AN EXTENSION OF SUB-FRACTIONAL BROWNIAN MOTION

In this paper, firstly, we introduce and study a self-similar Gaussian process with parameters H ∈ (0, 1) and K ∈ (0, 1] that is an extension of the well known sub-fractional Brownian motion introduced by Bojdecki et al. [4]. Secondly, by using a decomposition in law of this process, we prove the existence and the joint continuity of its local time. 2010 Mathematics Subject Classification: 60G18.


Introduction
The sub-fractional Brownian motion S H := {S H t ; t ≥ 0} (sfBm for short) with parameter H ∈ (0, 2) was introduced by Bojdecki et al. [4].It is a continuous centered Gaussian process, starting from zero, with covariance function The case H = 1 corresponds to the standard Brownian motion (Bm for short).Bojdecki et al. [4] have proved that the increments of sfBm satisfy On the other hand, Ruiz de Chávez and Tudor [8] have obtained for H ∈ (0, 1) the following decomposition in law of sfBm The case H ∈ (1, 2) is given by Bardina and Bascompte [1].They proved that 2Γ(2−H) , and the Bm W and the sfBm S H are independent.
They also proved that the process X H is Gaussian, centered, and that its covariance function is ).Moreover, Mendy [7] proved that there exists a constant The self similarity and stationarity of the increments are two main properties for which fBm enjoyed success as modeling tool in telecommunications and finance.The sfBm is an extension of Bm which preserves many properties of fBm, but not the stationarity of the increments.This property makes sfBm a possible candidate for models which involve long dependence, self similarity and non stationarity of increments.It is, thus, very natural to explore the existence of processes which keep some of the properties of sfBm, specially a decomposition in law that includes sfBm, but also enlarge our modelling tool kit.The same motivation is given by Houdré and Villa [6] in case of the bifractional Brownian motion (bfBm for short), which generalizes the fBm.Definition 1.1.We denote by S H,K := {S H,K t ; t ≥ 0} a centered Gaussian process, starting from zero, with covariance function where H ∈ (0, 1) and K ∈ (0, 1]. The case K = 1 corresponds to sfBm with parameter H ∈ (0, 1).Existence of S H,K can be shown in the following two ways: 1) Consider the process where {B H,K t ; t ∈ R} is the bfBm on the whole real line with parameters H ∈ (0, 1) and K ∈ (0, 1], introduced by Houdré and Villa [6].It is easy to see that Y t and S H,K have the same covariance function.Therefore S H,K exists.2) It can be shown by the following theorem which gives also a decomposition in law of the process.It will be useful in the sequel.Theorem 1.2. 1) For any H ∈ (0, 1) and K ∈ (0, 1], the function S(., .) is symmetric and positive definite.

Since the functions t
for all x ≥ 0. Therefore the function S(., .) is positive definite.
2) Using the fact that S H,K is a Gaussian process, it suffices to see that In the sequel C and C p denote constants which will be different even when they vary from one line to the next.

Local time of S H,K
We begin this section by the definition of local time.For a complete survey on local time, we refer to Geman and Horowitz [5] and the references therein.
Let X := {X t ; t ≥ 0} be a real-valued separable random process with Borel sample functions.For any Borel set B ⊂ R + , the occupation measure of X on B is defined as where λ is the one-dimensional Lebesgue measure on R + .If µ B is absolutely continuous with respect to Lebesgue measure on R, we say that X has a local time on B and define its local time, L(B, .), to be the Radon-Nikodym derivative of µ B .Here, x is the so-called space variable and B is the time variable.By standard monotone class arguments, one can deduce that the local time has a measurable modification that satisfies the occupation density formula: for every Borel set B ⊂ R + and every measurable function Sometimes, we write L(t, x) instead of L([0, t], x).
Here is the outline of the analytic method used by Berman [2] for the calculation of the moments of local time.
For fixed sample function at fixed t, the Fourier transform on x of L(t, x) is the function Using the density of occupation formula, we get Therefore, we may represent the local time as the inverse Fourier transform of this function, i.e., ( 4) e iu(Xs−x) ds du.
We end this section by the definition of the concept of local nondeterminism, (LND for short).Let J be an open interval on the t axis.Assume that {X t ; t ≥ 0} is a zero mean Gaussian process without singularities in any interval of the length δ, for some δ > 0, and without fixed zeros, i.e., there exists δ > 0, such that To introduce the concept of LND, Berman [3] defined the relative conditioning error, where for p ≥ 2, t 1 < • • • < t p are arbitrary ordered points in J.
We say that the process X is LND on J if for every p ≥ 2, lim This condition means that a small increment of the process is not almost relatively predictable on the basis of a finite number of observations from the immediate past.Berman [3] has proved, for Gaussian processes, that the LND is characterized as follows.
Proposition 2.1.A Gaussian process X is LND if and only if for every integer p ≥ 2, there exist two positive constants δ and C p such that for all orderer points t 1 < • • • < t p that are arbitrary points in J with t 0 = 0, t p − t 1 ≤ δ and (u 1 , . . ., u j ) ∈ R.
The purpose of this section is to present sufficient conditions for the existence of the local time of S H,K .Furthermore, using the LND approach, we show that the local time of S H,K has a jointly continuous version.
Theorem 2.3.Assume H ∈ (0, 1) and K ∈ (0, 1).On each (time For the proof of Theorem 2.3 we need the following lemma.This result on the regularity of the increments of S H,K will be the key for the existence and the regularity of local times.Lemma 2.4.Assume H ∈ (0, 1) and K ∈ (0, 1).There exists δ > 0 and, for any integer p ≥ 2, there exists a constant 0 < C p < +∞, such that for all s, t ≥ 0 such that |t − s| < δ.
Proof: By virtue of (3) and the elementary inequality (a+b) 2 ≤ 2a 2 +2b 2 , we have Since 0 < K < 1, we can choose δ > 0 small enough such that for all t, s ≥ 0 with |t − s| < δ, we have Indeed, it suffices to choose . Finally, with |t − s| < δ and Since S H,K is a centered Gaussian process, then the proof of Lemma 2.4.
Proof: By virtue of (3), we have Therefore, the elementary inequality (a + b) According to Remark 2.2, the sfBm S HK is LND on [0, 1], then there exist two constants δ > 0 and 0 < C p < +∞ such that for any Moreover (2) and the fact that H ∈ (0, 1) imply that Var In addition, the elementary inequality (a + b Therefore, it suffices now to choose This with Proposition 2.1 complete the proof of Proposition 2.5.Now, we are in position to give the main result of this section. Theorem 2.6.Assume H ∈ (0, 1) and K ∈ (0, 1) and let δ > 0 the constant appearing in Lemma 2.4.For any integer p ≥ 2 there exists a constant C p > 0 such that, for any t ≥ 0, any h ∈ (0, δ), all x, y ∈ R, and any 0 .
is sufficient for the local time L(t, u) of X to exist on [0, 1] a.s. and to be square integrable as a function of u.For any [a, b] ⊂ [0, ∞), and forI = [a , b ] ⊂ [a, b] such that |b − a | < δ, according to(5), we have,I I E[S H,K (t) − S H,K (s)]2 −1/2 ds dt < C I I |t − s| −HK ds dt.The last integral is finite because 0 < HK < 1.Then S H,K possesses, on any interval I ⊂ [a, b] with length |I| < δ, a local time which is square integrable as function of u.Finally, since [a, b] is a finite interval, we can obtain the local time on [a, b] by a patch-up procedure, i.e. we partition [a, b]