CENTRAL LIMIT THEOREM FOR THE THIRD MOMENT IN SPACE OF THE BROWNIAN LOCAL TIME INCREMENTS

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.


Introduction
Let {B t , t ≥ 0} be a standard one-dimensional Brownian motion. We denote by {L x t , t ≥ 0, x ∈ } a continuous version of its local time. The following central limit theorem for the L 2 modulus of continuity of the local time has been recently proved: where η is a N (0, 1) random variable independent of B and 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. 1 2 η as h tends to zero, where η is 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 as a stochastic integral. In comparison with the L 2 modulus 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 L 2 (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, 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 F h t defined in (2). Section 4 is devoted to the proof of Theorem 1, and the Appendix contains two technical lemmas.

Preliminaries on Malliavin Calculus
Let us recall some basic facts on the Malliavin calculus with respect the Brownian motion B = {B t , t ≥ 0}. We refer to [5] for a complete presentation of these notions. We assume that B is defined on a complete probability space (Ω, , P) such that is generated by B. Consider the set of smooth random variables of the form F = f B t 1 , . . . , B t n ,, where t 1 , . . . , t n ≥ 0, n ∈ and f is bounded and infinitely differentiable with bounded derivatives of all orders. The derivative operator D on a smooth random variable of this is defined by We denote by 1,2 the completion of with respect to the norm F 1,2 given by The classical Itô representation theorem asserts that any square integrable random variable can be expressed as is the filtration generated by B, and we obtain the Clark-Ocone formula (see [6])

Stochastic integral representations
Consider the random variable γ t defined rigorously as the limit in L 2 (Ω) where 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.
Then, γ ε t converges in L 2 (Ω) as tends to zero to the random variable Proof. By Clark-Ocone formula applied to γ ε t we obtain the integral representation where { t , t ≥ 0} denotes the filtration generated by the Brownian motion. Then, and for any r ≤ t As tends to zero this expression converges in which completes the proof.
Let us now obtain a stochastic integral representation for the third integrated moment Proof. Let us write We can pass to the limit as ε tends to zero the first We are going to apply the Clark-Ocone formula to the random variable We need to compute D E D r Φ (s) | r ds. To do this we decompose the region D, up to a set of zero Lebesgue measure, as Step 1 For the region D 1 we obtain We can write this in the following form Integrating with respect to the variable s 3 yields This expression can be written in terms of the local time: We make the change of variables y → h − y and z → z + h in the last integral and we obtain Taking the limit as tends to zero in L 2 (Ω × [0, t]) and integrating in x yields Step 2 For the region D 2 , taking first the conditional expectation with respect to s 2 which contains r , and integrating with respect to the law of B s 2 − B r we obtain

Integrating by parts yields
Letting tend to zero we obtain Hence, We have Substituting (6) into (5) yields Now we integrate in the variable u and we obtain Thus, which completes the proof.

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 from line to line.
Step 2 In order to handle the term h −2 t 0 (Φ (1) r + Φ (2) r )dB r we consider the function and we can write Applying Tanaka's formula to the time reversed Brownian motion {B r − B s , 0 ≤ s ≤ r}, we obtain where d B s denotes the backward Itô integral. Clearly, the only term that gives a nonzero contribution is We are going to use the following notation where 0 < σ < r. With this notation we want to find the limit in distribution of as h tends to zero. By Itô's formula, Therefore, Notice that, by Lemma 2 converges in L 2 (Ω) to −12γ t , which cancels with the limit obtained in Step 1.
To handle the first summand in the right-hand side of (10) we make the decomposition and Step 3 We claim that lim which implies that h −2 t 0 Γ r,h dB r converges to 0 in L 2 (Ω) as h tends to zero. Let us prove (13). We can write Clearly, for any p ≥ 1, converges in L p (Ω) to r 0 p t−r (B r − B s ) 2 ds, and, on the other hand, by Lemma 4 in the Appendix, sup 0≤σ≤r≤t sup x∈ |A h σ,r (x)| p converges to zero as h tend to zero, for any p ≥ 2. This completes the proof of (13).
Step 4 Finally, we will discuss the limit of the martingale 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 as h tends to zero 〈M h , B〉 t → 0, uniformly in compact sets, and Exchanging the order of integration we can write ∆ r,h as where Substituting the above equality in the expression of (Ψ h r,σ ) 2 yields Only the first term in the above sum will give a nonzero contribution to the limit. Let us consider first this term. We have As a consequence, As h tends to zero this converges to 1 4 Finally, the result follows from the inequalities for the local time proved by Barlow and Yor in [1].

Lemma 5.
Let δ h r,σ (x) be the random variable defined in (9). Then, for any p ≥ 2 there exists a constant C t,p such that for all 0 ≤ s ≤ t