# The Local Time of the Fractional Ornstein-Uhlenbeck Process

Authors: Guangjun Shen; Dongjin Zhu; Yong Ren; Xueping Ding
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the Hölder regularity of the local time of the fractional Ornstein-Uhlenbeck processXH={XtH,t≥0}. As a related problem, we study the collision local time of two independent fractional Ornstein-Uhlenbeck XHi={XtHi,t≥0}, i=1,2 with respective indices H1,H2∈(0,1).


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## Body

## 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 processX=(Xt)t≥0 starting from x∈ℝ is called Ornstein-Uhlenbeck process with coefficients v>0 if its infinitesimal generator is
(1)L=12v2d2dx2-xddx.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
(2)dXt=-Xtdt+vdBt,X0=x,
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 BH with Hurst index H∈(0,1) is a central Gaussian process with B0H=0 and the covariance function
(3)E[BtHBsH]=12[t2H+s2H-|t-s|2H]
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, BH coincides with the standard Brownian motion B. BH 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 BH. It has self-similar, long-range dependence, Hölder paths, and it has stationary increments. These properties make BH 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(4)dXt=-Xtdt+vdBtH,X0=x.Then (4) has a unique solution XtH={XtH,0≤t≤T}, which can be expressed as
(5)XtH=e-t(x+v∫0t‍esdBsH),
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 timeL(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. Section2 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 XHi={XtHi,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
(6)μB(A)=λ1{t∈B,X(t)∈A},
for all Borel set A⊆ℝ, where λ1 is the one-dimensional Lebesgue measure. If μB 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 μB. 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])
(7)XI-1(x)={t∈I:X(t)=x}.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 fixedt, the Fourier transform on x of L(t,x) is the function
(8)f(t,u)=∫ℝ‍eiuxL(t,x)dx.Using the density of occupation formula we have(9)f(t,u)=∫0t‍eiuX(s)ds.We can express the local timesL(t,x) as the inverse Fourier transform of f(t,u), namely,
(10)L(t,x)=12π∫-∞+∞‍(∫0t‍eiu(X(s)-x)ds)du.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])
(11)E(L(t+h,x)-L(t,x))n=1(2π)n∫[t,t+ω]n‍∫ℝn‍e-ix∑j=1nujE(ei∑j=1nujX(sj))×∏j=1n‍duj∏j=1n‍dsj,
and for every even integer n≥2,
(12)E(L(t+h,x)-L(t,x)-L(t+h,y)+L(t,y))n=1(2π)n∫[t,t+ω]n‍∫ℝn‍∏j=1n‍[e-ixuj-e-iyuj]×E(ei∑j=1nujX(sj))∏j=1n‍duj∏j=1n‍dsj.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
(13)E[(X(s)-X(t))2]>0,
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,
(14)limϵ→0inftn-t1≤ϵVn>0,
where Vn is the relative prediction error as follows:
(15)Vn=Var(X(tn)-X(tn-1)∣X(t1),…,X(tn-1))Var(X(tn)-X(tn-1)),and the infimum in (14) is taken over all ordered points t1<t2<⋯<tn in T with tn-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 Cn and δ (both may depend on n) such that
(16)Var(∑j=1n‍uj[X(tj)-X(tj-1)])≥Cn∑j=1n‍uj2Var[X(tj)-X(tj-1)],
for all ordered points 0=t0<t1<t2<⋯<tn in T with tn-t1<δ and all uj∈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 letCn 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, CH.


### 2.2.  Chaos Expansion

LetΩ 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,
(17)E(F2)=∫Ω‍|F(ω)|2P(dω)<∞.We can introduce the chaos expansion, which is an orthogonal decomposition ofL2(Ω,P). We refer to Hu [24], Nualart [3], and the references therein for more details. Let X:={Xt,t∈[0,T]} be a Gaussian process defined on the probality space (Ω,ℱ,P). If pn(x) is a polynomial of degree n in x, then we call pn(Xt) a polynomial function of X with t∈[0,T]. Let 𝒫n be the completion with respect to the L2(Ω,P) norm of the set {pm(Xt):0≤m≤n,t∈[0,T]}. Clearly, 𝒫n is a subspace of L2(Ω,P). If 𝒞n denotes the orthogonal complement of 𝒫n-1 in 𝒫n, then L2(Ω,P) is actually the direct sum of 𝒞n, that is,
(18)L2(Ω,P)=⨁n=0∞𝒞n.Namely, for any functionalF∈L2(Ω,P), there are Fn in 𝒞n, n=0,1,2,…, such that
(19)F=∑n=0∞‍Fn.The decomposition equation (19) is called the chaos expansionof F, and Fn is called the nth chaos of F. Clearly, we have
(20)E(|F|2)=∑n=0∞‍E(|Fn|2).
Recall that Meyer-Watanabe test function space 𝒰 (see Watanabe [25]) is defined as
(21)𝒰:={F∈L2(Ω,P):F=∑n=0∞‍Fn,∑n=0∞‍nE(|Fn|2)<∞},
and F∈L2(Ω,P) is said to be smooth if F∈𝒰.Now, forF∈L2(Ω,P), we define an operator Γu with u∈[0,1] by
(22)ΓuF:=∑n=0∞‍unFn.SetΘ(u):=ΓuF. Then Θ(1)=F. Define ΦΘ(u):=(d/du)(∥Θ(u)∥2), where ∥F∥2:=E(|F|2) for F∈L2(Ω,P). We have
(23)ΦΘ(u)=∑n=1∞‍nun-1E(|Fn|2).Note that∥Θ(u)∥2=E(|Θ(u)|2)=∑n=0∞‍E(un|Fn|2).Proposition 1.
LetF∈L2(Ω,P). Then F∈𝒰 if and only if ΦΘ(1)<∞.Consider two independent fractional Ornstein-UhlenbeckXHi={XtHi,t≥0}, i=1,2, with respective indices Hi∈(0,1). Let Hn(x) and x∈ℝ be the Hermite polynomials of degree n. That is,
(24)Hn(x)=(-1)n1n!ex2/2∂n∂xne-x2/2.Then,(25)etx-t2/2=∑n=0∞‍tnHn(x),
for all t∈𝒞 and x∈ℝ, this implies that
(26)exp(iuξ(XtH1-XtH2)+12u2ξ2Var(XtH1-XtH2))=∑n=0∞‍(iu)nσn(t,ξ)Hn(ξ(XtH1-XtH2)σ(t,ξ)),
where i=-1 and σ(t,ξ)=Var(XtH1-XtH2)ξ2 for ξ∈ℝ. Because of the orthogonality of {Hn(x),x∈ℝ}n∈ℤ+, we will get from (19) that
(27)(iu)nσn(t,ξ)Hn(ξ(XtH1-XtH2)σ(t,ξ))
is the nth chaos of exp(iuξ(XtH1-XtH2)+(1/2)u2ξ2Var(XtH1-BtH2)) for all t≥0.


## 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
(6)μB(A)=λ1{t∈B,X(t)∈A},
for all Borel set A⊆ℝ, where λ1 is the one-dimensional Lebesgue measure. If μB 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 μB. 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])
(7)XI-1(x)={t∈I:X(t)=x}.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 fixedt, the Fourier transform on x of L(t,x) is the function
(8)f(t,u)=∫ℝ‍eiuxL(t,x)dx.Using the density of occupation formula we have(9)f(t,u)=∫0t‍eiuX(s)ds.We can express the local timesL(t,x) as the inverse Fourier transform of f(t,u), namely,
(10)L(t,x)=12π∫-∞+∞‍(∫0t‍eiu(X(s)-x)ds)du.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])
(11)E(L(t+h,x)-L(t,x))n=1(2π)n∫[t,t+ω]n‍∫ℝn‍e-ix∑j=1nujE(ei∑j=1nujX(sj))×∏j=1n‍duj∏j=1n‍dsj,
and for every even integer n≥2,
(12)E(L(t+h,x)-L(t,x)-L(t+h,y)+L(t,y))n=1(2π)n∫[t,t+ω]n‍∫ℝn‍∏j=1n‍[e-ixuj-e-iyuj]×E(ei∑j=1nujX(sj))∏j=1n‍duj∏j=1n‍dsj.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
(13)E[(X(s)-X(t))2]>0,
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,
(14)limϵ→0inftn-t1≤ϵVn>0,
where Vn is the relative prediction error as follows:
(15)Vn=Var(X(tn)-X(tn-1)∣X(t1),…,X(tn-1))Var(X(tn)-X(tn-1)),and the infimum in (14) is taken over all ordered points t1<t2<⋯<tn in T with tn-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 Cn and δ (both may depend on n) such that
(16)Var(∑j=1n‍uj[X(tj)-X(tj-1)])≥Cn∑j=1n‍uj2Var[X(tj)-X(tj-1)],
for all ordered points 0=t0<t1<t2<⋯<tn in T with tn-t1<δ and all uj∈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 letCn 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, CH.


## 2.2.  Chaos Expansion

LetΩ 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,
(17)E(F2)=∫Ω‍|F(ω)|2P(dω)<∞.We can introduce the chaos expansion, which is an orthogonal decomposition ofL2(Ω,P). We refer to Hu [24], Nualart [3], and the references therein for more details. Let X:={Xt,t∈[0,T]} be a Gaussian process defined on the probality space (Ω,ℱ,P). If pn(x) is a polynomial of degree n in x, then we call pn(Xt) a polynomial function of X with t∈[0,T]. Let 𝒫n be the completion with respect to the L2(Ω,P) norm of the set {pm(Xt):0≤m≤n,t∈[0,T]}. Clearly, 𝒫n is a subspace of L2(Ω,P). If 𝒞n denotes the orthogonal complement of 𝒫n-1 in 𝒫n, then L2(Ω,P) is actually the direct sum of 𝒞n, that is,
(18)L2(Ω,P)=⨁n=0∞𝒞n.Namely, for any functionalF∈L2(Ω,P), there are Fn in 𝒞n, n=0,1,2,…, such that
(19)F=∑n=0∞‍Fn.The decomposition equation (19) is called the chaos expansionof F, and Fn is called the nth chaos of F. Clearly, we have
(20)E(|F|2)=∑n=0∞‍E(|Fn|2).
Recall that Meyer-Watanabe test function space 𝒰 (see Watanabe [25]) is defined as
(21)𝒰:={F∈L2(Ω,P):F=∑n=0∞‍Fn,∑n=0∞‍nE(|Fn|2)<∞},
and F∈L2(Ω,P) is said to be smooth if F∈𝒰.Now, forF∈L2(Ω,P), we define an operator Γu with u∈[0,1] by
(22)ΓuF:=∑n=0∞‍unFn.SetΘ(u):=ΓuF. Then Θ(1)=F. Define ΦΘ(u):=(d/du)(∥Θ(u)∥2), where ∥F∥2:=E(|F|2) for F∈L2(Ω,P). We have
(23)ΦΘ(u)=∑n=1∞‍nun-1E(|Fn|2).Note that∥Θ(u)∥2=E(|Θ(u)|2)=∑n=0∞‍E(un|Fn|2).Proposition 1.
LetF∈L2(Ω,P). Then F∈𝒰 if and only if ΦΘ(1)<∞.Consider two independent fractional Ornstein-UhlenbeckXHi={XtHi,t≥0}, i=1,2, with respective indices Hi∈(0,1). Let Hn(x) and x∈ℝ be the Hermite polynomials of degree n. That is,
(24)Hn(x)=(-1)n1n!ex2/2∂n∂xne-x2/2.Then,(25)etx-t2/2=∑n=0∞‍tnHn(x),
for all t∈𝒞 and x∈ℝ, this implies that
(26)exp(iuξ(XtH1-XtH2)+12u2ξ2Var(XtH1-XtH2))=∑n=0∞‍(iu)nσn(t,ξ)Hn(ξ(XtH1-XtH2)σ(t,ξ)),
where i=-1 and σ(t,ξ)=Var(XtH1-XtH2)ξ2 for ξ∈ℝ. Because of the orthogonality of {Hn(x),x∈ℝ}n∈ℤ+, we will get from (19) that
(27)(iu)nσn(t,ξ)Hn(ξ(XtH1-XtH2)σ(t,ξ))
is the nth chaos of exp(iuξ(XtH1-XtH2)+(1/2)u2ξ2Var(XtH1-BtH2)) 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{XtH,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
(28)limsuph→0supxL(t+h,x)-L(t,x)h1-H(loglog(h-1))H≤C1a.s.(29)limsuph→0supx,tL(t+h,x)-L(t,x)h1-H(log(h-1))H≤C2a.s.Proof.
Lett≥0 be a fixed point. Following the Fourier analytic approach of Berman [26], we have
(30)E[L(t+h,x)-L(t,x)]n=1(2π)n×∫[t,t+h]n∫ℝnE(exp(i∑j=1n‍uj(XsjH-XtH)))×∏j=1nduj∏j=1ndsj.
LetΔXsH=XsH-XtH, s≥0, and denote by R(s1,s2,…,sn) the covariance matrix of (ΔXs1H,…,ΔXsnH) for different s1,…,sn, then we have
(31)detR(s1,s2,…,sn)=Var(ΔXs1H)Var(ΔXs2H∣ΔXs1H)⋯Var(ΔXsnH∣ΔXs1H,…,ΔXsn-1H).
By Yan et al. [11], one can write the fractional Ornstein-Uhlenbeck process starting from zero as
(32)XtH=v∫0t‍F(t,u)dBu,0≤t≤T,
whereB is a standard Brownian motion with B0=0, and for 0<u<t(33)F(t,u)=(H-12)κHe-tu1/2-H×∫ut‍sH-1/2(s-u)H-3/2esds,
with1/2<H<1, κH=(2HΓ((3/2)-H)/Γ(H+(1/2))Γ(2-2H))1/2, and
(34)F(t,u)=κHu1/2-H×(-e-t∫ut‍(s-u)H-1/2sH-1/2esds+tH-1/2(t-u)H-1/2+21-2He-t×∫ut‍(s-u)H-1/2sH-3/2esdse-t∫ut‍),
with0<H<1/2.
For anyr,s∈[t,t+h] such that r<s, we have
(35)Var(ΔXsH∣ΔXuH,u≤r)≥Var(ΔXsH-ΔXrH∣Bu,u≤r)=Var(XsH-XrH∣Bu,u≤r)=Var(XsH∣Bu,u≤r),
where the last equality follows from the fact thatXrH is measurable with respect to σ(Bu,u≤r). Moreover, we can write
(36)XsH=v∫0s‍F(s,u)dBu=v∫0r‍F(s,u)dBu+v∫rs‍F(s,u)dBu.
Hence, by using the measurability of∫0r‍F(s,u)dBu with respect to σ(Bu,u≤s), we have
(37)Var(XsH∣Bu,u≤r)=Var(v∫rs‍F(s,u)dBu∣Bu,u≤r)=Var(v∫rs‍F(s,u)dBu)≥CH(s-r)2H,
where, to obtain the second equality, we have used the fact that∫rs‍F(s,u)dBu is independent of σ(Bu,u≤s) (by the independence of the increments of the Brownian motion). Combining (31), inequation (35), and inequation (37), we have
(38)detR(s1,s2,…,sn)≥CH∏j=1n‍(sj-sj-1)2H>0,
wheres0=0. Hence, the change of variable V=R1/2U, U=(u1,…,un) implies that
(39)∫ℝn‍E(exp(i∑j=1n‍uj(XsjH-XtH)))∏j=1n‍duj=∫ℝn‍E(exp(i∑j=1n‍ujΔXsjH))∏j=1n‍duj=(2π)n/2(detR(s1,s2,…,sn))1/2.
Hence,(40)E[L(t+h,x)-L(t,x)]n=n!(2π)n/2×∫t<s1<⋯<sn<t+h‍1(detR(s1,s2,…,sn))1/2ds1⋯dsn≤CHn!(2π)n/2∫t<s1<⋯<sn<t+h‍∏j=1n‍1(sj-sj-1)Hds1⋯dsn≤CHn!(2π)n/2hn(1-H)(Γ(1-H))nΓ(1+n(1-H)).
Following from Stirling’s formula, we haven!/Γ(1+n(1-H))≤Ann!H, n≥2, for a suitable finite number A. So
(41)E(L(t+h,x)-L(t,x)h1-H)n≤Cnn!H.
Following, we first prove that for anyK>0, there exists a positive and finite constant B>0, depending on t, such that for sufficiently small u(42)P(L(t+h,x)-L(t,x)≥Bh1-HuH)≤e-(K/u).
First consideru of the form u=1/n. By Chebyshev’s inequality and inequation (41), we have
(43)P(L(t+h,x)-L(t,x)≥Bh1-HnH)≤E(L(t+h,x)-L(t,x)Bh1-HnH)n≤CnBn(1n)nHn!H(by Stirling’s formula)≤CnBn(2πn)H/2e-Hn=exp(n(log(CB)-H)+H2(logn+log2π)).
ChooseB>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 un+1<u<un and since n≥1, n/(m+1)≥1/2. This proves inequation (42).
On the other hand, if we takeu(h)=1/loglog(1/h) and consider hn of the form 2-n, then inequation (42) implies
(44)P(L(t+hn,x)-L(t,x)≥Bhn1-H(loglog(1hn))H)≤n-2,
for largen. So, following that Borel-Cantelli lemma and monotonicity arguments, we have
(45)L(t+h,x)-L(t,x)h1-H≤B(loglog(1hn))Ha.s.
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-UhlenbeckXHi={XtHi,t≥0}, i=1,2. It is defined formally by the following expression:
(46)ℓT=∫0T‍δ0(XtH1-XtH2)dt,where δ0 is the Dirac delta function. It is a measure of the amount of time for which the trajectories of the two processes, XtH1 and XtH2, 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 ℓT exists in L2. We approximate the Dirac delta function by the heat kernel
(47)pɛ(x)=12πɛe-x2/2ɛ≡12π∫ℝ‍eixξe-ɛ(ξ2/2)dξ.
For ɛ>0 we define
(48)ℓɛ,T=∫0T‍pϵ(XtH1-XtH2)dt=12π∫0T‍∫ℝ‍eiξ(XtH1-XtH2)·e-ɛ(ξ2/2)dξdt,and a natural question to study is that of the behavior ofℓɛ,T as ɛ tends to zero.Theorem 3.
ForHi∈(0,1), i=1,2. Then ℓɛ,T converges in L2(Ω,ℱ,P), as ɛ↓0. Moreover, the limit is denoted by ℓT, then ℓT∈L2(Ω,ℱ,P).Proof.
First we claim thatℓɛ,T∈L2(Ω,ℱ,P) for every ɛ>0. By (48) we have
(49)E(ℓɛ,T2)=14π2∬0T∫ℝ2Eeiξ(XtH1-XtH2)+iη(XsH1-XsH2)×e-((ɛ(ξ2+η2))/2)dξdηdsdt=14π2∬0T∫ℝ2e-(1/2)σ2e-((ɛ(ξ2+η2))/2)dξdηdsdt,
where σ2 denotes the variance of random variable ξ(XtH1-XtH2)+η(XsH1-XsH2), that is,
(50)σ2:=Var(ξ(XtH1-XtH2)+η(XsH1-XsH2)).
According to the property of local nondeterminism (see Theorem  3.1 in [11]), we have
(51)σ2=Var(ξ(XtH1-XsH1)-ξ(XtH2-XsH2)+(ξ+η)(XsH1-XsH2))≥C[ξ2((t-s)2H1+(t-s)2H2)+(ξ+η)2(s2H1+s2H2)].
Thus, we have
(52)14π2∬0T∫ℝ2e-(1/2)σ2e-(ɛ(ξ2+η2))/2dξdηdsdt≤∫0T‍∫0t‍∫ℝ2‍e-(C/2)[ξ2((t-s)2H1+(t-s)2H2)+(ξ+η)2(s2H1+s2H2)]dξdηdsdt=C∫0T‍∫0t‍[((t-s)2H1+(t-s)2H2)&((t-s)2H1)s×(s2H1+s2H2)]-1/2dsdt≤C∫0T‍∫0t‍(t-s)-(1/2)(H1+H2)s-(1/2)(H1+H2)dsdt<∞,because of Hi∈(0,1), which yields
(53)E(ℓɛ,T2)<∞,
for allɛ∈(0,1].
Second, we claim that the sequence{ℓɛ,T,ɛ>0} is of Cauchy in L2(Ω,ℱ,P). For any θ,ɛ>0 we have
(54)E(|ℓε,T-ℓθ,T|2)=14π2∬0T∫ℝ2Eeiξ(XtH1-XtH2)+iη(XsH1-XsH2)·(e-(ε/2)ξ2-e-(θ/2)ξ2)×(e-(ε/2)η2-e-(θ/2)η2)dξdηdsdt≤14π2supξ∈ℝ(1-e-((|ε-θ|2|ξ|2)/2))2×∬0T∫ℝ2e-(1/2)σ2dξdηdsdt.
Thus, dominated convergence theorem yields(55)E(|ℓɛ,T-ℓθ,T|2)⟶0,
asɛ→0 and θ→0, which leads to ℓɛ,T is a Cauchy sequence in L2(Ω,ℱ,P). Consequently, limɛ→0ℓɛ,T exists in L2(Ω,ℱ,P). This completes the proof.For the increments of collision local time we have the following.Theorem 4.
LetH1,H2∈(0,1) and β=min{H1,H2}. Then the collision local time ℓT satisfies the following estimate:
(56)E(|ℓt-ℓs|2)≤CH1,H2(t-s)2-2β,
for alls,t, s<t.Proof.
For any0≤r, l≤T we denote
(57)σr,l2:=Var(ξ(XrH1-XrH2)+η(XlH1-XlH2)).
Then the property of local nondeterminism (see Theorem  3.1 in [11]) yields
(58)σr,l2≥C[ξ2((r-l)2H1+(r-l)2H2)+(ξ+η)2(l2H1+l2H2)]
for a constantC>0. It follows from (48) that for 0≤s≤t≤T(59)E(|ℓɛ,t-ℓɛ,s|2)=2(2π)2∫st‍∫sr‍drdl∫ℝ2‍e-(1/2)σr,l2e-(ɛ/2)(ξ2+η2)dξdη≤C∫st‍dr∫sr‍(r-l)-βl-βdl≤C(t-s)2-2β.
Thus, Theorem 3 and Fatou’s lemma yield
(60)E(|ℓt-ℓs|2)=E(limɛ→0|ℓɛ,t-ℓɛ,s|2)≤liminfɛ→0E(|ℓɛ,t-ℓɛ,s|2)≤C(t-s)2-2β.
This completes the proof.Letλt=Var(XtH1-XtH2) for t≥0 and
(61)ρs,t=E[(XtH1-XtH2)(XsH1-XsH2)],fors,t≥0.Lemma 5 (An and Yan [28]).
For anyx∈[-1,1) we have
(62)∑n=1∞‍(2n-1)!!(2n-2)!!xn=x(1-x)-3/2,
where(2n-2)!!=1·3·5⋯(2n-1) and (2n-1)!!=2·4·6⋯(2n-2).By Cauchy-Schwartz’s inequality, we haveρs,t2≤λsλt. Hence,
(63)ρs,t2(λsλt-ρs,t2)3/2=ρs,t2λsλt(1-ρs,t2λsλt)-3/2(1λsλt)1/2=∑n=1∞‍(2n-1)!!(2n-2)!!(ρs,t2λsλt)n(1λsλt)1/2,for allt,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.
LetℓT,T≥0 be the collision local time process of two independent fractional Ornstein-Uhlenbeck XHi={XtHi,t≥0},i=1,2, with respective indices Hi∈(0,1). Then ℓT is smooth in the sense of the Meyer-Watanabe if and only if
(64)min{H1,H2}<13.Proof.
By Yan et al. [11], we have (65)λtλs-ρs,t2≍(s2H1+s2H2)(t2H1+t2H2)-12(t2H1+s2H1-|t-s|2H1+t2H2+s2H2-|t-s|2H2)≍(s2H1+s2H2)[(t-s)2H1+(t-s)2H2],
where the notationF≍G means that there are positive constants c1 and c2 so that
(66)c1G(x)≤F(x)≤c2G(x),
in the common domain of definition forF and G. Hence, following Theorem 2 in An and Yan [28], we have ∬0T(ρs,t2/(λsλt-ρs,t2)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, ℓT is smooth in the sense of the Meyer-Watanabe if and only if
(67)∬0Tρs,t2(λtλs-ρs,t2)-3/2dsdt<∞.
In fact, forɛ>0, T≥0 we denote
(68)Θɛ(u,T,ℓɛ,T):=E(|Γuℓɛ,T|2),
andΘ(u,T,ℓT):=E(|ΓuℓT|2). Thus, by Proposition 1 to prove that (67) holds if and only if ΦΘ(1)<∞. Clearly, we have
(69)ℓɛ,T=∫0T‍pɛ(XtH1-XtH2)dt=12π∫0T‍∫ℝ‍eiξ(XtH1-XtH2)·e-(1/2)ɛξ2dξdt=12π∫0T‍∫ℝ‍e-(1/2)ξ2Var(XtH1-XtH2)×e-(1/2)ɛξ2∑n=0∞‍inσn(t,ξ)Hn×(ξ(XtH1-XtH2)σ(t,ξ))dξdt≡∑n=0∞‍Fn.
Notice that
(70)ΦΘɛ(1)=∑n=0∞‍nE(|Fn|2)=∑n=0∞‍n4π2×E[∬0T∫ℝ2exp(-12ɛ(|ξ|2+|η|2))×σn(t,ξ)σn(s,η)·exp(-12((XsH1-XsH2)ξ2Var(XtH1-XtH2)+η2Var(XsH1-XsH2))(-12((XsH1-XsH2)ξ2Var(XtH1-XtH2))·Hn(ξ(XtH1-XtH2)σ(t,ξ))×Hn(η(XsH1-XsH2)σ(s,η))dξdηdsdt∫0T‍∫0T‍]=∑n=1∞‍14π2(2n-1)!×[∬0T∫ℝ2(ξη)2n×[E((XtH1-XtH2)(XsH1-XsH2))]2n·exp(12-(12(ξ2Var((XtH1-XtH2)+ɛ)+η2Var((XsH1-XsH2)+η2Var((XsH1-XsH2)+ɛ(XsH1-XsH2))(ξ2Var((XtH1-XtH2)+ɛ))×12))dξdηdsdt∫0T‍∫0T‍∫ℝ2‍(ξη)2n],
for allT≥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])
(71)E(Hn(X)Hm(Y))={0,m≠n,1n![E(XY)]n,m=n.
We obtain
(72)ΦΘɛ(1)=∑n=1∞‍(Γ(n+1/2))222n+14π2(2n-1)!×∬0Tρs,t2n((λs+ɛ)(λt+ɛ))n+(1/2)dsdt=∑n=1∞‍12π(2n-1)!!(2n-2)!!×∬0Tρs,t2n((λs+ɛ)(λt+ɛ))n+(1/2)dsdt=12π∬0Tρs,t2((λs+ɛ)(λt+ɛ)-ρs,t2)3/2dsdt,
where we have used the following equality:
(73)∫ℝ‍ξ2nexp(-ξ2(Var(XtH1-XtH2)+ɛ)2)dξ=2n+(1/2)Γ(n+12)(Var(XtH1-XtH2)+ɛ)-(n+(1/2)).
Hence, we have
(74)limɛ→0ΦΘɛ(1)=12π∫∫0Tρs,t2(λsλt-ρs,t2)3/2dsdt,
for allT≥0. This completes the proof.



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*Source: 375480-2013-09-19.xml*