The Local Time of the Fractional Ornstein-Uhlenbeck Process

Abstract

We investigate the Hölder regularity of the local time of the fractional Ornstein-Uhlenbeck process 𝑋^𝐻={𝑋^𝐻_𝑡,𝑡≥0}. As a related problem, we study the collision local time of two independent fractional Ornstein-Uhlenbeck 𝑋^𝐻𝑖={𝑋^𝐻𝑖_𝑡,𝑡≥0}, 𝑖=1,2 with respective indices 𝐻₁,𝐻₂∈(0,1).

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 𝑋=(𝑋_𝑡)_𝑡≥0 starting from 𝑥∈ℝ is called Ornstein-Uhlenbeck process with coefficients 𝑣>0 if its infinitesimal generator is 1𝐿=2𝑣²𝑑²𝑑𝑥²𝑑−𝑥.𝑑𝑥 (1)

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 𝑑𝑋_𝑡=−𝑋_𝑡𝑑𝑡+𝑣𝑑𝐵_𝑡,𝑋₀=𝑥, (2) where 𝐵 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 𝐵^𝐻 with Hurst index 𝐻∈(0,1) is a central Gaussian process with 𝐵^𝐻₀=0 and the covariance function 𝐸𝐵^𝐻_𝑡𝐵^𝐻_𝑠=12𝑡^2𝐻+𝑠^2𝐻−|𝑡−𝑠|^2𝐻 (3) for all 𝑡,𝑠⩾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 𝐻=1/2, 𝐵^𝐻 coincides with the standard Brownian motion 𝐵. 𝐵^𝐻 is neither a semimartingale nor a Markov process unless 𝐻=1/2, and so many of the powerful techniques from stochastic analysis are not available when dealing with 𝐵^𝐻. It has self-similar, long-range dependence, Hölder paths, and it has stationary increments. These properties make 𝐵^𝐻 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 𝑑𝑋_𝑡=−𝑋_𝑡𝑑𝑡+𝑣𝑑𝐵^𝐻_𝑡,𝑋₀=𝑥. (4)

Then (4) has a unique solution 𝑋^𝐻_𝑡={𝑋^𝐻_𝑡,0≤𝑡≤𝑇}, which can be expressed as 𝑋^𝐻_𝑡=𝑒^−𝑡𝑥+𝑣^𝑡₀𝑒^𝑠𝑑𝐵^𝐻_𝑠, (5) 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 𝐻=1/2, the fractional Ornstein-Uhlenbeck process is the classical Ornstein-Uhlenbeck process 𝑋 with parameter 𝑣 starting at 𝑥∈ℝ. An advantage of using fractional Ornstein-Uhlenbeck process is to realize stationary long range dependent processes.

The intuitive idea of local time 𝐿(𝑡,𝑥) for a stochastic process 𝑋 is that 𝐿(𝑡,𝑥) measures the amount of time 𝑋 spends at the level 𝑥 during the interval [0,𝑡]. 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-Uhlenbeck 𝑋^𝐻𝑖={𝑋^𝐻𝑖_𝑡,𝑡≥0}, 𝑖=1,2 with respective indices 𝐻₁,𝐻₂∈(0,1).

2. Preliminaries

2.1. 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 [16–18]. Let 𝑋(𝑡) be any Borel function on ℝ with values in ℝ. For any Borel set 𝐵⊂ℝ, the occupation measure of 𝑋 is defined by 𝜇_𝐵(𝐴)=𝜆₁{𝑡∈𝐵,𝑋(𝑡)∈𝐴}, (6) for all Borel set 𝐴⊆ℝ, where 𝜆₁ is the one-dimensional Lebesgue measure. If 𝜇_𝐵 is absolutely continuous with respect to the Lebesgue measure 𝜆₁ on ℝ, we say that 𝑋(𝑡) has a local time on 𝐵 and define its local time 𝐿(𝐵,𝑥) to be the Radon-Nikodym derivative of 𝜇_𝐵. If 𝐵=[0,𝑡], we simply write 𝐿(𝐵,𝑥) as 𝐿(𝑡,𝑥). If 𝐼=[0,𝑇] and 𝐿(𝑡,𝑥) is continuous as a function of (𝑡,𝑥)∈𝐼×ℝ, then we say that 𝑋 has a jointly continuous local time on 𝐼. In this latter case, the set function 𝐿(⋅,𝑥) can be extended to be a finite Borel measure on the level set (see Adler [14, Theorem  8.6.1]) 𝑋_𝐼⁻¹(𝑥)={𝑡∈𝐼∶𝑋(𝑡)=𝑥}. (7)

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 𝑡, the Fourier transform on 𝑥 of 𝐿(𝑡,𝑥) is the function 𝑓(𝑡,𝑢)=_ℝ𝑒^𝑖𝑢𝑥𝐿(𝑡,𝑥)𝑑𝑥. (8)

Using the density of occupation formula we have 𝑓(𝑡,𝑢)=^𝑡₀𝑒^{𝑖𝑢𝑋(𝑠)}𝑑𝑠. (9)

We can express the local times 𝐿(𝑡,𝑥) as the inverse Fourier transform of 𝑓(𝑡,𝑢), namely, 𝐿()=1𝑡,𝑥2𝜋^+∞_−∞^𝑡₀𝑒^{𝑖𝑢(𝑋(𝑠)−𝑥)}𝑑𝑠𝑑𝑢. (10)

It follows from (10) that for any 𝑥,𝑦∈ℝ, 𝑡,𝑡+𝜔∈[0,𝑇] and any integer 𝑛≥2, we have (see, e.g., Boufoussi et al. [20, 21]) 𝐸(𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥))^𝑛=1(2𝜋)^𝑛_{[𝑡,𝑡+𝜔]}^𝑛_ℝ^𝑛𝑒^{−𝑖𝑥∑𝑛𝑗=1𝑢𝑗}𝐸𝑒^{𝑖∑𝑛𝑗=1𝑢𝑗𝑋(𝑠𝑗)}×^𝑛_𝑗=1𝑑𝑢_𝑗^𝑛_𝑗=1𝑑𝑠_𝑗, (11) and for every even integer 𝑛≥2, 𝐸𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)−𝐿𝑡+ℎ,𝑦+𝐿𝑡,𝑦^𝑛=1(2𝜋)^𝑛_{[𝑡,𝑡+𝜔]}^𝑛_ℝ^𝑛𝑛_𝑗=1𝑒^{−𝑖𝑥𝑢𝑗}−𝑒^{−𝑖𝑦𝑢𝑗}×𝐸𝑒^{𝑖∑𝑛𝑗=1𝑢𝑗𝑋(𝑠𝑗)𝑛}_𝑗=1𝑑𝑢_𝑗^𝑛_𝑗=1𝑑𝑠_𝑗. (12)

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 𝑋={𝑋(𝑡),𝑡∈ℝ₊} be a real-valued, separable Gaussian process with mean 0 and let 𝑇⊂ℝ₊ be an open interval. Assume that 𝐸[𝑋(𝑡)²]>0 for all 𝑡∈𝑇 and there exists 𝛿>0 such that 𝐸(𝑋(𝑠)−𝑋(𝑡))²>0, (13) for 𝑠,𝑡∈𝑇 with 0<|𝑠−𝑡|<𝛿.

Recall from Berman [22] that 𝑋 is called locally nondeterministic on 𝑇 if for every integer 𝑛≥2, lim_𝜖→0inf_{𝑡𝑛−𝑡1≤𝜖}𝑉_𝑛>0, (14) where 𝑉_𝑛 is the relative prediction error as follows: 𝑉_𝑛=Var𝑋𝑡_𝑛−𝑋𝑡_𝑛−1∣𝑋𝑡₁,…,𝑋𝑡_𝑛−1Var𝑋𝑡_𝑛−𝑋𝑡_𝑛−1, (15)

and the infimum in (14) is taken over all ordered points 𝑡₁<𝑡₂<⋯<𝑡_𝑛 in 𝑇 with 𝑡_𝑛−𝑡₁≤𝜖. Roughly speaking, (14) 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=1𝑢_𝑗𝑋𝑡_𝑗−𝑋𝑡_𝑗−1≥𝐶_𝑛^𝑛_𝑗=1𝑢²_𝑗Var𝑋𝑡_𝑗−𝑋𝑡_𝑗−1, (16) for all ordered points 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑛 in 𝑇 with 𝑡_𝑛−𝑡₁<𝛿 and all 𝑢_𝑗∈ℝ (1≤𝑗≤𝑛). 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 𝐶_𝑛 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 𝐶, 𝐶_𝐻.

2.2. Chaos Expansion

Let Ω be the space of continuous ℝ¹-valued functions 𝜔 on [0,𝑇]. Then Ω is a Banach space with respect to the supreme norm. Let ℱ be the 𝜎-algebra on Ω. Let 𝑃 be the probability measure on the measurable space (Ω,ℱ). Let 𝔼 denote the expectation on this probability space. The set of all square integrable functionals is denoted by 𝐿²(Ω,𝑃), that is, 𝐸𝐹²=_Ω|𝐹(𝜔)|²𝑃(𝑑𝜔)<∞. (17)

We can introduce the chaos expansion, which is an orthogonal decomposition of 𝐿²(Ω,𝑃). We refer to Hu [24], Nualart [3], and the references therein for more details. Let 𝑋∶={𝑋_𝑡,𝑡∈[0,𝑇]} be a Gaussian process defined on the probality space (Ω,ℱ,𝑃). If 𝑝_𝑛(𝑥) is a polynomial of degree 𝑛 in 𝑥, then we call 𝑝_𝑛(𝑋_𝑡) a polynomial function of 𝑋 with 𝑡∈[0,𝑇]. Let 𝒫_𝑛 be the completion with respect to the 𝐿²(Ω,𝑃) norm of the set {𝑝_𝑚(𝑋_𝑡)∶0≤𝑚≤𝑛,𝑡∈[0,𝑇]}. Clearly, 𝒫_𝑛 is a subspace of 𝐿²(Ω,𝑃). If 𝒞_𝑛 denotes the orthogonal complement of 𝒫_𝑛−1 in 𝒫_𝑛, then 𝐿²(Ω,𝑃) is actually the direct sum of 𝒞_𝑛, that is, 𝐿²(Ω,𝑃)=^∞_𝑛=0𝒞_𝑛. (18)

Namely, for any functional 𝐹∈𝐿²(Ω,𝑃), there are 𝐹_𝑛 in 𝒞_𝑛, 𝑛=0,1,2,…, such that 𝐹=^∞_𝑛=0𝐹_𝑛. (19)

The decomposition equation (19) is called the chaos expansion of 𝐹, and 𝐹_𝑛 is called the 𝑛th chaos of 𝐹. Clearly, we have 𝐸|𝐹|²=^∞_𝑛=0𝐸𝐹_𝑛². (20) Recall that Meyer-Watanabe test function space 𝒰 (see Watanabe [25]) is defined as 𝒰∶=𝐹∈𝐿²(Ω,𝑃)∶𝐹=^∞_𝑛=0𝐹_𝑛,^∞_𝑛=0𝑛𝐸𝐹_𝑛²<∞, (21) and 𝐹∈𝐿²(Ω,𝑃) is said to be smooth if 𝐹∈𝒰.

Now, for 𝐹∈𝐿²(Ω,𝑃), we define an operator Γ_𝑢 with 𝑢∈[0,1] by Γ_𝑢𝐹∶=^∞_𝑛=0𝑢^𝑛𝐹_𝑛. (22)

Set Θ(𝑢)∶=Γ_√𝑢𝐹. Then Θ(1)=𝐹. Define Φ_Θ(𝑢)∶=(𝑑/𝑑𝑢)(‖Θ(𝑢)‖²), where ‖𝐹‖²∶=𝐸(|𝐹|²) for 𝐹∈𝐿²(Ω,𝑃). We have Φ_Θ(𝑢)=^∞_𝑛=1𝑛𝑢^𝑛−1𝐸𝐹_𝑛². (23)

Note that ‖Θ(𝑢)‖²=𝐸(|Θ(𝑢)|²)=∑^∞_𝑛=0𝐸(𝑢^𝑛|𝐹_𝑛|²).

Proposition 1. Let 𝐹∈𝐿²(Ω,𝑃). Then 𝐹∈𝒰 if and only if Φ_Θ(1)<∞.

Consider two independent fractional Ornstein-Uhlenbeck 𝑋^𝐻𝑖={𝑋^𝐻𝑖_𝑡,𝑡≥0}, 𝑖=1,2, with respective indices 𝐻_𝑖∈(0,1). Let 𝐻_𝑛(𝑥) and 𝑥∈ℝ be the Hermite polynomials of degree 𝑛. That is, 𝐻_𝑛(𝑥)=(−1)^𝑛1𝑒𝑛!^𝑥2/2𝜕^𝑛𝜕𝑥^𝑛𝑒^−𝑥2/2. (24)

Then, 𝑒^{𝑡𝑥−𝑡2/2}=^∞_𝑛=0𝑡^𝑛𝐻_𝑛(𝑥), (25) for all 𝑡∈𝒞 and 𝑥∈ℝ, this implies that exp𝑖𝑢𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+12𝑢²𝜉²Var𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡=^∞_𝑛=0(𝑖𝑢)^𝑛𝜎^𝑛(𝑡,𝜉)𝐻_𝑛𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝜎(),𝑡,𝜉 (26) where 𝑖=√−1 and 𝜎(𝑡,𝜉)=Var(𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡)𝜉² for 𝜉∈ℝ. Because of the orthogonality of {𝐻_𝑛(𝑥),𝑥∈ℝ}_𝑛∈ℤ+, we will get from (19) that (𝑖𝑢)^𝑛𝜎^𝑛(𝑡,𝜉)𝐻_𝑛𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝜎(𝑡,𝜉) (27) is the 𝑛th chaos of exp(𝑖𝑢𝜉(𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡)+(1/2)𝑢²𝜉²Var(𝑋^𝐻1_𝑡−𝐵^𝐻2_𝑡)) for all 𝑡≥0.

3. Local Time of Fractional Ornstein-Uhlenbeck Process

In this section, we offer the Hölder regularity of the local time of fractional Ornstein-Uhlenbeck process.

Theorem 2. Let {𝑋^𝐻_𝑡,𝑡≥0} be the fractional Ornstein-Uhlenbeck process. Then, for every 𝑡∈ℝ⁺ and any 𝑥∈ℝ, there exist positive and finite constants 𝐶₁ and 𝐶₂ such that limsup_ℎ→0sup_𝑥𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)ℎ^1−𝐻loglogℎ^−1𝐻≤𝐶₁𝑎.𝑠.limsup_ℎ→0sup_𝑥,𝑡𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)ℎ^1−𝐻logℎ^−1𝐻≤𝐶₂𝑎.𝑠. (28)(29)

Proof. Let 𝑡≥0 be a fixed point. Following the Fourier analytic approach of Berman [26], we have 𝐸[𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)]^𝑛=1(2𝜋)^𝑛×_{[]𝑡,𝑡+ℎ𝑛ℝ𝑛}𝐸exp𝑖^𝑛_𝑗=1𝑢_𝑗𝑋^𝐻_𝑠𝑗−𝑋^𝐻_𝑡×^𝑛_𝑗=1𝑑𝑢_𝑗^𝑛_𝑗=1𝑑𝑠_𝑗. (30)
Let Δ𝑋^𝐻_𝑠=𝑋^𝐻_𝑠−𝑋^𝐻_𝑡, 𝑠≥0, and denote by 𝑅(𝑠₁,𝑠₂,…,𝑠_𝑛) the covariance matrix of (Δ𝑋^𝐻_𝑠1,…,Δ𝑋^𝐻_𝑠𝑛) for different 𝑠₁,…,𝑠_𝑛, then we have det𝑅𝑠₁,𝑠₂,…,𝑠_𝑛=VarΔ𝑋^𝐻_𝑠1VarΔ𝑋^𝐻_𝑠2∣Δ𝑋^𝐻_𝑠1⋯VarΔ𝑋^𝐻_𝑠𝑛∣Δ𝑋^𝐻_𝑠1,…,Δ𝑋^𝐻_𝑠𝑛−1. (31)
By Yan et al. [11], one can write the fractional Ornstein-Uhlenbeck process starting from zero as 𝑋^𝐻_𝑡=𝑣^𝑡₀𝐹(𝑡,𝑢)𝑑𝐵_𝑢,0≤𝑡≤𝑇, (32)
where 𝐵 is a standard Brownian motion with 𝐵₀=0, and for 0<𝑢<𝑡𝐹()=1𝑡,𝑢𝐻−2𝜅_𝐻𝑒^−𝑡𝑢^1/2−𝐻×^𝑡_𝑢𝑠^𝐻−1/2(𝑠−𝑢)^𝐻−3/2𝑒^𝑠𝑑𝑠, (33)
with 1/2<𝐻<1, 𝜅_𝐻=(2𝐻Γ((3/2)−𝐻)/Γ(𝐻+(1/2))Γ(2−2𝐻))^1/2, and 𝐹(𝑡,𝑢)=𝜅_𝐻𝑢^1/2−𝐻×−𝑒^−𝑡𝑡_𝑢(𝑠−𝑢)^𝐻−1/2𝑠^𝐻−1/2𝑒^𝑠𝑑𝑠+𝑡^𝐻−1/2(𝑡−𝑢)^𝐻−1/2+2𝑒1−2𝐻^−𝑡×^𝑡_𝑢(𝑠−𝑢)^𝐻−1/2𝑠^𝐻−3/2𝑒^𝑠𝑑𝑠, (34)
with 0<𝐻<1/2.
For any 𝑟,𝑠∈[𝑡,𝑡+ℎ] such that 𝑟<𝑠, we have VarΔ𝑋^𝐻_𝑠∣Δ𝑋^𝐻_𝑢,𝑢≤𝑟≥VarΔ𝑋^𝐻_𝑠−Δ𝑋^𝐻_𝑟∣𝐵_𝑢𝑋,𝑢≤𝑟=Var^𝐻_𝑠−𝑋^𝐻_𝑟∣𝐵_𝑢𝑋,𝑢≤𝑟=Var^𝐻_𝑠∣𝐵_𝑢,𝑢≤𝑟, (35)
where the last equality follows from the fact that 𝑋^𝐻_𝑟 is measurable with respect to 𝜎(𝐵_𝑢,𝑢≤𝑟). Moreover, we can write 𝑋^𝐻_𝑠=𝑣^𝑠₀𝐹(𝑠,𝑢)𝑑𝐵_𝑢=𝑣^𝑟₀𝐹(𝑠,𝑢)𝑑𝐵_𝑢+𝑣^𝑠_𝑟𝐹(𝑠,𝑢)𝑑𝐵_𝑢. (36)
Hence, by using the measurability of ∫^𝑟₀𝐹(𝑠,𝑢)𝑑𝐵_𝑢 with respect to 𝜎(𝐵_𝑢,𝑢≤𝑠), we have Var𝑋^𝐻_𝑠∣𝐵_𝑢,𝑢≤𝑟=Var𝑣^𝑠_𝑟𝐹(𝑠,𝑢)𝑑𝐵_𝑢∣𝐵_𝑢𝑣,𝑢≤𝑟=Var^𝑠_𝑟𝐹(𝑠,𝑢)𝑑𝐵_𝑢≥𝐶_𝐻(𝑠−𝑟)^2𝐻, (37)
where, to obtain the second equality, we have used the fact that ∫^𝑠_𝑟𝐹(𝑠,𝑢)𝑑𝐵_𝑢 is independent of 𝜎(𝐵_𝑢,𝑢≤𝑠) (by the independence of the increments of the Brownian motion). Combining (31), inequation (35), and inequation (37), we have det𝑅𝑠₁,𝑠₂,…,𝑠_𝑛≥𝐶_𝐻^𝑛_𝑗=1𝑠_𝑗−𝑠_𝑗−1^2𝐻>0, (38)
where 𝑠₀=0. Hence, the change of variable 𝑉=𝑅^1/2𝑈, 𝑈=(𝑢₁,…,𝑢_𝑛) implies that _ℝ𝑛𝐸exp𝑖^𝑛_𝑗=1𝑢_𝑗𝑋^𝐻_𝑠𝑗−𝑋^𝐻_𝑡^𝑛_𝑗=1𝑑𝑢_𝑗=_ℝ𝑛𝐸exp𝑖^𝑛_𝑗=1𝑢_𝑗Δ𝑋^𝐻_𝑠𝑗^𝑛_𝑗=1𝑑𝑢_𝑗=(2𝜋)^𝑛/2det𝑅𝑠₁,𝑠₂,…,𝑠_𝑛^1/2. (39)
Hence, 𝐸[𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)]^𝑛=𝑛!(2𝜋)^𝑛/2×_{𝑡<𝑠1<⋯<𝑠𝑛<𝑡+ℎ}1det𝑅𝑠₁,𝑠₂,…,𝑠_𝑛^1/2𝑑𝑠₁⋯𝑑𝑠_𝑛≤𝐶_𝐻𝑛!(2𝜋)^{𝑛/2𝑡<𝑠}₁^<⋯<𝑠_𝑛^{𝑛<𝑡+ℎ}_𝑗=11𝑠_𝑗−𝑠_𝑗−1^𝐻𝑑𝑠₁⋯𝑑𝑠_𝑛≤𝐶_𝐻𝑛!(2𝜋)^𝑛/2ℎ^{𝑛(1−𝐻)}(Γ(1−𝐻))^𝑛Γ((.1+𝑛1−𝐻)) (40)
Following from Stirling’s formula, we have𝑛!/Γ(1+𝑛(1−𝐻))≤𝐴^𝑛𝑛!^𝐻, 𝑛≥2, for a suitable finite number 𝐴. So 𝐸𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)ℎ^1−𝐻𝑛≤𝐶^𝑛𝑛!^𝐻. (41)
Following, we first prove that for any 𝐾>0, there exists a positive and finite constant 𝐵>0, depending on 𝑡, such that for sufficiently small 𝑢𝑃𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)≥𝐵ℎ^1−𝐻𝑢^𝐻≤𝑒^{−(𝐾/𝑢)}. (42)
First consider 𝑢 of the form 𝑢=1/𝑛. By Chebyshev’s inequality and inequation (41), we have 𝑃𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)≥𝐵ℎ^1−𝐻𝑛^𝐻𝐿()()≤𝐸𝑡+ℎ,𝑥−𝐿𝑡,𝑥𝐵ℎ^1−𝐻𝑛^𝐻𝑛≤𝐶^𝑛𝐵^𝑛1𝑛^𝑛𝐻𝑛!^𝐻≤𝐶byStirling'sformula_𝑛𝐵^𝑛(2𝜋𝑛)^𝐻/2𝑒^−𝐻𝑛𝑛𝐶=explog𝐵+𝐻−𝐻2log𝑛+log2𝜋. (43)
Choose 𝐵>𝐶 and 𝑛₀ large such that for any 𝑛≥𝑛₀, to dominate (43) by 𝑒^−2𝐾𝑛. Moreover, for 𝑢 sufficiently small, there exists 𝑛≥𝑛₀ such that 𝑢_𝑛+1<𝑢<𝑢_𝑛 and since 𝑛≥1, 𝑛/(𝑚+1)≥1/2. This proves inequation (42).
On the other hand, if we take 𝑢(ℎ)=1/loglog(1/ℎ) and consider ℎ_𝑛 of the form 2^−𝑛, then inequation (42) implies 𝑃𝐿𝑡+ℎ_𝑛,𝑥−𝐿(𝑡,𝑥)≥𝐵ℎ_𝑛^1−𝐻1loglogℎ_𝑛^𝐻≤𝑛⁻², (44)
for large 𝑛. So, following that Borel-Cantelli lemma and monotonicity arguments, we have 𝐿(𝑡+ℎ,𝑥)−𝐿(𝑡,𝑥)ℎ^1−𝐻1≤𝐵loglogℎ_𝑛^𝐻a.s. (45) This completes the proof of inequation (28). we can obtain inequation (29) in the similar manner.

4. 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 𝑋^𝐻𝑖={𝑋^𝐻𝑖_𝑡,𝑡≥0}, 𝑖=1,2. It is defined formally by the following expression: ℓ_𝑇=^𝑇₀𝛿₀𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝑑𝑡, (46)where 𝛿₀ is the Dirac delta function. It is a measure of the amount of time for which the trajectories of the two processes, 𝑋^𝐻1_𝑡 and 𝑋^𝐻2_𝑡, collide on the time interval [0,𝑇]. The collision local time for fractional Brownian motion has been studied by Jiang and Wang [27]. We shall show that the random variable ℓ_𝑇 exists in 𝐿². We approximate the Dirac delta function by the heat kernel 𝑝_𝜀(𝑥)=1√𝑒2𝜋𝜀^{−𝑥2/2𝜀}≡12𝜋_ℝ𝑒^𝑖𝑥𝜉𝑒^{−𝜀(𝜉2/2)}𝑑𝜉. (47) For 𝜀>0 we define ℓ_𝜀,𝑇=^𝑇₀𝑝_𝜖𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡=1𝑑𝑡2𝜋^𝑇_0ℝ𝑒^{𝑖𝜉(𝑋𝐻1𝑡−𝑋𝐻2𝑡)}⋅𝑒^{−𝜀(𝜉2/2)}𝑑𝜉𝑑𝑡, (48)

and a natural question to study is that of the behavior of ℓ_𝜀,𝑇 as 𝜀 tends to zero.

Theorem 3. For 𝐻_𝑖∈(0,1), 𝑖=1,2. Then ℓ_𝜀,𝑇 converges in 𝐿²(Ω,ℱ,𝑃), as 𝜀↓0. Moreover, the limit is denoted by ℓ_𝑇, then ℓ_𝑇∈𝐿²(Ω,ℱ,𝑃).

Proof. First we claim that ℓ_𝜀,𝑇∈𝐿²(Ω,ℱ,𝑃) for every 𝜀>0. By (48) we have 𝐸ℓ²_𝜀,𝑇=14𝜋^2𝑇_0ℝ²𝐸𝑒^{𝑖𝜉(𝑋𝐻1𝑡−𝑋𝐻2𝑡)+𝑖𝜂(𝑋𝐻1𝑠−𝑋𝐻2𝑠)}×𝑒^{−((𝜀(𝜉2+𝜂2))/2)}=1𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡4𝜋^2𝑇_0ℝ²𝑒^{−(1/2)𝜎2}𝑒^{−((𝜀(𝜉2+𝜂2))/2)}𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡, (49) where 𝜎² denotes the variance of random variable 𝜉(𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡)+𝜂(𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠), that is, 𝜎²∶=Var𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+𝜂𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠. (50) According to the property of local nondeterminism (see Theorem  3.1 in [11]), we have 𝜎²=Var𝜉𝑋^𝐻1_𝑡−𝑋^𝐻1_𝑠−𝜉𝑋^𝐻2_𝑡−𝑋^𝐻2_𝑠+𝜉+𝜂𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠≥𝐶𝜉²(𝑡−𝑠)^2𝐻1+(𝑡−𝑠)^2𝐻2+𝜉+𝜂²𝑠^2𝐻1+𝑠^2𝐻2. (51) Thus, we have 14𝜋^2𝑇_0ℝ²𝑒^{−(1/2)𝜎2}𝑒^{−(𝜀(𝜉2+𝜂2))/2}≤𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡_{𝑇0𝑡0ℝ2}𝑒^{−(𝐶/2)[𝜉2((𝑡−𝑠)12𝐻+(𝑡−𝑠)22𝐻)+(𝜉+𝜂)2(𝑠12𝐻+𝑠22𝐻)]}𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡=𝐶^𝑇₀^𝑡₀(𝑡−𝑠)^2𝐻1+(𝑡−𝑠)^2𝐻2×𝑠^2𝐻1+𝑠^2𝐻2−1/2𝑑𝑠𝑑𝑡≤𝐶^𝑇₀^𝑡₀(𝑡−𝑠)^{−(1/2)(𝐻1+H2)}𝑠^{−(1/2)(𝐻1+H2)}𝑑𝑠𝑑𝑡<∞, (52)because of 𝐻_𝑖∈(0,1), which yields 𝐸ℓ²_𝜀,𝑇<∞, (53)
for all 𝜀∈(0,1].
Second, we claim that the sequence {ℓ_𝜀,𝑇,𝜀>0} is of Cauchy in 𝐿²(Ω,ℱ,𝑃). For any 𝜃,𝜀>0 we have 𝐸ℓ_𝜀,𝑇−ℓ_𝜃,𝑇²=14𝜋^2𝑇_0ℝ²𝐸𝑒^{𝑖𝜉(𝑋𝐻1𝑡−𝑋𝐻2𝑡)+𝑖𝜂(𝑋𝐻1𝑠−𝑋𝐻2𝑠)}⋅𝑒^{−(𝜀/2)𝜉2}−𝑒^{−(𝜃/2)𝜉2}×𝑒^{−(𝜀/2)𝜂2}−𝑒^{−(𝜃/2)𝜂2}≤1𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡4𝜋²sup_𝜉∈ℝ1−𝑒^{−((|𝜀−𝜃|2|𝜉|2)/2)2}×^𝑇_0ℝ2𝑒^{−(1/2)𝜎2}𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡. (54)
Thus, dominated convergence theorem yields 𝐸ℓ_𝜀,𝑇−ℓ_𝜃,𝑇²⟶0, (55)
as 𝜀→0 and 𝜃→0, which leads to ℓ_𝜀,𝑇 is a Cauchy sequence in 𝐿²(Ω,ℱ,𝑃). Consequently, lim_𝜀→0ℓ_𝜀,𝑇 exists in 𝐿²(Ω,ℱ,𝑃). This completes the proof.

For the increments of collision local time we have the following.

Theorem 4. Let 𝐻₁,𝐻₂∈(0,1) and 𝛽=min{𝐻₁,𝐻₂}. Then the collision local time ℓ_𝑇 satisfies the following estimate: 𝐸ℓ_𝑡−ℓ_𝑠²≤𝐶_𝐻1,𝐻2(𝑡−𝑠)^2−2𝛽, (56)
for all 𝑠,𝑡, 𝑠<𝑡.

Proof. For any 0≤𝑟, 𝑙≤𝑇 we denote 𝜎²_𝑟,𝑙∶=Var𝜉𝑋^𝐻1_𝑟−𝑋^𝐻2_𝑟+𝜂𝑋^𝐻1_𝑙−𝑋^𝐻2_𝑙. (57) Then the property of local nondeterminism (see Theorem  3.1 in [11]) yields 𝜎²_𝑟,𝑙≥𝐶𝜉²(𝑟−𝑙)^2𝐻1+(𝑟−𝑙)^2𝐻2+𝜉+𝜂²𝑙^2𝐻1+𝑙^2𝐻2 (58)
for a constant 𝐶>0. It follows from (48) that for 0≤𝑠≤𝑡≤𝑇𝐸ℓ_𝜀,𝑡−ℓ_𝜀,𝑠²=2(2𝜋)^{2𝑡𝑠𝑟𝑠}𝑑𝑟𝑑𝑙_ℝ2𝑒^{−(1/2)𝜎2𝑟,𝑙}𝑒^{−(𝜀/2)(𝜉2+𝜂2)}𝑑𝜉𝑑𝜂≤𝐶^𝑡_𝑠𝑑𝑟^𝑟_𝑠(𝑟−𝑙)^−𝛽𝑙^−𝛽()𝑑𝑙≤𝐶𝑡−𝑠^2−2𝛽. (59) Thus, Theorem 3 and Fatou’s lemma yield 𝐸ℓ_𝑡−ℓ_𝑠²=𝐸lim_𝜀→0ℓ_𝜀,𝑡−ℓ_𝜀,𝑠²≤liminf_𝜀→0𝐸ℓ_𝜀,𝑡−ℓ_𝜀,𝑠²≤𝐶(𝑡−𝑠)^2−2𝛽. (60) This completes the proof.

Let 𝜆_𝑡=Var(𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡) for 𝑡≥0 and 𝜌_𝑠,𝑡=𝐸𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠, (61)

for 𝑠,𝑡≥0.

Lemma 5 (An and Yan [28]). For any 𝑥∈[−1,1) we have ^∞_𝑛=1(2𝑛−1)!!()𝑥2𝑛−2!!^𝑛=𝑥(1−𝑥)^−3/2, (62)
where (2𝑛−2)!!=1⋅3⋅5⋯(2𝑛−1) and (2𝑛−1)!!=2⋅4⋅6⋯(2𝑛−2).

By Cauchy-Schwartz’s inequality, we have 𝜌²_𝑠,𝑡≤𝜆_s𝜆_𝑡. Hence, 𝜌²_𝑠,𝑡𝜆_𝑠𝜆_𝑡−𝜌²_𝑠,𝑡^3/2=𝜌²_𝑠,𝑡𝜆_𝑠𝜆_𝑡𝜌1−^2𝑠,𝑡𝜆_𝑠𝜆_𝑡^−3/21𝜆_𝑠𝜆_𝑡^1/2=^∞_𝑛=1(2𝑛−1)!!()𝜌2𝑛−2!!𝜆_𝑠𝜆^2𝑠,𝑡_𝑡^𝑛1𝜆_𝑠𝜆_𝑡^1/2, (63)

for all 𝑡,𝑠≥0 and 𝑠≠𝑡.

Below, we consider the smoothness of the collision local time. Our main object is to explain and prove the following theorem.

Theorem 6. Let ℓ_𝑇,𝑇≥0 be the collision local time process of two independent fractional Ornstein-Uhlenbeck 𝑋^𝐻𝑖={𝑋^𝐻𝑖_𝑡,𝑡≥0},𝑖=1,2, with respective indices 𝐻_𝑖∈(0,1). Then ℓ_𝑇 is smooth in the sense of the Meyer-Watanabe if and only if min𝐻₁,𝐻₂<13. (64)

Proof. By Yan et al. [11], we have 𝜆_𝑡𝜆_𝑠−𝜌²_𝑠,𝑡≍𝑠^2𝐻1+𝑠^2𝐻2𝑡^2𝐻1+𝑡^2𝐻2−12𝑡^2𝐻1+𝑠^2𝐻1−|𝑡−𝑠|^2𝐻1+𝑡^2𝐻2+𝑠^2𝐻2−|𝑡−𝑠|^2𝐻2≍𝑠^2𝐻1+𝑠^2𝐻2(𝑡−𝑠)^2𝐻1+(𝑡−𝑠)^2𝐻2, (65)
where the notation 𝐹≍𝐺 means that there are positive constants 𝑐₁ and 𝑐₂ so that 𝑐₁𝐺(𝑥)≤𝐹(𝑥)≤𝑐₂𝐺(𝑥), (66)
in the common domain of definition for 𝐹 and 𝐺. Hence, following Theorem 2 in An and Yan [28], we have ∬^𝑇₀(𝜌²_𝑠,𝑡/(𝜆_𝑠𝜆_𝑡−𝜌²_𝑠,𝑡)^3/2)𝑑𝑠𝑑𝑡<∞ if and only if min{𝐻₁,𝐻₂}<1/3. Therefore, in order to prove Theorem 6, it only needs to prove: for 𝑇≥0, ℓ_𝑇 is smooth in the sense of the Meyer-Watanabe if and only if ^𝑇₀𝜌²_𝑠,𝑡𝜆_𝑡𝜆_𝑠−𝜌²_𝑠,𝑡^−3/2𝑑𝑠𝑑𝑡<∞. (67)
In fact, for 𝜀>0, 𝑇≥0 we denote Θ_𝜀𝑢,𝑇,ℓ_𝜀,𝑇∶=𝐸Γ_√𝑢ℓ_𝜀,𝑇², (68)
and Θ(𝑢,𝑇,ℓ_𝑇)∶=𝐸(|Γ_√𝑢ℓ_𝑇|²). Thus, by Proposition 1 to prove that (67) holds if and only if Φ_Θ(1)<∞. Clearly, we have ℓ_𝜀,𝑇=^𝑇₀𝑝_𝜀𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡=1𝑑𝑡2𝜋^𝑇_0ℝ𝑒^{𝑖𝜉(𝑋𝐻1𝑡−𝑋𝐻2𝑡)}⋅𝑒^{−(1/2)𝜀𝜉2}=1𝑑𝜉𝑑𝑡2𝜋^𝑇_0ℝ𝑒^{−(1/2)𝜉2Var(𝑋𝐻1𝑡−𝑋𝐻2𝑡)}×𝑒^{−(1/2)𝜀𝜉2∞}_𝑛=0𝑖^𝑛𝜎^𝑛(𝑡,𝜉)𝐻_𝑛×𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝜎()≡𝑡,𝜉𝑑𝜉𝑑𝑡_∞𝑛=0𝐹_𝑛. (69) Notice that Φ_Θ𝜀(1)=^∞_𝑛=0𝑛𝐸𝐹_𝑛²=^∞_𝑛=0𝑛4𝜋²×𝐸^𝑇_0ℝ2−1exp2𝜀𝜉²+𝜂²×𝜎^𝑛(𝑡,𝜉)𝜎^𝑛−1𝑠,𝜂⋅exp2𝜉²Var𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+𝜂²Var𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠⋅𝐻_𝑛𝜉𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝜎(𝑡,𝜉)×𝐻_𝑛𝜂𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠𝜎=𝑠,𝜂𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡_∞𝑛=114𝜋²()!×2𝑛−1_𝑇0ℝ2𝜉𝜂^2𝑛×𝐸𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠^2𝑛⋅exp−𝜉²Var𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+𝜀+𝜂²Var𝑋^𝐻1_𝑠−𝑋^𝐻2_𝑠×1+𝜀2𝑑𝜉𝑑𝜂𝑑𝑠𝑑𝑡, (70)
for all 𝑇≥0, where we have used the following fact: For two random variables 𝑋, 𝑌 with joint Gaussian distribution such that 𝐸(𝑋)=𝐸(𝑌)=0 and 𝐸(𝑋²)=𝐸(𝑌²)=1 we have (see, for example, Nualart [3]) 𝐸𝐻_𝑛(𝑋)𝐻_𝑚(𝑌)=10,𝑚≠𝑛,[𝐸(𝑛!𝑋𝑌)]^𝑛,𝑚=𝑛. (71) We obtain Φ_Θ𝜀(1)=^∞_𝑛=1(Γ(𝑛+1/2))²2^2𝑛+14𝜋²(2𝑛−1)!×^𝑇₀𝜌^2𝑛_𝑠,𝑡𝜆_𝑠+𝜀𝜆_𝑡+𝜀^𝑛+(1/2)=𝑑𝑠𝑑𝑡_∞𝑛=11()2𝜋2𝑛−1!!(2𝑛−2)!!×^𝑇₀𝜌^2𝑛_𝑠,𝑡𝜆_𝑠+𝜀𝜆_𝑡+𝜀^𝑛+(1/2)=1𝑑𝑠𝑑𝑡2𝜋^𝑇₀𝜌²_𝑠,𝑡𝜆_𝑠+𝜀𝜆_𝑡+𝜀−𝜌²_𝑠,𝑡^3/2𝑑𝑠𝑑𝑡, (72) where we have used the following equality: _ℝ𝜉^2𝑛−𝜉exp²Var𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+𝜀2𝑑𝜉=2^𝑛+(1/2)Γ1𝑛+2Var𝑋^𝐻1_𝑡−𝑋^𝐻2_𝑡+𝜀^{−(𝑛+(1/2))}. (73) Hence, we have lim_𝜀→0Φ_Θ𝜀(1)=12𝜋^𝑇₀𝜌²_𝑠,𝑡𝜆_𝑠𝜆_𝑡−𝜌²_𝑠,𝑡^3/2𝑑𝑠𝑑𝑡, (74)
for all 𝑇≥0. This completes the proof.

Acknowledgments

The authors would like to thank anonymous earnest referee whose remarks and suggestions greatly improved the presentation of our paper. Guangjun Shen is partially supported by the National Natural Science Foundation of China (11271020), the Natural Science Foundation of Anhui Province (1208085MA11, 1308085QA14), and the Key Natural Science Foundation of Anhui Educational Committee (KJ2011A139), Dongjin Zhu is partially supported by the Key Natural Science Foundation of the Anhui Educational Committee (KJ2012ZD01, KJ2013A133) and the Philosophy and Social Science Planning Foundation of Anhui Province (AHSK11-12D128), and Yong Ren is partially supported by the National Natural Science Foundation of China (11371029).