Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion

Department of Mathematics, College of Science, Bengbu University, 1866 Caoshan Rd., Bengbu 233030, China College of Information Science and Technology, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, China Ocean College, Zhejiang University, Zhejiang 310012, China Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, No. 500 Dong-Chuan Road, Zhongshan, Shanghai 200241, China


Introduction
Given H ∈ (0, 1) and K ∈ [0, 1], the bifractional Brownian motion with the indices H and K is a mean zero Gaussian process B � B H,K t , t ≥ 0 such that B H,K 0 � 0 and for all (s, t ≥ 0). is process was first introduced by Houdré and Villa [1]. More works for bifractional Brownian motion and their application can be found in [2][3][4][5][6][7][8][9][10] and the references therein. Clearly, the process is a fractional Brownian motion with Hurst the parameter H when K � 1. Particularly, the process is a Brownian motion when (K � 1) and (H � (1/2)).
Since (B H,K t ) is neither a Markov process nor a semimartingale unless (K � 1) and (H � (1/2)), 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 selfsimilarity, but its increments are not stationary.
Let B � (B 1 , . . . , B d ) be a d-dimensional bifractional Brownian motion with the index (HK ≥ (1/2)). at is to say, each component of B is an independent one-dimensional bifractional Brownian motion with the index (HK ≥ (1/2)). Let R t be the bifractional Bessel process defined by R t � ��������������� � (B 1 t ) 2 + · · · + (B d t ) 2 . ere is an extensive literature on this process for the standard Brownian motion case (K � 1 and H � (1/2)) and the fractional Brownian motion case (K � 1) (see [11][12][13][14][15]). For (d ≥ 2, HK > (1/2)), by the Itô formula for the bifractional Brownian motion, we have (see Alós et al. [16] and Es-Sebaiy and Tudor [3]) where stochastic integrals are interpreted in the divergence sense and δ denotes the Dirac delta function. When K � 1 and H � (1/2), the process is a standard Brownian motion by Lévy's characterization theorem. Given K � 1, the fractional Brownian motion case was researched by Hu and Nualart [11]. So, it is natural and interesting to research the process X � X t , t ≥ 0 for more general H and K. 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. e 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. erefore, it seems interesting to study bifractional Bessel processes driven by bifractional Brownian motions. e 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

Preliminaries
In this paper, we assume that ((1/2) < HK < 1) is arbitrary but fixed and let B � B H,k t , 0 ≤ t ≤ T be a bifractional Brownian motion with the index H and K, which is defined on the complete probability space (Ω, F, and P). One can construct a stochastic calculus of variations with respect to the bifractional Brownian motion B H,k 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 B H,k satisfies the estimates: One can write its covariance as follows: where erefore, Since R 1 is of the class C 2 ([0, T] 2 ) and (z 2 /zt zs)R 1 (t, s) is always negative, R 1 is the distribution function and has (z 2 /zt zs)R 1 (t, s) for density. R 2 is the distribution function with density (z 2 /zr zs)R 1 (t, s) � 2(2HK − 1)HK|t − s| 2HK− 2 and belongs to L 1 ([0, T] 2 ). It follows that there exist two positive constants c H,K and C H,K which satisfy As a Gaussian process of B H,k , we can construct a stochastic calculus of variations with respect to this process. Suppose that H is the completion of the space E which is generated by 1 [0,T] , t ∈ [0, T] with respect to the following inner product: en, φ ∈ E↦B(φ) is an isometry from E to the Gaussian space generated by B which can be extended to H. We can write this Hilbert space H as follows: where We can define the spaces of measurable functions as follows: where It is easy to see that E is dense in |H| and |H| is a Banach space. Suppose that S is the set of smooth functional where f ∈ C ∞ b (R n ) and φ i ∈ H. e Malliavin derivative D of the above functional F is given as follows: e derivative operator D is a closable operator from space L 2 (Ω) into space L 2 (Ω; H). We denote D 1,2 , the closure of S, with respect to norm e divergence integral δ is the adjoint operator of D. δ(u) can be defined by the duality relationship: for any u ∈ D 1,2 . For any u ∈ D 1,2 , one has (D 1,2 ⊂ Dom(δ)) and where expressing the Skorokhod integral of a process u.

Case of One Dimension
We study the stochastic process X � X t , t ≥ 0 defined by If K � 1 and H � (1/2), X t is a standard Brownian motion from Levy's characterization theorem. It is then natural to study any parameter H and K. Next, we first prove X is an HK-self-similar process for any HK ≥ (1/2).

Proposition 1.
e stochastic process X � X t , t ≥ 0 is HK-self-similar.
Proof. Together with the HK-self-similarity property of the bifractional Brownian motion and Tanaka formula (4), for any a > 0, one can obtain where � d denotes that both stochastic processes have the same distributions. is proof is completed.
Proof. For ε > 0, we denote (25) en, which is a density function of the bifractional Brownian motion where

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As ε ⟶ 0, by taking the limit of (27) in the space L 2 (Ω), one can obtain sign B H,K t � ∞ m�0 a n (t) where e proof is completed. In this paper, the notation F≍G implies that there are two positive constants c 1 and c 2 such that where C denotes a generic positive constant and F and G have the common domain. Proof. By Stirling's formula we have e proof is completed. where e above proposition is the chaos expansion of and implies the following result, which can be proved by the method similar to Proposition 3.
Theorem 1. e stochastic process X of (21) is short-range dependent. Before proving this theorem, a lemma given by Yan et al. [9] is stated.
Now, we only need to estimate a k and b k . For a k , by the orthogonal decomposition, where x ⟶ 0. By Lemma 1, we obtain as t ⟶ ∞ and s ∈ (0, 1), which implies So, the term a k behaves as k 2HK− 3 o(k − HK ). Now, we evaluate the second term b n . For s < t, using Lemma 1, one can obtain where h(y, z) is the density function of (B H,K e proof is completed.

Case of Multidimension
We now consider the d-dimensional bifractional Brownian motion B � (B 1 t , . . . , B d t ) t ≥ 0 with the index HK ≥ (1/2), which implies the components (B i , i � 1, . . . , d) are independent bifractional Brownian motions with the same index HK ≥ (1/2). As in Section 2, we can define the derivative and divergence operators, D i and δ i , with respect to each component B i . Suppose that (D (52) be a bifractional Bessel process. In the following, we research the stochastic process: e next theorem can be proved similar to Es-Sebaiy and Tudor [3].
be a d-dimensional bifractional Brownian motion with (2HK > 1) and f be a function of class C 2 (R d ). en, (55) e 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].
Together with the definition of the derivative operator and the self-similarity of the bifractional Brownian motion, one can obtain Step 2. We now prove (56). Note that f: which is not differentiable at the origin. So, we cannot apply the Itô formula (55) to f. But, if one considers the square of the bifractional Bessel process then one can apply the Itô formula (55), and we have For any ε > 0, g ε (y) ∈ C 2 (R), and lim ε⟶0 g ε (y) � � � y √ for any x ≥ 0. Applying (55) to g ε (R 2 t ), we obtain 6 Mathematical Problems in Engineering where Together with t 0 (s 2HK− 1 /R s ) < ∞ a.s. and the bounded convergence theorem, one can obtain For the third term, by the substituting u � (ρ/s H,K ) and Fubini's theorem, we can obtain that is, Finally, we show that in L (1/HK) (Ω). We have as ε ⟶ 0. On the other hand, one can obtain e distribution of (B 1 1 , . . . , B d 1 ) in spherical coordinates yields Mathematical Problems in Engineering 7 where C d,H,K > 0 is a constant which depends on H, K, and d.
For any r ∈ [0, T], We have by the bounded convergence theorem, that is, is proves the desired convergence (68), and the proposition follows.

Proposition 7. Stochastic process X which is given by (54) is HK-self-similar.
Proof. Set a > 0. Together with the HK-self-similarity property of the bifractional Brownian motion and (56), we can obtain For h ∈ H ⊗n , we denote where where So, is completes the proof.

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Let f j (x) � (x j / ���������� � x 2 1 + · · · + x 2 d ); then, f j (tx) � f j (x). So, for such f j , one can obtain and the chaos expansion of e theorem is proved. □ Theorem 4. e stochastic process X is short-range dependent.
Proof. Let For every K ≥ 1, by the formula, we can decompose ρ k as For ρ k,1 , one can use the decomposition where where N is independent of B s and is denoted as ad-dimensional standard normal random variable. By Lemma 13 in [11], Mathematical Problems in Engineering us, which implies that the term ρ k,1 behaves as k H− 3 . For ρ k,2 , one has Since behaves as Mt − HK as t ⟶ ∞, where We see that the term ρ k,2 also behaves as k HK− 3 , and the theorem follows.
and note that the function t ⟶ σ(t) is increasing since So, Φ ε ∈ C 2 (R), where Φ ε (x) � 0 for x > ε and Φ ε (x) � 1 for x < ε. us, one can obtain which implies that A + � A − a.s. and we set L x x (X) � A + t . is completes the proof. Combining this corollary with Es-Sebaiy and Tudor [3], we can obtain the following results.