Integration with respect to local time and Ito's formula for smooth nondegenerate martingales

We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\^o's s formula as an integral over space and time with respect to local time.


Introduction
We consider a continuous nondegenerate martingale X = {X t , t ∈ [0, 1]} of the form X t = t 0 u s dW s where W = {W t , t ∈ [0, 1]} is a standard Brownian motion and u is an adapted stochastic process. Let F : R × [0, 1] → R be an absolutely continuous function with partial derivatives satisfying some local integrability properties. The main aim of this paper is to obtain an Itô's formula for F (X t , t) where the term corresponding usually to the second order derivative is expressed as an integral over space and time with respect to local time.
We will prove this results when u satisfies (locally) the assumptions (H1) For all t ∈ [0, 1], u t belongs to the space D 3,2 and for all p ≥ 2 |u t | ≥ ρ > 0 for some constant ρ and for all t ∈ [0, 1]. Moret and Nualart (2000) consider an Itô's formula for this class of nondegenerate martingales. Their main result reads as follows: Theorem 0.1 (Moret and Nualart, 2000) Let u be a process satisfying (H1) and (H2). Set X = t 0 usdW s . Then for any funcion f ∈ L 2 loc (R) the quadratic covariation [f (X), X] exists and the following Itô's formula holds for all t ∈ [0, 1], where F (x) = F (0) + x 0 f (y)dy. Moret (1999), gave an extension of this last result for functions F depending also on t. They consider a new hypothesis on functions f : (C) f (·, t) ∈ L 2 loc (R) and for all compact set K ⊂ R f (x, t) is continuous in t as a function of [0, T ] to L 2 (K) Then, their result is the following: Theorem 0.2 (Moret, 1999) Let u be a process satisfying (H1) and (H2). Set X = t 0 usdW s . Let F (x, t) be an absolutely continuous function in x such that the partial derivative f (·, t) satisfies (C). Then, the quadratic covariation [f (X, ·), X] exists and the following Itô's formula holds exists uniformly in probability for (D n ) n a sequence of smooth partitions of [0, 1].
In these two results, following the ideas of Föllmer, Protter and Shiryayev (1995) for the Brownian motion, the additional term is written as a quadratic covariation. Jolis (1997, 2002) extended the results of Föllmer et al. (1995) to the case of the elliptic and hypoelliptic diffusions.
Nevertheless, it is important to point out the differences between the work of Moret and Nualart (2000) and Föllmer et al. (1995). One of the keys of their proofs is to obtain some a priori estimates on the Riemann sums. In Föllmer et al. (1995) these estimates are obtained using the semimartingale expression of the time-reversed Brownian motion and well-known bounds for the density of the Brownian motion. Moret and Nualart (2000) used another approach, using Malliavin calculus in order to obtain sharp estimates for the density of the process X t and avoiding the time-reversed arguments.
We want to express the quadratic variation term as an integral with respect to the local time. There are several papers where the integrals with respect to local time are used in Itô's formula. In 1981, Bouleau and Yor obtained the following extension of the Itô's formula : Theorem 0.3 (Bouleau and Yor, 1981) Let X = (X t ) t≥0 be a continuous semimartingale and let F : R −→ R be an absolutely continuous function with derivative f . Assume that f is a mesurable locally bounded function. Then: Eisenbaum (2000Eisenbaum ( , 2001 defined an integral in time and space with respect to the local time of the Brownian motion. Using this integral, the quadratic covariation in the formula given in Föllmer et al. can be expressed as an integral with respect to the local time. She obtained the following result: Theorem 0.4 (Eisenbaum, 2000 and2001) Let W = (W t ) 0≤t≤1 be a standard Brownian motion and F a function defined on R × [0, 1] such that there exist first order Radon-Nikodym derivatives ∂F ∂t and ∂F ∂x such that for every A ∈ R + , Then, This result has been extended by Bardina and Rovira (2007) for elliptic diffusion processes. In our papers we will follow the ideas Eisenbaum (2000Eisenbaum ( ,2001, assuming on the function F the hypothesis considered in Theorem 0.4. In the papers of Eisenbaum (2000Eisenbaum ( ,2001, as well as in Föllmer et al. (1995) or in the extension of Bardina and Rovira (2007), one of the main ingredients is the study of the time reversed process and the relationship between the quadratic covariation and the forward and backward stochastic integrals. We show that we can adapt the methods of Eisebaum without using the time reversed process and the backward integral. We will follow the methods of Moret and Nualart (2000) and we will use Malliavin calculus to obtain the necessary estimates for the Riemann sums .
In our paper, the existence of the quadratic covariation is not one of our main objectives. Nevertheless, it will be an important tool in our computations. We recall its definition.
Definition 1 Given two stochastic processes Y = {Y t , t ∈ [0, 1]} and Z = {Z t , t ∈ [0, 1]} we define their quadratic covariation as the stochastic process [Y, Z] given by the following limit in probability, if it exists, where D n is a sequence of partitions of [0, 1].
We will assume that the partitions D n satisfy (M) lim n sup ti∈Dn (t i+1 − t i ) = 0, M := sup n sup ti∈Dn ti+1 ti < ∞. We impose this condition in order to avoid certain possibly exploding Riemann sums.
Other extensions for Itô's formula has been obtained recently. Among others, there is the paper of Dupoiron et al. (2004) for uniformly elliptic diffusions and Dirichet processes, the work of Ghomrasni and Peskir (2006) for continuous semimartingales, the paper of Flandoli, Russo and Wolf (2004) for a Lyons-Zheng process or the work of Di Nunno, Meyer-Brandis, Øksendal and Proske (2005) for Lévy processes.
The paper is organized as follows. In Section 1 we give some basic definitions and results on Malliavin calculus, recalling some results obtained in Moret and Nualart (2000). In Section 2 we define the space where we are able to construct an integral in the plane with respect to the local time of a nondegenerate Brownian martingale. Finally, Section 3 is devoted to present our main result the extension of Itô's formula.
Along the paper we will denote all the constants by C, C p , K or K p , unless they may change from line to line.

Preliminaries
Let (Ω, F , P ) be the canonical probability space of a standard Brownian motion W = {W t , 0 ≤ t ≤ 1}, that is, Ω is the space of all continuous functions ω : [0, 1] → R vanishing at 0, P is the standard Wiener measure on Ω and F is the completion of the Borel σ-field of Ω with respect to Let S be the set of smooth random variables of the form The Malliavin derivative of order N ≥ 2 is defined by iteration, as follows.
For any real number p ≥ 1 and any integer N ≥ 1 we denote by D N,p the completion of the set S with respect to the norm The domain of the derivative operator D is the space D 1,2 . The divergence operator δ is the adjoint of the derivative operator. The domain of the operator δ, denoted by Dom δ, is the set of processes u ∈ L 2 ([0, T ] × Ω) such that there exists a square integrable random variable δ(u) verifying for any F ∈ S. The operator δ is an extension of Itô's stochastic integral and we will make use of the notation δ(u) = 1 0 u s dW s . We will recall some useful results from Moret and Nualart (2000). We refer the reader to this paper for their proof and a detailed account of these results. We also refer to Nualart (1995Nualart ( , 2006 for any other property about operators D and δ. a,b] belongs to Dom δ. Then Y has an absolutely continuous distribution with density p that satisfies the inequality Proof: It follows from Proposition 1 and (2.6) in Moret and Nualart (2000). 2 The following Proposition is also a slight modification of Corollary 2 of Moret-Nualart (2000).
Proof: See Corollary 2 in Moret-Nualart (2000). The same proof works using a dominated convergence argument. 2 where C 0 is a constant does not depend on Z.
Proof: See Lemma 10 in Moret-Nualart (2000). 2 2 Stochastic integration with respect to local time of the martingale Following the ideas of Eisenbaum (2000), we consider first the space of functions for whose elements we can define a stochastic integration with respect to local time. Let f be a measurable function from R × [0, 1] into R. We define the norm · by Consider the set of functions It is easy to check that H is a Banach space. Let us consider X a nondegenerate martingale of the type X t = t 0 u s dW s where u is an adapted stochastic process satisfying hypotheses (H1) and (H2). Let us show now how to define a stochastic integration over the plane with respect to the local time L of the process X for the elements of H.
Let f ∆ be an elementary function, where (x k ) 1≤k≤m1 is a finite sequence of real numbers, (s l ) 1≤l≤m2 is a subdivision of [0, 1], (f kl ) 1≤k≤m1; 1≤l≤m2 is a sequence of real numbers and finally, It is easy to check that the elementary functions are dense in H.
We define the integration for the elementary function f ∆ with respect to the local time L of the martingale X as follows Let f be a function of H. Let us consider (f n ) n∈N a sequence of elementary functions converging to f in H. We will check that the sequence where d a L a t denotes the integral with respect to a −→ L a t .
Proof: It follows easily from Theorem 0.3 and Theorem 0.1.
and C does not depend on f 1 and f 2 .
Proof: When f 1 = f 2 = f ∈ C ∞ K (R), this inequality is checked in the proof of Proposition 14 in Moret and Nualart (2000). The same proof also works when f 1 = f 2 with f 1 , f 2 ∈ C ∞ K (R). Now, fixed our functions f 1 , f 2 let us consider sequences f n 1 ↑ f 1 and f n 2 ↑ f 2 with f i n ∈ C ∞ K (R) for all n and i ∈ {1, 2}. Then, the result can be obtained by a dominated convergence argument. Proof: Let f ∆ be an elementary function. From Theorem 0.2 and Lemma 2.1 it is easy to get that the quadratic covariation [f (X, .), X] t exists and that The key of the proof is to check that for all elementary function f ∆ where the constant does not depend on f ∆ . Notice that, where in the last inequality we have used Fatou's lemma. Along the study of I 1 and I 2 we will make use of the methods presented in the proofs of Propositions 13 and 14 in Moret and Nualart (2000). For the sake of completeness, we will give the main steps of our proofs in the study of I 2 . For the other terms, we will refer the reader to the paper of Moret and Nualart (2000).
By the isometry, and using Propositions 1.2 and 1.3 Using the condition (M) over the partitions, we have that, by bounded convergence, On the other hand, Following now the methods of Proposition 14 of Moret and Nualart (1999) and using again Propositions 1.2 and 1.3 as we did in the study of I 2 , we get that By similar computations to those of the term I 2 we obtain that lim inf Let us study now I 1,2 . Using Lemma 2.2, notice that Following again the methods of the proof of Proposition 14 of Moret and Nualart (1999) -more precisely, the proof of inequalities (5.36) and (5.37)-the last expression is bounded by ti,tj ∈Dn,ti<tj ≤t := C(I 1,2,1 + I 1,2,2 ).
Since tj ∈Dn,ti<tj ≤t we get that And this clearly yields that Finally we have to consider I 1,2,2 . First of all, notice that I 1,2,2 := ti,tj ∈Dn,ti<tj ≤t we obtain the bound I 1,2,2 ≤ I 1,2,2,1 + I 1,2,2,2 , where Now, since we can write using an argument of bounded convergence we have that On the other hand, observe that fixed l, there exists only one j (that we will denote by j(l)) such that t j(l) < s l ≤ t j(l)+1 . So, ti,tj ∈Dn,ti<tj ≤t we obtain easily that lim n→∞ I 1,2,2,2 = 0.
So, putting together (3)-(11), we have proved (2). Now, given f ∈ H, let us consider {f n } n∈N a sequence of elementary functions converging to f in H, and we define Clearly, this limit exists. Indeed, for any ε > 0 there exists n 0 such that for any n, m ≥ n 0 , f n − f m < ε and using inequality (2) we obtain that Moreover, using again inequality (2), it is clear that the definition does not depend on the choice of the sequence (f n ). Indeed, given (f 1 n ) n∈N and (f 2 n ) n∈N two sequences converging to f in H, we have that goes to zero when n tends to infinity.
The following results is an obvious consequence of Theorem 0.2 and Remark 2.4.
Corollary 2.5 Let u be a process satisfying (H1) and (H2). Set X = t 0 usdW s . Consider a sequence of partitions D n of partitions of [0, 1] verifying conditions (M). Let F (x, t) be an absolutely continuous function in x such that the partial derivative f (·, t) satisfies (C). Then, if f ∈ H, we have the following extension for the Itô's formula: F (X s , ds).

Itô's formula extension
Now we can state the main result of this paper.

Assume that these derivatives satisfy that for every
Then, for all t ∈ [0, 1], Proof: Using localization arguments we can assume that F has compact support and Let g ∈ C ∞ be a function with compact support from R to R + such that R g(s)ds = 1. We define, for any n ∈ N, g n (s) = ng (ns) and Then F n ∈ C ∞ (R × [0, 1]). Hence, by the usual Itô's formula, for every ε > 0, we can write Using the arguments of Azéma et al. (1998) we will study the convergence of (12). Since F is a continuous function with compact support, it is easy to check that (F n (X t , t)) n∈N converges in probability to F (X t , t).
On the other hand Hence, ∂F ∂t ∈ L 1 (R × [0, 1]). Under our hypothesis over the martingale X, it follows from Proposition 1.1 and Lemma 1.3 that for any t ∈ [0, 1], the random variable X t is absolutely continuous with density p t satisfying the estimate Then, it is easy to see that On the other hand, following the same type of arguments that in (13), we are able to write Letting ε to zero, the proof is finished. 2 Remark 3.2 Notice that under the hypotheses of Theorem 3.1, it is possible that ∂F ∂x does not belong to the space H. In this case, using the localization arguments, we can always assume that ∂F ∂x (x, s)I (ε,t) (s), x ∈ R, s ∈ [0, 1] belongs to H for any ε > 0 and we can define  (14) exist.