Abstract

Let be a -dimensional bifractional Brownian motion and be the bifractional Bessel process with the index . The Itô formula for the bifractional Brownian motion leads to the equation . In the Brownian motion case and , is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process is not a bifractional Brownian motion unless and . We also study some other properties and their application of this stochastic process.

1. Introduction

Given and , the bifractional Brownian motion with the indices and is a mean zero Gaussian process such that andfor all . This process was first introduced by Houdré and Villa [1]. More works for bifractional Brownian motion and their application can be found in [210] and the references therein. Clearly, the process is a fractional Brownian motion with Hurst the parameter when . Particularly, the process is a Brownian motion when and . Since is neither a Markov process nor a semimartingale unless and , a lot of powerful techniques from classical stochastic analysis are not available to deal with it. As the generalization of the fractional Brownian motion, the bifractional Brownian motion also admits Hölder paths and self-similarity, but its increments are not stationary.

Let be a -dimensional bifractional Brownian motion with the index . That is to say, each component of is an independent one-dimensional bifractional Brownian motion with the index . Let be the bifractional Bessel process defined by .

There is an extensive literature on this process for the standard Brownian motion case and the fractional Brownian motion case (see [1115]). For , by the Itô formula for the bifractional Brownian motion, we have (see Alós et al. [16] and Es-Sebaiy and Tudor [3])and for and , one also haswhere stochastic integrals are interpreted in the divergence sense and denotes the Dirac delta function. When and , the processis a standard Brownian motion by Lévy’s characterization theorem. Given , the fractional Brownian motion case was researched by Hu and Nualart [11]. So, it is natural and interesting to research the process for more general . Since there is no characterization as convenient as Lévy’s characterization theorem for general bifractional Brownian motion, to prove a stochastic process is a bifractional Brownian motion or not is difficult. The method used here is essentially based on Hu and Nualart [11] and Shen et al. [17]. It is not difficult to find that the bifractional Brownian motion has the nonavailability of convenient stochastic integral representations and more complexity of dependence structures than an fractional Brownian motion and a subfractional Brownian motion. Therefore, it seems interesting to study bifractional Bessel processes driven by bifractional Brownian motions.

The rest of the paper is organized as follows. In Section 2, we present some preliminaries for the bifractional Brownian motion. In Section 3, some properties to the process are studied. In Section 4, we consider the process with . In Section 5, we consider the local time and Tanaka formula of the process .

2. Preliminaries

In this paper, we assume that is arbitrary but fixed and let be a bifractional Brownian motion with the index , which is defined on the complete probability space . One can construct a stochastic calculus of variations with respect to the bifractional Brownian motion by the Malliavin calculus method (see Alòs et al. [16] and Nualart [18]). We next recall the basic definitions and results for this calculus.

Bifractional Brownian motion satisfies the estimates:

One can write its covariance as follows:where

Therefore,

Since is of the class and is always negative, is the distribution function and has for density. is the distribution function with density and belongs to . It follows that there exist two positive constants and which satisfy

Denote

As a Gaussian process of , we can construct a stochastic calculus of variations with respect to this process. Suppose that is the completion of the space which is generated by with respect to the following inner product:

Then, is an isometry from to the Gaussian space generated by which can be extended to . We can write this Hilbert space as follows:where . We can define the spaces of measurable functions as follows:where

It is easy to see that is dense in and is a Banach space. Suppose that is the set of smooth functionalwhere and . The Malliavin derivative of the above functional is given as follows:

The derivative operator is a closable operator from space into space . We denote , the closure of , with respect to norm

The divergence integral is the adjoint operator of . can be defined by the duality relationship:for any . For any , one has andwhereexpressing the Skorokhod integral of a process .

3. Case of One Dimension

We study the stochastic process defined by

If and , is a standard Brownian motion from Levý’s characterization theorem. It is then natural to study any parameter . Next, we first prove is an -self-similar process for any .

Proposition 1. The stochastic process is -self-similar.

Proof. Together with the -self-similarity property of the bifractional Brownian motion and Tanaka formula (4), for any , one can obtainwhere denotes that both stochastic processes have the same distributions. This proof is completed.
For stochastic process , we first obtain the Wiener chaos expansion. Let be the multiple Wiener integral of the stochastic process .

Proposition 2. For any , one can obtainwhere

Proof. For , we denoteThen,which is a density function of the bifractional Brownian motion and in as . By Stroock’s formula, one can obtainwhereAs , by taking the limit of (27) in the space , one can obtainwhere , which impliesThe proof is completed.
In this paper, the notation implies that there are two positive constants and such thatwhere denotes a generic positive constant and and have the common domain.

Proposition 3. The random variable belongs to the Sobolev–Watanabe space for any and .

Proof. By Stirling’s formulawe haveThe proof is completed.

Proposition 4. For any , one haswhere and

The above proposition is the chaos expansion of and implies the following result, which can be proved by the method similar to Proposition 3.

Proposition 5. For any and , the random variable belongs to the Sobolev–Watanabe space . Now, we consider the stochastic process :where .

Definition 1. We say a stochastic process is long-range dependent (resp. short-range dependent) if for each ,

Theorem 1. The stochastic process of (21) is short-range dependent. Before proving this theorem, a lemma given by Yan et al. [9] is stated.

Lemma 1. Let and , one defineswhere . Then, we have

Remark 1. The proof of estimate (39) uses the following two inequalities:where and . It is not difficult to prove inequality (40), which is stronger than the well-known inequalitybecause for all .

Proof of Theorem 1. For , one can obtainNow, we only need to estimate and . For , by the orthogonal decomposition,wherein which independent of . SetSince. By Lemma 1, we obtainas and , which impliesSo, the term behaves as . Now, we evaluate the second term . For , using Lemma 1, one can obtainwhere is the density function of . So,The proof is completed.

4. Case of Multidimension

We now consider the -dimensional bifractional Brownian motion with the index , which implies the components are independent bifractional Brownian motions with the same index . As in Section 2, we can define the derivative and divergence operators, and , with respect to each component . Suppose that are the associated Sobolev spaces. Similarly, denotes the set of processes in which satisfies

Letbe a bifractional Bessel process. In the following, we research the stochastic process:

The next theorem can be proved similar to Es-Sebaiy and Tudor [3].

Theorem 2. Let be a d-dimensional bifractional Brownian motion with and be a function of class . Then,

The following proposition gives an integral representation for bifractional Bessel processes and can be proved along the lines of the proof of Proposition 5.2. in Guerra and Nualart [19].

Proposition 6. Suppose that is a bifractional Bessel process associated to the d-dimensional bifractional Brownian motion with index . For each , one can obtain and

Proof. Step1. We prove are well defined which only proves for each . Since for each , one can obtainTogether with the definition of the derivative operator and the self-similarity of the bifractional Brownian motion, one can obtainsince . So, the integral is well defined since for each .Step 2. We now prove (56). Note that is defined bywhich is not differentiable at the origin. So, we cannot apply the Itô formula (55) to . But, if one considers the square of the bifractional Bessel processthen one can apply the Itô formula (55), and we haveSetFor any , , and for any . Applying (55) to , we obtainwhereTogether with a.s. and the bounded convergence theorem, one can obtainFor the third term, by the substituting and Fubini’s theorem, we can obtainthat is,Finally, we show thatin . We haveas . On the other hand, one can obtainThe distribution of in spherical coordinates yieldswhere is a constant which depends on , and . For any ,We haveby the bounded convergence theorem, that is,This proves the desired convergence (68), and the proposition follows.

Proposition 7. Stochastic process which is given by (54) is -self-similar.

Proof. Set . Together with the -self-similarity property of the bifractional Brownian motion and (56), we can obtainFor , we denote

Theorem 3. Suppose , are with polynomial growth and smooth functions. Then, the stochastic process has the following chaos expansion:where

Proof. For each , using Stroock’s formula, we can obtainwhereSo,This completes the proof.
Let ; then, . So, for such , one can obtainThen, can be denoted byand the chaos expansion of isThe theorem is proved.

Theorem 4. The stochastic process is short-range dependent.

Proof. LetFor every , by the formula, we can decompose asFor , one can use the decompositionwherewhere is independent of and is denoted as a-dimensional standard normal random variable. By Lemma 13 in [11],Thus,which implies that the term behaves as .
For , one hasSincebehaves as as , whereWe see that the term also behaves as , and the theorem follows.

5. The Local Times of

Now, we consider the local times of the stochastic process defined by

Lemma 2. Let . Then, for all , we have

Proof. Using the Itô formula, one can obtainand note that the function is increasing sinceSo,for all . Now, let us prove , a.s. We only need to show thatis nondecreasing. Letwhere for and for . Thus, one can obtainSo, for all , one can obtainwhich implies that and . Therefore,is nondecreasing. This completes the proof.

Theorem 5. Let the stochastic process be defined byand let be a convex function with polynomial growth. Then, there is a continuous increasing process which satisfieswhere denotes the left-hand derivative of .

Proof. If , then this is the Itô formula, andtogether with Lemma 2, implies that the stochastic process is increasing.
Now, let . For and , one setsIt is easy to see that and has polynomial growth. So, for all , there exists a continuous increasing process such thatNote thatand one can obtain as in probability. So, converges to a stochastic process which, as a limit of increasing stochastic processes, is itself an increasing stochastic process andwhere can be chosen to be a.s. continuous. The proof is completed.

Corollary 1. For the process and all , there exists a local time such that

Proof. Note that the left derivative of the function is equal to . By Theorem 5, one can obtainwhere is a continuous increasing stochastic process. Similarly, there exists a continuous increasing stochastic process which satisfiesTherefore, one can obtainwhich implies that a.s. and we set . This completes the proof.
Combining this corollary with Es-Sebaiy and Tudor [3], we can obtain the following results.

Corollary 2. Suppose that is the local time of the process and is the weighted local time of the bifractional Brownian motion defined by

Then, we have

Proof. By the Tanaka formula, one can obtainwhich implies that (117) holds.

Corollary 3. For any and , we haveMoreover, let be a convex function with polynomial growth; one can obtain the following Itô–Tanaka formula:where denotes the left derivative of and signed measure which is defined byFinally, one can prove that the local time of the processexists by the same method and can obtain the similar results.

6. Conclusions

This paper presents theorems and propositions associated with respect to the stochastic process , where is a -dimensional bifractional Brownian motion and . Since there is no Lévy’s characterization theorem for a general bifractional Brownian motion, to prove whether a stochastic processis a bifractional Brownian motion or not is difficult. Theorems 1 and 4 prove is short-range dependent in one-dimensional case and multidimensional case, respectively. Theorem 5 considers the local times of the stochastic process of in one-dimensional case. Theorem 2 gives the the following chaos expansion of the stochastic process . Moreover, significance results associated with the above theorem are given.

Data Availability

All the data generated during this study are included within this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

Acknowledgments

Xichao Sun acknowledges the National Natural Science Foundation of China (11971101 and 11426036), Natural Science Foundation of Anhui Province (1808085MA02), and Quality Engineering of Anhui Education (2018jyxm0497 and 2020zdxsjg214). Ming Li acknowledges the National Natural Science Foundation of China under grant nos. 61672238, 61272402, and 61070214.