The Local Time of the Fractional Ornstein-Uhlenbeck Process

and Applied Analysis 3 means that a small increment of the process X is not almost relatively predictable based on a finite number of observations from the immediate past. It follows from Berman [22, Lemma 2.3] that (14) is equivalent to the following property which says that X has locally approximately independent increments: for any positive integer n ≥ 2, there exist positive constants C n and δ (both may depend on n) such that Var( n ∑ j=1 u j [X (t j ) − X (t j−1 )])


Introduction
The Brownian motion and the Ornstein-Uhlenbeck process are the two most well-studied and widely applied stochastic processes.The Einstein-Smoluchowski theory may be seen as an idealized Ornstein-Uhlenbeck theory, and predictions of either cannot be distinguished by the experiment.However, if the Brownian particle is under the influence of an external force, the Einstein-Smoluchowski theory breaks down, while the Ornstein-Uhlenbeck theory remains successful.It is well known that a diffusion process  = (  ) ≥0 starting from  ∈ R is called Ornstein-Uhlenbeck process with coefficients V > 0 if its infinitesimal generator is The Ornstein-Uhlenbeck process (see, e.g., Revuz and Yor [1]) has a remarkable history in physics.It is introduced to model the velocity of the particle diffusion process, and later it has been heavily used in finance, and thus in econophysics.It can be constructed as the unique strong solution of Itô stochastic differential equation where  is a standard Brownian motion starting at 0. Recently, as an extension of Brownian motion, fractional Brownian motion has become an object of intense study, due to its interesting properties and its applications in various scientific areas including condensed matter physics, biological physics, telecommunications, turbulence, image processing, finance, and econophysics (see, e.g., Gouyet [2], Nualart [3], Biagini et al. [4], Mishura [5], Willinger et al. [6], and references therein).Recall that fractional Brownian motion   with Hurst index  ∈ (0, 1) is a central Gaussian process with   0 = 0 and the covariance function for all ,  ⩾ 0. This process was first introduced by Kolmogorov and studied by Mandelbrot and van Ness [7], where a stochastic integral representation in terms of a standard Brownian motion was established.For  = 1/2,   coincides with the standard Brownian motion .  is neither a semimartingale nor a Markov process unless  = 1/2, and so many of the powerful techniques from stochastic analysis are not available when dealing with   .It has self-similar, longrange dependence, Hölder paths, and it has stationary increments.These properties make   an interesting tool for many applications.
On the other hand, extensions of the classical Ornstein-Uhlenbeck process have been suggested mainly on demand of applications.The fractional Ornstein-Uhlenbeck process is an extension of the Ornstein-Uhlenbeck process, where fractional Brownian motion is used as integrator Then (4) has a unique solution    = {   , 0 ≤  ≤ }, which can be expressed as and the solution is called the fractional Ornstein-Uhlenbeck process.More work for the process can be found in Cheridito et al. [8], Lim and Muniandy [9], Metzler and Klafter [10], and Yan et al. [11,12].Clearly, when  = 1/2, the fractional Ornstein-Uhlenbeck process is the classical Ornstein-Uhlenbeck process  with parameter V starting at  ∈ R.An advantage of using fractional Ornstein-Uhlenbeck process is to realize stationary long range dependent processes.The intuitive idea of local time (, ) for a stochastic process  is that (, ) measures the amount of time  spends at the level  during the interval [0, ].Moreover, since the work of Varadhan [13], the local time of stochastic processes has become an important subject.Therefore, it seems interesting to study the local time of fractional Ornstein-Uhlenbeck process, a rather special class of Gaussian processes.
In this paper, we focus our attention on the Hölder regularity of the local time of fractional Ornstein-Uhlenbeck process.
The rest of this paper is organized as follows.Section 2 contains a brief review on the local times of Gaussian processes and the approach of chaos expansion of the Gaussian process.In Section 3, we give Hölder regularity of the local time.In Section 4, as a related problem, we study the so-called collision local time of two independent fractional Ornstein-

Local Times and Local Nondeterminism.
We recall briefly the definition of local time.For a comprehensive survey on local times of both random and nonrandom vector fields, we refer to Alder [14], Geman and Horowitz [15], and Xiao [16][17][18].Let () be any Borel function on R with values in R. For any Borel set  ⊂ R, the occupation measure of  is defined by This fact has been used by many authors to study fractal properties of level sets, inverse image, and multiple times of stochastic processes.For example, Xiao [16] and Hu [19] have studied the Hausdorff dimension, and exact Hausdorff and packing measure of the level sets of iterated Brownian motion, respectively.
For a fixed sample function at fixed , the Fourier transform on  of (, ) is the function Using the density of occupation formula we have We can express the local times (, ) as the inverse Fourier transform of (, ), namely, It follows from (10) that for any ,  ∈ R, ,  +  ∈ [0, ] and any integer  ≥ 2, we have (see, e.g., Boufoussi et al. [20,21]) and for every even integer  ≥ 2, ( ( + ℎ, ) −  (, ) −  ( + ℎ, ) +  (, )) The concept of local nondeterminism was first introduced by Berman [22] to unify and extend his methods for studying local times of real-valued Gaussian processes.Let  = {(),  ∈ R + } be a real-valued, separable Gaussian process with mean 0 and let  ⊂ R + be an open interval.Assume that [() 2 ] > 0 for all  ∈  and there exists  > 0 such that for ,  ∈  with 0 < | − | < .Recall from Berman [22] that  is called locally nondeterministic on  if for every integer  ≥ 2, where   is the relative prediction error as follows: and the infimum in ( 14) is taken over all ordered points Abstract and Applied Analysis 3 means that a small increment of the process  is not almost relatively predictable based on a finite number of observations from the immediate past.It follows from Berman [22,Lemma 2.3] that ( 14) is equivalent to the following property which says that  has locally approximately independent increments: for any positive integer  ≥ 2, there exist positive constants   and  (both may depend on ) such that for all ordered points We refer to Nolan [23,Theorem 2.6] for a proof of the above equivalence in a much more general setting.
For simplicity throughout this paper we let   stand for a positive constant depending only on the subscripts and its value may be different in different appearances, and this assumption is also adaptable to ,   .

Local Time of Fractional Ornstein-Uhlenbeck Process
In this section, we offer the Hölder regularity of the local time of fractional Ornstein-Uhlenbeck process.
On the other hand, if we take (ℎ) = 1/ log log(1/ℎ) and consider ℎ  of the form 2 − , then inequation (42) implies for large .So, following that Borel-Cantelli lemma and monotonicity arguments, we have This completes the proof of inequation (28).we can obtain inequation (29) in the similar manner.

Existence and Smoothness of Collision Local Time
In this section we will study the so-called collision local time of two independent fractional Ornstein-Uhlenbeck    = { It is defined formally by the following expression: where  0 is the Dirac delta function.It is a measure of the amount of time for which the trajectories of the two processes,   1

𝑡 and 𝑋
2  , collide on the time interval [0, ].The collision local time for fractional Brownian motion has been studied by Jiang and Wang [27].We shall show that the random variable ℓ  exists in  2 .We approximate the Dirac delta function by the heat kernel For  > 0 we define and a natural question to study is that of the behavior of ℓ , as  tends to zero.
For the increments of collision local time we have the following.
Proof.For any 0 ≤ ,  ≤  we denote Then the property of local nondeterminism (see Theorem 3.1 in [11]) yields for a constant  > 0. It follows from (48) that for 0 ≤  ≤  ≤  (60) This completes the proof.