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 \cite{CLMR},
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

Hu, Yaozhong

Nualart, David

English

arXiv

This is the published version, also available here: http://dx.doi.org/10.1214/ECP.v14-1511.The purpose of this note is to prove a central limit theorem for the L2-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

KU ScholarWorks (Univ. of Kansas)

Elect. Comm. in Probab. 14 (2009), 529–539 ELECTRONICCOMMUNICATIONSin PROBABILITYSTOCHASTIC INTEGRAL REPRESENTATION OF THE L2 MODU-LUS OF BROWNIAN LOCAL TIME AND A CENTRAL LIMIT THEO-REMYAOZHONG HUDepartment of Mathematics, University of Kansas, Lawrence, Kansas, 66045email: hu@math.ku.eduDAVID NUALART1Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045email: nualart@math.ku.eduSubmitted August 19, 2009, accepted in final form November 11, 2009AMS 2000 Subject classification: 60H07, 60F05, 60J55, 60J65Keywords: Malliavin calculus, Clark-Ocone formula, Brownian local time, Knight theorem, centrallimit theorem, Tanaka formula.AbstractThe purpose of this note is to prove a central limit theorem for the L2-modulus of continuity of theBrownian local time obtained in [3], using techniques of stochastic analysis. The main ingredientsof the proof are an asymptotic version of Knight’s theorem and the Clark-Ocone formula for theL2-modulus of the Brownian local time.1 IntroductionLet B = {Bt , t ≥ 0} be a standard Brownian motion, and denote by {Lt(x), t ≥ 0, x ∈ R} its localtime. In [3] the authors have proved the following central limit theorem for the L2modulus ofcontinuity of the local time:Theorem 1. For each fixed t > 0,h−32∫R(Lt(x + h)− Lt(x))2d x − 4thL−→ 8Çαt3η, (1.1)as h tends to zero, whereαt =∫R(Lt(x))2d x , (1.2)and η is a N(0, 1) random variable independent of B.1D. NUALART IS SUPPORTED BY THE NSF GRANT DMS0604207.529530 Electronic Communications in ProbabilityWe make use of the notationGt(h) =∫R(Lt(x + h)− Lt(x))2d x . (1.3)It is proved in [3, Lemma 8.1] that E Gt(h)= 4th+O(h2). Therefore, we can replace the term4th in (1.1) by E Gt(h).The proof of Theorem 1 is done in [3] by the method of moments. The purpose of this paper isto provide a simple proof of this result. Our method is based on an asymptotic version of Knight’stheorem (see Revuz and Yor [7], Theorem (2.3), page 524) combined with the techniques ofstochastic analysis and Malliavin calculus. The main idea is to apply the Clark-Ocone stochasticintegral representation formula to express Gt(h) − E Gt(h)as a stochastic integral. Then, bymeans of simple estimates using Hölder’s inequality, it is proved that the leading term is a martin-gale, to which we can apply an asymptotic version of Knight’ theorem. An important ingredientis to show the convergence of the quadratic variation of this martingale, which will be derived byusing Tanaka’s formula and backward Itô stochastic integrals.The paper is organized as follows. In the next section we recall some preliminaries on Malliavincalculus and we establish a stochastic integral representation for the random variable Gt(h). Then,Section 3 is devoted to the proof of Theorem 1.2 Stochastic integral representation of the L2-modulus of con-tinuityLet us introduce some basic facts on the Malliavin calculus with respect the the Brownian motionB = {Bt , t ≥ 0}. We refer to [4] for a complete presentation of these notions. We assume that Bis defined on a complete probability space (Ω,F , P) such that F is generated by B. Consider theset S of smooth random variables of the formF = fBt1 , . . . , Btn, (2.4)where t1, . . . , tn ≥ 0, f ∈ C∞b (Rn) (the space of bounded functions which have bounded deriva-tives of all orders) and n ∈ N. The derivative operator D on a smooth random variable of the form(2.4) is defined byDt F =n∑i=1∂ f∂ x iBt1 , . . . , BtnI[0,t i](t),which is an element of L2 (Ω× [0,∞)). We denote by D1,2 the completion of S with respect tothe norm ‖F‖1,2 given by‖F‖21,2 = EF2+ E∫ ∞0 Dt F2 d t.The classical Itô representation theorem asserts that any square integrable random variable canbe expressed asF = E[F]+∫ ∞0ut dBt ,Brownian local time and a central limit theorem 531where u = {ut , t ≥ 0} is a unique adapted process such that E∫∞0u2t d t<∞. If F belongs toD1,2, then ut = E[Dt F |Ft], where {Ft , t ≥ 0} is the filtration generated by B, and we obtain theClark-Ocone formula (see [5])F = E[F]+∫ ∞0E[Dt F |Ft]dBt . (2.5)The random variable Gt(h) =∫R(Lt(x + h) − Lt(x))2d x can be expressed in terms of the self-intersection local time of Brownian motion. In fact,Gt(h) =∫R∫ t0δ(Bu + x + h)du−∫ t0δ(Bu + x)du2d x= −2∫ t0∫ t0δ(Bv − Bu + h)dudv−∫ t0∫ t0δ(Bv − Bu)dudv= −2∫ t0∫ v0 δ(Bv − Bu + h) +δ(Bv − Bu − h)− 2δ(Bv − Bu)dudv .The rigorous justification of above argument can be made easily by approximating the Dirac deltafunction by the heat kernel pε(x) =1p2πεe−x2/2ε as ε tends to zero. That is, Gt(h) is the limit inL2(Ω) as ε tends to zero ofGεt (h) =−2∫ t0∫ v0 pε(Bv − Bu + h) + pε(Bv − Bu − h)− 2pε(Bv − Bu)dudv. (2.6)Applying Clark-Ocone formula we can derive the following stochastic integral representation forGt(h).Proposition 2. The random variable Gt(h) defined in (1.3) can be expressed asGt(h) = E(Gt(h)) +∫ t0ut,h(r)dBr ,whereut,h(r) = 4∫ r0∫ h0 pt−r(Br − Bu −η)− pt−r(Br − Bu +η)dηdu+4∫ r0I[0,h](Bu − Br)− I[0,h](Br − Bu)du. (2.7)Proof For any u< v and any h ∈ R we can writeDr pε(Bv − Bu + h) = p′ε(Bv − Bu + h)I[u,v](r),and for any u< r < vEDr p′ε(Bv − Bu + h)|Fr= Ep′ε(pv− rη+ Br − Bu + h)= p′v−r+ε(Br − Bu + h),532 Electronic Communications in Probabilitywhere η denotes a N(0, 1) random variable independent of B. Therefore, from Clark-Ocone for-mula (2.5) and Equation (2.6) we obtainGεt (h) = E(Gεt (h)) +∫ t0uεt,h(r)dBr ,whereuεt,h(r) = −2∫ tr∫ r0p′v−r+ε(Br − Bu + h) + p′v−r+ε(Br − Bu − h)+2p′v−r+ε(Br − Bu)dudv.This expression can be written asuεt,h(r) =−2∫ tr∫ r0∫ h0(p′′v−r+ε(Br − Bu +η)− p′′v−r+ε(Br − Bu −η)dηdudv.Using the fact that p′′t (x) = 2∂ pt∂ t(x) we obtainuεt,h(r) = −4 ∫ r0∫ h0 (pt−r+ε(Br − Bu +η)− pt−r+ε(Br − Bu −η)dηdu−∫ r0∫ h0 (pε(Br − Bu +η)− pε(Br − Bu −η)dηdu!.Letting ε tend to zero we get that uεt,h(r) converges in L2(Ω× [0, t]) to ut,h(r) as h tends to zero,which implies the desired result.From Proposition 2 we can make the following decompositionut,h(r) = ût,h(r) + 4Ψh(r),whereût,h(r) = −4∫ r0∫ h0 pt−r(Br − Bu +η)− pt−r(Br − Bu −η)dηdu= −4∫ r0∫ h0∫ η−ηp′t−r(Br − Bu + ξ)dξdηdu (2.8)andΨh(r) =−∫ r0I[0,h](Br − Bu)− I[0,h](Bu − Br)du. (2.9)As a consequence, we finally obtainGt(h)−E(Gt(h)) =∫ t0ût,h(r)dBr + 4∫ t0Ψh(r)dBr . (2.10)Brownian local time and a central limit theorem 5333 Proof of Theorem 1The 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.Step 1 We claim that the stochastic integral∫ t0ût,h(r)dBr makes no contribution to the limit (1.1).That is,h−3/2∫ t0ût,h(r)dBrconverges in L2(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 thatE∫ t0|ût,h(r)|2dr≤ Ch4 ,for all h> 0.Proof From (2.8) we can writeE|ût,h(r)|2=∫ r0∫ r0∫ h0∫ h0∫ η1−η1∫ η2−η2E(p′t−r(Br − Bu1 + ξ1)×p′t−r(Br − Bu2 + ξ2))dξ1dξ2dη1dη2du1du2.By a symmetry argument, it suffices to integrate in the region 0< u1 < u2 < r. SetΦ(u1, u2,ξ1,ξ2) = Ep′t−r(Br − Bu1 + ξ1)p′t−r(Br − Bu2 + ξ2).Then,Φ(u1, u2,ξ1,ξ2) = Ep′t−r(Br − Bu2 + Bu2 − Bu1 + ξ1)p′t−r(Br − Bu2 + ξ2)= Ep′t−r+u2−u1(Br − Bu2 + ξ1)p′t−r(Br − Bu2 + ξ2)=∫Rpr−u2(z)p′t−r+u2−u1(z+ ξ1)p′t−r(z+ ξ2)dz≤ ‖pr−u2‖p1‖p′t−r+u2−u1‖p2‖p′t−r‖p3 ,where 1p1+ 1p2+ 1p3= 1. It is easy to see that‖pr−u2‖p1 ≤ C(r − u2)− 12+ 12p1 ,‖p′t−r+u2−u1‖p2 ≤ C(t − r + u2 − u1)−1+ 12p2 ≤ C(u2 − u1)−1+ 12p2 ,‖p′t−r‖p3 ≤ C(t − r)−1+ 12p3 ,for some constant C > 0. ThusE|ût,h(r)|2≤ C∫ r0∫ u20∫ h0∫ h0∫ η1−η1∫ η2−η2(r − u2)− 12+ 12p1×(u2 − u1)−1+ 12p2 (t − r)−1+12p3 dξ1dξ2dη1dη2du1du2≤ C h4 .This proves the proposition.534 Electronic Communications in ProbabilityStep 2 Taking into account Proposition 3 and Equation (2.10), in order to show Theorem 1 itsuffices to show the following convergence in law:h−32∫ t0Ψh(r)dBrL→ 2ηÇαt3,where η is a standard normal random variable independent of B, αt has been defined in (1.2),and Ψh(r) is given by (2.9). Notice thatMht = h− 32∫ t0Ψh(r)dBris a martingale with quadratic variation¬Mh¶t= h−3∫ t0Ψ2h(r)dr.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∫ t0Ψ2h(r)dr →43αt , (3.11)and¬Mh, B¶t= h−3/2∫ t0Ψh(r)dr → 0, (3.12)as h tends to zero, where the convergence (3.12) is uniform in compact sets. In fact, let Bh bethe Brownian motion such that Mht = Bh〈Mh〉t. Then, from Theorem (2.3) pag. 524 in [7], andthe convergences (3.11) and (3.12), we deduce that (B, Bh, 〈Mh〉t) converges in distribution to(B,β , 43αt), where β is a Brownian motion independent of B. This implies that Mht = Bh〈Mh〉tconverges in distribution to β 43αt, which yields the desired result.Before proving (3.11) and (3.12) we will express Ψh(r) using Tanaka’s formula. By the occupationformula for the Brownian motion we can writeΨh(r) = −∫RI[0,h](Br − x)− I[0,h](x − Br)Lr(x)d x=∫ h0 Lr(Br + y)− Lr(Br − y)d y.We can express the difference Lr(Br − y) − Lr(Br + y) by means of Tanaka’s formula for theBrownian motion {Br − Bs, 0≤ s ≤ r}:Lr(Br + y)− Lr(Br − y) = y + (Br − y)+ − (Br + y)+−∫ r0IBr−Bs+y>0 − IBr−Bs−y>0dbBs,Brownian local time and a central limit theorem 535where dbBs denote the backward stochastic Itô integral and y > 0. Integrating in the variable yyieldsΨh(r) =h22−∫ h0(Br + y)+ − (Br − y)+d y−∫ h0∫ r0I y>|Br−Bs |dbBsd y. (3.13)By stochastic Fubini’s theorem∫ h0∫ r0I y>|Br−Bs |dbBsd y =∫ r0(h− |Br − Bs|)+dbBs. (3.14)Hence,Ψh(r) =h22−∫ h0(Br + y)+ − (Br − y)+d y−∫ r0(h− |Br − Bs|)+dbBs. (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,¬Mh, B¶tconverges to zero in L2(Ω) uniformly in compact sets as h tendsto zero.Proof In view of (3.13) it suffices to show thatsup0≤t≤t1h−3/2∫ t0∫ r0(h− |Br − Bs|)+dbBsdrconverges to zero in L2(Ω) as h tends to zero, for any t1 > 0. For any p ≥ 2 and any 0 ≤ s < t wecan write by Fubini’s theorem and Burkholder’s inequalityE ∫ ts∫ r0(h− |Br − Bv |)+dbBvdrp!≤ 2p−1¨E ∫ s0∫ ts(h− |Br − Bv |)+drdbBvp!+E ∫ ts∫ tv(h− |Br − Bv |)+drdbBvp!«≤ cp¨E∫ s0∫ ts(h− |Br − Bv |)+dr2dvp/2+E∫ ts∫ tv(h− |Br − Bv |)+dr2dvp/2«= cp(I1 + I2).536 Electronic Communications in ProbabilityThe term I1 can be expressed using occupation formula as followsI1 = E∫ s0∫R(h− |x − Bv |)+(Lt(x)− Ls(x))d x2dvp/2≤ sp/2h2pEsupx|Lt(x)− Ls(x)|p.By the inequalities for local time proved, for instance, in [1] we obtainI1 ≤ cph2p|t − s|p/2.Similarly,I2 = E∫ ts∫R(h− |x − Bv |)+(Lt(x)− Lv(x))d x2dvp/2≤ h2p|t − s|p/2 sups≤v≤tEsupx|Lt(x)− Lv(x)|p≤ cph2p|t − s|p.Finally, a standard application of the Garsia-Rudemich-Rumsey lemma allows us to conclude.Step 4 We are going to show thath−3∫ t0Ψh(r)2drL2(Ω)→43αt , (3.16)as h tends to zero. Notice thatαt = 2∫ t0∫ v0δ0(Bv − Bu)dudvis the self-intersection local time of B, and Equation (3.16) provides an approximation for thisself-intersection local time which has its own interest.Taking into account (3.13) and (3.14), the convergence (3.16) will follow fromh−3∫ t0∫ r0(h− |Br − Bs|)+dbBs2drL2(Ω)→43αt , (3.17)as h tends to zero. By Itô’s formula we can write∫ r0(h− |Br − Bs|)+dbBs2= 2∫ r0∫ rs(h− |Br − Bu|)+dbBu×(h− |Br − Bs|)+dbBs +∫ r0(h− |Br − Bs|)+2 ds. (3.18)Finally, (3.17) follows form (3.18) and the next two lemmas.Brownian local time and a central limit theorem 537Lemma 5. We have∫ t0∫ r0(h− |Br − Bs|)+2h3dsdrL2(Ω)→43αt ,as h tends to zero.Proof Notice thatαt =∫RLt(x)2d x =∫ t0∫RLr(x)Ldr(x)d x =∫ t0Lr(Br)dr,and∫ t0∫ r0(h− |Br − Bs|)+2h3dsdr =∫ t0∫R(h− |Br − x |)+2h3Lr(x)d x .As a consequence, taking into account that∫R(h− |Br − x |)+2h3d x =∫R[(h− |x |)+]2h3d x =43,we obtain∫ t0∫ r0(h− |Br − Bs|)+2h3dsdr −43αt≤∫ t0∫R(h− |Br − x |)+2h3Lr(x)− Lr(Br) d xdr≤43∫ t0sup|x−y|<hLr(x)− Lr(y) dr,which clearly converges to zero in L2 by the properties of the Brownian local time (see, for in-stance, [2]).Lemma 6. We have1h6E∫ t0∫ r0∫ rs(h− |Br − Bu|)+dbBu(h− |Br − Bs|)+dbBsdr2→ 0,as h tends to zero.Proof By the isometry property of the backward Itô integral we can writeBh : =1h6E∫ t0∫ r0∫ rs(h− |Br − Bu|)+dbBu(h− |Br − Bs|)+dbBsdr2=1h6E∫ t0∫ ts(h− |Br − Bs|)+∫ rs(h− |Br − Bu|)+dbBudr2ds≤ EB1h B2h,538 Electronic Communications in ProbabilitywhereB1h =∫ t0∫ ts(h− |Br − Bs|)+h2dr2dsandB2h = sup0<s<r<t∫ rs(h− |Br − Bu|)+hdbBu2.As in Lemma 5, we can show that B1h converges to94∫ t0 Lt(Bs)− Ls(Bs)2 ds, and the convergenceholds in Lp(Ω) for any p ≥ 2. Here we use∫R(h− |x |)+h2d x =32.On the other hand,∫ rs(h−|Br−Bu|)+hdbBu can be expressed using again Tanaka’s formula:1h∫ rs(h− |Br − Bu|)+dbBu =∫ h0∫ rsI y>|Br−Bu|dbBud y=1h∫ h0 Lr(Br − y)− Lr(Br + y)d y−1h∫ h0 Ls(Br − y)− Ls(Br + y)d y +h2+1h∫ h0(Br − Bs + y)+ − (Br − Bs − y)+d y.Therefore,1h∫ rs(h− |Br − Bu|)+dbBu≤ sups<r<tsup|x−y|≤2hLr(x)− Lr(y)+O(h),which also converges to zero in Lp(Ω) for any p ≥ 2.References[1] Barlow, M. T.; Yor, M. (Semi-) martingale inequalities and local times. Z. Wahrsch. Verw.Gebiete 55 (1981), no. 3, 237–254. MR0608019[2] Borodin, A. N. Brownian local time. Russian Math. Surveys 44 (1989), 1–51. MR0998360[3] Chen, X., Li, W., Marcus, M. B. and Rosen, J. A CLT for the L2 modulus of continiuty ofBrownian local time. Preprint.[4] Nualart, D. The Malliavin Calculus and Related Topics. Second edition. Springer Verlag, Berlin,2006. MR2200233[5] Ocone, D. Malliavin calculus and stochastic integral representation of diffusion processes.Stochastics 12 (1984), 161–185. MR0749372Brownian local time and a central limit theorem 539[6] Pitman, J.; Yor, M. Asymptotic laws of planar Brownian motion. Ann. Probab. 14 (1986), no.3, 733–779. MR0841582[7] Revuz, D.; Yor, M. Continuous martingales and Brownian motion. Third edition. Springer-Verlag, Berlin, 1999. MR1725357[8] Rosen, J. A stochastic calculus proof of the CLT for the L2 modulus of continuity of local time.http://arxiv.org/pdf/0910.2922.pdf.

Stochastic integral representation of the $L^{2}$ modulus of Brownian local time and a central limit theorem

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Stochastic integral representation of the L2L^{2}L2 modulus of Brownian local time and a central limit theorem

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Stochastic integral representation of the $L^{2}$ modulus of Brownian local time and a central limit theorem