About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 375480, 9 pages
http://dx.doi.org/10.1155/2013/375480
Research Article

The Local Time of the Fractional Ornstein-Uhlenbeck Process

Department of Mathematics, Anhui Normal University, 1 East Beijing Road, Wuhu 241000, China

Received 11 June 2013; Revised 11 August 2013; Accepted 11 August 2013

Academic Editor: Mark McKibben

Copyright © 2013 Guangjun Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the Hölder regularity of the local time of the fractional Ornstein-Uhlenbeck process . As a related problem, we study the collision local time of two independent fractional Ornstein-Uhlenbeck , with respective indices .

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 starting from is called Ornstein-Uhlenbeck process with coefficients 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 is a central Gaussian process with and the covariance function for all . 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 , coincides with the standard Brownian motion . is neither a semimartingale nor a Markov process unless , and so many of the powerful techniques from stochastic analysis are not available when dealing with . It has self-similar, long-range 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 , 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 , the fractional Ornstein-Uhlenbeck process is the classical Ornstein-Uhlenbeck process with parameter starting at . 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 . 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 , with respective indices .

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 [1618]. Let be any Borel function on with values in . For any Borel set , the occupation measure of is defined by for all Borel set , where is the one-dimensional Lebesgue measure. If is absolutely continuous with respect to the Lebesgue measure on , we say that has a local time on and define its local time to be the Radon-Nikodym derivative of . If , we simply write as . If and is continuous as a function of , then we say that has a jointly continuous local time on . In this latter case, the set function can be extended to be a finite Borel measure on the level set (see Adler [14, Theorem  8.6.1])

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 , and any integer , we have (see, e.g., Boufoussi et al. [20, 21]) and for every even integer ,

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 be a real-valued, separable Gaussian process with mean 0 and let be an open interval. Assume that for all and there exists such that for with .

Recall from Berman [22] that is called locally nondeterministic on if for every integer , where is the relative prediction error as follows:

and the infimum in (14) is taken over all ordered points in with . Roughly speaking, (14) 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 , there exist positive constants and (both may depend on ) such that for all ordered points in with and all (). 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 , .

2.2. Chaos Expansion

Let be the space of continuous -valued functions on . Then is a Banach space with respect to the supreme norm. Let be the -algebra on . Let be the probability measure on the measurable space . Let denote the expectation on this probability space. The set of all square integrable functionals is denoted by , that is,

We can introduce the chaos expansion, which is an orthogonal decomposition of . We refer to Hu [24], Nualart [3], and the references therein for more details. Let be a Gaussian process defined on the probality space . If is a polynomial of degree in , then we call a polynomial function of with . Let be the completion with respect to the norm of the set . Clearly, is a subspace of . If denotes the orthogonal complement of in , then is actually the direct sum of , that is,

Namely, for any functional , there are in , , such that

The decomposition equation (19) is called the chaos expansion of , and is called the th chaos of . Clearly, we have Recall that Meyer-Watanabe test function space (see Watanabe [25]) is defined as and is said to be smooth if .

Now, for , we define an operator with by

Set . Then . Define , where for . We have

Note that .

Proposition 1. Let . Then if and only if .

Consider two independent fractional Ornstein-Uhlenbeck , , with respective indices . Let and be the Hermite polynomials of degree . That is,

Then, for all and , this implies that where and for . Because of the orthogonality of , we will get from (19) that is the th chaos of for all .

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 be the fractional Ornstein-Uhlenbeck process. Then, for every and any , there exist positive and finite constants and such that

Proof. Let be a fixed point. Following the Fourier analytic approach of Berman [26], we have
Let , , and denote by the covariance matrix of for different , then we have
By Yan et al. [11], one can write the fractional Ornstein-Uhlenbeck process starting from zero as
where is a standard Brownian motion with , and for
with , , and
with .
For any such that , we have
where the last equality follows from the fact that is measurable with respect to . Moreover, we can write
Hence, by using the measurability of with respect to , we have
where, to obtain the second equality, we have used the fact that is independent of (by the independence of the increments of the Brownian motion). Combining (31), inequation (35), and inequation (37), we have
where . Hence, the change of variable , implies that
Hence,
Following from Stirling’s formula, we have, , for a suitable finite number . So
Following, we first prove that for any , there exists a positive and finite constant , depending on , such that for sufficiently small
First consider of the form . By Chebyshev’s inequality and inequation (41), we have
Choose and large such that for any , to dominate (43) by . Moreover, for sufficiently small, there exists such that and since , . This proves inequation (42).
On the other hand, if we take and consider of the form , 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.

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 , . It is defined formally by the following expression: where is the Dirac delta function. It is a measure of the amount of time for which the trajectories of the two processes, and , collide on the time interval . The collision local time for fractional Brownian motion has been studied by Jiang and Wang [27]. We shall show that the random variable exists in . We approximate the Dirac delta function by the heat kernel For we define

and a natural question to study is that of the behavior of as tends to zero.

Theorem 3. For , . Then converges in , as . Moreover, the limit is denoted by , then .

Proof. First we claim that for every . By (48) we have where denotes the variance of random variable , that is, According to the property of local nondeterminism (see Theorem  3.1 in [11]), we have Thus, we have because of , which yields
for all .
Second, we claim that the sequence is of Cauchy in . For any we have
Thus, dominated convergence theorem yields
as and , which leads to is a Cauchy sequence in . Consequently, exists in . This completes the proof.

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

Theorem 4. Let and . Then the collision local time satisfies the following estimate:
for all , .

Proof. For any , we denote Then the property of local nondeterminism (see Theorem  3.1 in [11]) yields
for a constant . It follows from (48) that for Thus, Theorem 3 and Fatou’s lemma yield This completes the proof.

Let for and

for .

Lemma 5 (An and Yan [28]). For any we have
where and .

By Cauchy-Schwartz’s inequality, we have . Hence,

for all and .

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

Theorem 6. Let be the collision local time process of two independent fractional Ornstein-Uhlenbeck , with respective indices . Then is smooth in the sense of the Meyer-Watanabe if and only if

Proof. By Yan et al. [11], we have
where the notation means that there are positive constants and so that
in the common domain of definition for and . Hence, following Theorem 2 in An and Yan [28], we have if and only if . Therefore, in order to prove Theorem 6, it only needs to prove: for , is smooth in the sense of the Meyer-Watanabe if and only if
In fact, for , we denote
and . Thus, by Proposition 1 to prove that (67) holds if and only if . Clearly, we have Notice that
for all , where we have used the following fact: For two random variables , with joint Gaussian distribution such that and we have (see, for example, Nualart [3]) We obtain where we have used the following equality: Hence, we have
for all . 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).

References

  1. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, New York, NY, USA, 1999. View at MathSciNet
  2. J.-F. Gouyet, Physics and Fractal Structures, Springer, Berlin, Germany, 1996. View at MathSciNet
  3. D. Nualart, The Malliavin Calculus and Related Topics, Springer, New York, NY, USA, 2006. View at MathSciNet
  4. F. Biagini, Y. Hu, B. Øksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, vol. 1929 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  6. W. Willinger, M. S. Taqqu, and V. Teverovsky, “Stock market prices and long-range dependence,” Finance Stochastics, vol. 3, no. 1, pp. 1–13, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. B. B. Mandelbrot and J. W. van Ness, “Fractional Brownian motions, fractional noises and applications,” SIAM Review, vol. 10, pp. 422–437, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. P. Cheridito, H. Kawaguchi, and M. Maejima, “Fractional Ornstein-Uhlenbeck processes,” Electronic Journal of Probability, vol. 8, article 3, pp. 1–14, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. C. Lim and S. V. Muniandy, “Generalized Ornstein-Uhlenbeck processes and associated self-similar processes,” Journal of Physics A, vol. 36, no. 14, pp. 3961–3982, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. Yan, Y. Lu, and Z. Xu, “Some properties of the fractional Ornstein-Uhlenbeck process,” Journal of Physics A, vol. 41, no. 14, Article ID 145007, 17 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Yan and M. Tian, “On the local times of fractional Ornstein-Uhlenbeck process,” Letters in Mathematical Physics, vol. 73, no. 3, pp. 209–220, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. R. S. Varadhan, “Appendix to euclidean quantum field theory by K. Symanzk,” in Local Quantum Theory, R. Jost, Ed., Academic Press, New York, NY, USA, 1969.
  14. R. J. Adler, The Geometry of Random Fields, John Wiley & Sons, New York, NY, USA, 1981. View at Zentralblatt MATH · View at MathSciNet
  15. D. Geman and J. Horowitz, “Occupation densities,” The Annals of Probability, vol. 8, no. 1, pp. 1–67, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Xiao, “Local times and related properties of multidimensional iterated Brownian motion,” Journal of Theoretical Probability, vol. 11, no. 2, pp. 383–408, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Xiao, “Properties of local-nondeterminism of Gaussian and stable random fields and their applications,” Annales de la Faculté des Sciences de Toulouse Série 6, vol. 15, no. 1, pp. 157–193, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. Y. Xiao, “Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields,” Probability Theory and Related Fields, vol. 109, no. 1, pp. 129–157, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Y. Hu, “Hausdorff and packing measures of the level sets of iterated Brownian motion,” Journal of Theoretical Probability, vol. 12, no. 2, pp. 313–346, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Boufoussi, M. Dozzi, and R. Guerbaz, “On the local time of multifractional Brownian motion,” Stochastics, vol. 78, no. 1, pp. 33–49, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. B. Boufoussi, M. Dozzi, and R. Guerbaz, “Sample path properties of the local time of multifractional Brownian motion,” Bernoulli, vol. 13, no. 3, pp. 849–867, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. M. Berman, “Local nondeterminism and local times of Gaussian processes,” Indiana University Mathematics Journal, vol. 23, pp. 69–94, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. P. Nolan, “Local nondeterminism and local times for stable processes,” Probability Theory and Related Fields, vol. 82, no. 3, pp. 387–410, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Y. Hu, “Self-intersection local time of fractional Brownian motions—via chaos expansion,” Journal of Mathematics of Kyoto University, vol. 41, no. 2, pp. 233–250, 2001. View at Zentralblatt MATH · View at MathSciNet
  25. S. Watanabe, Stochastic Differential Equation and Malliavin Calculus, Tata Institute of Fundamental Reaearch, Springer, New York, NY, USA, 1984.
  26. S. M. Berman, “Local times and sample function properties of stationary Gaussian processes,” Transactions of the American Mathematical Society, vol. 137, pp. 277–299, 1969. View at Publisher · View at Google Scholar · View at MathSciNet
  27. Y. Jiang and Y. Wang, “On the collision local time of fractional Brownian motions,” Chinese Annals of Mathematics B, vol. 28, no. 3, pp. 311–320, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. An and L. Yan, “Smoothness for the collision local time of fractional Brownian motion,” Preprint.