltx−Lsx). (20)
Finally,
sup x
(Lxt −Lsx)≤sup x
Z t
s
δx(Bu−Bs)du,
andRstδx(Bu−Bs)duhas the same distribution asLxt−s, or, by the scaling properties of the local time, aspt−s L1x/pt−s, so
Esup x
(Lxt −Lsx)p≤(t−s)p2Esup
x (L1x)p.
AknowledgementWe would like to thank Jay Rosen for his interest and his helpful remarks on this paper.
References
[1] Barlow, M. T.; Yor, M. Semimartingale inequalities via the Garsia-Rodemich-Rumsey lemma, and applications to local times.J. Funct. Anal.49(1982), no. 2, 198–229. MR0680660
[2] Borodin, A. N. Brownian local time.Russian Math. Surveys44(1989), 1–51. MR0998360
[3] Chen, X., Li, W., Marcus, M. B. and Rosen, J: A CLT for the L2 modulus of continiuty of
[4] Hu, Y.; Nualart, D. Stochastic integral representation of theL2modulus of Brownian local time and a central limit theorem.Electron. Commun. Probab.14(2009), 529–539. MR2564487
[5] Nualart, D.The Malliavin Calculus and Related Topics. Second edition. Springer Verlag, Berlin, 2006. MR2200233
[6] Ocone, D. Malliavin calculus and stochastic integral representation of diffusion processes.
Stochastics12(1984), 161–185. MR0749372
[7] Perkins, E. Local time is a semimartingale.Z. Wahrsch. Verw. Gebiete 60(1982), 79–117. MR0661760
[8] Pitman, J.; Yor, M. Asymptotic laws of planar Brownian motion.Ann. Probab.14(1986), no. 3, 733–779. MR0841582
[9] Revuz, D.; Yor, M. Continuous martingales and Brownian motion. Third edition. Springer-Verlag, Berlin, 1999. MR1725357
[10] Rogers, L. C. G.; Walsh, J. B. The exact 4/3-variation of a process arising from Brownian motion.Stochastics Stochastics Rep.51(1994), no. 3-4, 267–291. MR1378160
[11] Rosen, L. Derivatives of self-intersection local times. To appear inSéminaire de Probabilités.
