On the local time of sub-fractional Brownian motion

S = {S t , t ≥ 0} be a sub-fractional Brownian motion with H ∈ (0, 1). We establish the existence, the joint continuity and the Hölder regularity of the local time L of S . We will also give Chung’s form of the law of iterated logarithm for S . This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].


Introduction
The intuitive idea of a local time L(t, x) for a process X is that L(t, x) measures the amount of time X spends at the level x during the interval [0, t].We are concerned in this paper with the existence and regularity of the local time of the sub-fractional Brownian motion (Sub-fBm).We will also give Chung's form of the law of iterated logarithm for S H . Sub-fractional Brownian motion S H = {S H t , t ≥ 0} is a centered Gaussian process with covariance function where H ∈ (0, 2).This process was introduced by Bojdecky et al [8] as an intermediate process between standard Brownian motion and fractional Brownian motion.Recall that fractional Brownian motion (fBm for short) B H = {B H t , t ≥ 0} is a centered Gaussian process with covariance function where H ∈ (0, 2).Note that both fBm and Sub-fBm are standard Brownian motion for H = 1.For H = 1, Sub-fBm preserves some of main properties of fBm, such as long-range dependence, but its increments are not stationary, they are more weakly correlated on non-overlapping intervals than fBm ones, and their covariance decays polynomially at a higher rate as the distance between the intervals tends to infinity.For a more detailed discussion of Sub-fBm and its properties we refer the reader to Bojdecky et al [8].Some properties of this process have also been studied in Tudor [21] and [22].
In [10] the authors obtain the following equality in law where 2 dW θ and standard Brownian motion W and fractional Brownian motion B H are independents.The centered Gaussian process Lei and Nualart [17] in order to obtain a decomposition of bifractional Brownian motion into the sum of a transformation of X H t and a fBm.We will establish our results by using an approach based on the concept of local nondeterminism (LND for simplicity), introduced by Berman [6] to unify and extend his earlier works on the local times of stationnaire Gaussian processes.The joint continuity as well as Hölder conditions in both the space and the (time) set variable of the local time of locally nondeterministic (LND) Gaussian process and fields have been studied by Berman [4] and [6], Pitt [20], Kôno [15], Geman and Horowitz [13], and recently by Csörgo, Lin and Shao [11] and [23].Recently, Boufoussi, Dozzi and Guerbaz [9] and Guerbaz [14] have studied respectively the local time of the multifractional Brownian motion (mBm) and the local time of the filtered white noises.Th multifractional Brownian motion extend the fBm in the sens that its Hurst parameter is not more constant, but a Hölder function of time.The paper is organized as follows.Section 2 contains a brief review on the local times of Gaussian processes and Berman's concept of local nondeterminism.In section 3 we prove the existence of a square integrable version of the local time, the joint continuity and Hölder regularity in time and in space.Chung's form of the law of iterated logarithm for Sub-fBm is obtained in section 4, which is applied to derive a lower bound for local moduli of continuity of local times of Sub-fBm.Will use C, C 1 , . . . to denote unspecified positive finite constants which may not necessary be the same at each occurrence.

Preliminaries
We recall some aspects of local times and we refer to the paper of Geman and Horowitz [13] for an insightful survey local times.Let X = {X(t), t ≥ 0} be a real valued separable random process with Borel sample functions.For any Borel set B of the real line, the occupation measure of X is defined as follows and λ is the Lebesgue measure on R + .If µ(A, .) is absolutely continuous with respect to the Lebesgue measure on R, we say that X has local times on A and define its local time, L(A, .), as the Radon-Nikodym derivative of µ(A, .).Here x is the so-called space variable, and A is the time variable.The existence of jointly continuous local time reveals information on the fluctuation of the sample paths of process itself [1, Chap 8].There are several approach for proving the joint continuity of the local times, one of them is the Fourier analytic method developed by Berman to extend his early works on the local times of stationary Gaussian processes.The main tool used in Berman's approach (see Berman [6]) is the local nondeterminism.We give a brief review of the concept of local nondeterminism, more informations on the subject can be found in [6].Let J be an open interval on t axis.Assume that {X(t), t ≥ 0} is a zero mean Gaussian process without singularities in any interval of length δ, for some δ > 0, and without fixed zeros; i.e. there exists δ > 0 such that To introduce the concept of local nondeterminism, Berman defined the relative conditioning error, where, for m ≥ 2, t 1 , . . ., t m are arbitrary points in J ordered according to their indices, i.e.
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 has proved, for Gaussian processes, that the local nondeterminism as characterized as follows.
Proposition 2.1.X is LND if and only if for every integer m ≥ 2, there exists positive constants C and δ (both may depend on m) such that ) The proof of this proposition is given in [6], Lemmas 2.1 and 8.1.

Local time of sub-fractional Brownian motion
The propose of this section is to present sufficient conditions for the existence of the local times of sub-fractional Brownian motion.Furthermore, using the local nondeterminism approach, we show that the local times have a jointly continuous version.

Square integrability
For the proof of Theorem 3.1, we need the following lemma.This result on the regularity of the increments of the Sub-fBm will be the key for the existence and the regularity of local times.

Lemma 3.2.
There exists δ > 0 and, for any integer m ≥ 1, there exists Proof.We use the decomposition of the Sub-fBm given by Ruiz de Chavez and Tudor [10] : where 2 dW θ and standard Brownian motion W and fractional Brownian motion B H are independents.
Making use of the theorem on finite increments for the function v → e −θv , for v ∈ (s, t), there exists α ∈ (s, t) such that where Since 0 < H < 1, we can choose δ small enough such that for all s, t ≥ 0 and |t − s| < δ we have Since S H is a centered Gaussian process then we obtain the result.
Proof of Theorem 3.1.Fix T > 0. It is well known (see Berman [4]) that, for a jointly measure zero-mean Gaussian process X = {X t , t ∈ [0, T ]} with bounded variance, the variance condition x) where a 0 = a and a n = b.

LND Property of Sub-fBm
In order to study joint continuity of local time we prove the LND of Sub-fBm.
Proof.It is sufficient to prove that the sub-fBm S H satisfies Proposition 2.1.
By using the elementary inequality According to Kôno et al. [16], the fBm B H is local nondeterministic on [0, T ], then by Proposition 2.1, there exists two constants δ m > 0 and C m > 0 such that for any Moreover, we have This last inequality imply that (3.6) becomes In addition we have Therefore it suffices now to choose and the theorem is proved.

Joint continuity and Hölder regularity
Let T > 0 and H([0, T ]) be the family of interval I ⊂ [0, T ] of length at most δ (the constant appearing in Lemma 3.2).In this paragraph we will apply some results of Berman on LND process to prove the joint continuity of local times of the Sub-fBm.The main result is the following.
The proof of Theorem 3.4 relies on the following upper bounds for the moments of the local times.

.13)
Proof.We will proof only (3.13), the proof of (3.12) is similar.It follows from (25.7) in Geman and Horowitz [13](see also Boufoussi et al. [9]) that for any x, y ∈ R, t, t + h ∈ [0, +∞[ and for every even integer m ≥ 2, Using the elementary inequality |1 − e iθ | ≤ 2 1−ξ |θ| ξ for all 0 < ξ < 1 and any θ ∈ R, we obtain where in order to apply the LND property of S H , we replaced the integration over the domain [t, t + h] by over the subset We deal now with the inner multiple integral over the u's.Change the variable of integration by mean of the transformation Then the linear combination in the exponent in (3.14) is transformed according to where t 0 = 0. Since S H is a Gaussian process, the characteristic function in (3.14) has the form Moreover, the last product is at most equal to a finite sum of 2 m−1 terms of the form m j=1 |x j | ξε j , where ε j = 0, 1 or 2 and m j=1 ε j = m.Let us write for simply σ . Combining the result of Proposition 2.1, (3.15) and (3.16), we get that the integral in (3.14) is dominated by the sum over all possible of (ε 1 , . . ., ε m ) ∈ {0, 1, 2} m of the following where C m is the constant given in Proposition 2.1.The change of variable x j = v j σ j converts the last integral to Let us denote Consequently According to Lemma 3.2, for h sufficient small, namely 0 < h < inf(δ, 1), we have It follows that the integral on the right hand side of (3.17) is bounded, up to a constant, by Since, (t j − t j−1 ) < 1, for all j ∈ {2, . . ., m}, we have Since by hypothesis ξ < 1 2H − 1 2 , the integral in (3.19) is finite.Moreover, by an elementary calculation( cf.Ehm [12]), for all m ≥ 1, h > 0 and b j < 1, , where s 0 = t.It follows that (3.19) is dominated by , Proof of Theorem 3.4.Since L(0, x) = 0 for all x ∈ R, hence if we replace t and t + h by 0 and t respectively in 3.13, we obtain The jointly continuity of the local time straightforward from (3.12), (3.13) and (3.20) and classical parameter Kolmogorov's theorem (c.f.Berman [5], Theorem 5.1).
The Hölder condition (i) of Theorem 3.1 follows of (3.13) and one parameter Kolmogorov's theorem (see also the proof Theorem 2 in Pitt [20]).We turn out to the proof of (ii).According to Theorem 3.1 in Berman [7], the inequalities (3.12), (3.13) and (3.20) imply that (ii) holds for any Letting ξ tends to zero, we obtain the desired result.
As a classical consequence, we have the following result on the Hausdorff dimension of the level set.We refer to Adler [1] and Baraka et al. [3] for definition and results for the fractional Brownian motion.
For the sake of simplicity, let Λ = sup s∈[t,t+δ] (X H (t) − X H (s)).By (4.4), we obtain It follows that Consequently, (4.4) becomes ) and u n = δ H+ξ n .Therefore, according to (4.5), we have It follows from the Borel-Cantelli lemma that there exists n 0 = n(ω) such that for all n ≥ n 0 , sup s∈ [ u) of X exists on [0, T ] almost surely and be square integrable as a function of u.For any [a, b] ⊂ [0, +∞[ and for I = [a , b ] ⊂ [a, b] such that |b − a | < δ, according to Lemma 3.2 we have, s| −H dsdt.The last integral is finite because 0 < H < 1.Then according to Geman and Horowitz [13, Theorem 22.1], the conclusion of the theorem holds for any interval I ⊂ [a, b] with length |I| < δ.Finally, since [a, b] is finite interval, we can obtain the local time on [a, b] by a standard patch-up procedure i.e. we partition [a, b] t,t+δn] |X H (t) − X H (s)| ≤ δ H+ξ n almost On the local time of sub-fractional Brownian motion surely.Furthermore, for δ n+1 ≤ δ ≤ δ n , we have almost surely Consequently (4.2) is proved.This completes the proof of the Theorem.Remark 4.2.The main interest of the previous proof is that it can be used to generalized many other LIL known for the fBm to the Sub-fBm.For example, we have the LIL given in Li and Shao[18, equation (7.5)], for the fBm to the Sub-fBm as followsThe Chung laws are known to be linked to the optimality of the moduli of continuity of local times of stochastic processes.More precisely:|S H (t) − S H (s)| we obtain the lemma.