# Some Properties of Bifractional Bessel Processes Driven by Bifractional Brownian Motion

Authors: Xichao Sun; Rui Guo; Ming Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

LetB=Bt1,…,Btdt≥0 be a d-dimensional bifractional Brownian motion and Rt=Bt12+⋯+Btd2 be the bifractional Bessel process with the index 2HK≥1. The Itô formula for the bifractional Brownian motion leads to the equation Rt=∑i=1d∫0tBsi/RsdBsi+HKd−1∫0ts2HK−1/Rsds. In the Brownian motion case K=1 and H=1/2, Xt≔∑i=1d∫0tBsi/RsdBsi,d≥1 is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process Xt is not a bifractional Brownian motion unless K=1 and H=1/2. We also study some other properties and their application of this stochastic process.


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## Body

## 1. Introduction

GivenH∈0,1 and K∈0,1, the bifractional Brownian motion with the indices H and K is a mean zero Gaussian process B=BtH,K,t≥0 such that B0H,K=0 and(1)EBsH,KBtH,K=Rt,s≔12Kt2H+s2HK−t−s2HK,for all s,t≥0. This process was first introduced by Houdré and Villa [1]. More works for bifractional Brownian motion and their application can be found in [2–10] 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 BtH,K 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 self-similarity, but its increments are not stationary.LetB=B1,…,Bd be a d-dimensional bifractional Brownian motion with the index HK≥1/2. That is to say, each component of B is an independent one-dimensional bifractional Brownian motion with the index HK≥1/2. Let Rt be the bifractional Bessel process defined by Rt=Bt12+⋯+Btd2.There is an extensive literature on this process for the standard Brownian motion caseK=1andH=1/2 and the fractional Brownian motion case K=1 (see [11–15]). 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])(2)Rt=∑i=1d∫0tBsiRsdBsi+HKd−1∫0ts2HK−1Rsds,and for d=1 and HK≥1/2, one also has(3)BtH,K=∫0tsignBsH,KdBsH,K+HK∫0tδBsH,Ks2HK−1ds,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(4)Xt=∫0tsignBsH,KdBsH,K,d=1,∑i=1d∫0tBsiRsdBsi,d≥2is 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=Xt,t≥0 for more general HandK. 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 Section2, we present some preliminaries for the bifractional Brownian motion. In Section 3, some properties to the process ∫0tsignBtH,KdBtH,K are studied. In Section 4, we consider the process ∑i=1d∫0tBsi/RsdBsi with d≥2. In Section 5, we consider the local time and Tanaka formula of the process ∫0tsignBtH,KdBtH,K.


## 2. Preliminaries

In this paper, we assume that1/2<HK<1 is arbitrary but fixed and let B=BtH,k,0≤t≤T be a bifractional Brownian motion with the index HandK, which is defined on the complete probability space Ω,ℱ,andP. One can construct a stochastic calculus of variations with respect to the bifractional Brownian motion BH,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 motionBH,k satisfies the estimates:(5)2−Kt−s2HK≤EBt−Bs2≤21−Kt−s2HK.One can write its covariance as follows:(6)Rt,s=R1t,s+R2t,s,where(7)R1t,s=12Ks2H+t2HK−s2HK+t2HK,R2t,s=12Kt2HK+s2HK−t−s2HK.Therefore,(8)∂2∂t∂sR1t,s=4H2KK−12−Kt2H−1s2H−1t2H+s2HK−2,∂2∂t∂sR2t,s=22HK−1HKt−s2HK−2.SinceR1 is of the class C20,T2 and ∂2/∂t∂sR1t,s is always negative, R1 is the distribution function and has ∂2/∂t∂sR1t,s for density. R2 is the distribution function with density ∂2/∂r∂sR1t,s=22HK−1HKt−s2HK−2 and belongs to L10,T2. It follows that there exist two positive constants cH,K and CH,K which satisfy(9)cH,Kt−s2HK−2≤∂2∂t∂sRt,s≤CH,Kt−s2HK−2.Denote(10)ϕt,s=2HK−1HKt−s2HK−2,fors,t≥0.As a Gaussian process ofBH,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 10,T,t∈0,T with respect to the following inner product:(11)10,s,10,tℋ=Rs,t.Then,φ∈ℰ↦Bφ 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:(12)ℋ=φ:0,T⟶ℝ|φℋ<∞,where φℋ2≔∫0Tφsφrϕs,rdsdr. We can define the spaces of measurable functions as follows:(13)ℋ=φ:0,T⟶ℝ|φℋ<∞,where(14)φℋ2≔∫0T∫0Tφsφrϕs,rdsdr<∞.It is easy to see thatℰ is dense in ℋ and ℋ is a Banach space. Suppose that S is the set of smooth functional(15)F=fBH,Kφ1,BH,Kφ2,…,BH,Kφ2,where f∈Cb∞ℝn and φi∈ℋ. The Malliavin derivative D of the above functional F is given as follows:(16)DF=∑j=1n∂f∂xjBH,Kφ1,BH,Kφ2,…,BH,Kφnφj.The derivative operatorD is a closable operator from space L2Ω into space L2Ω;ℋ. We denote D1,2, the closure of S, with respect to norm(17)F1,2≔EF2+EDFℋ2.The divergence integralδ is the adjoint operator of D. δu can be defined by the duality relationship:(18)EFδu=EDF,uℋ,for any u∈D1,2. For any u∈D1,2, one has D1,2⊂Domδ and(19)Eδu2=Euℋ2+E∫0,T4DξurDηusϕξ,sϕη,rdsdrdξdη≤Euℋ2+E∫0,T4DξurDηusϕξ,sϕη,rdsdrdξdη,where(20)δu=∫0TusdBsH,K,expressing the Skorokhod integral of a process u.


## 3. Case of One Dimension

We study the stochastic processX=Xt,t≥0 defined by(21)Xt=∫0tsignBsH,KdBsH,K.IfK=1 and H=1/2, Xt is a standard Brownian motion from Levý’s characterization theorem. It is then natural to study any parameter HandK. Next, we first prove X is an HK-self-similar process for any HK≥1/2.Proposition 1.
The stochastic processX=Xt,t≥0 is HK-self-similar.Proof.
Together with theHK-self-similarity property of the bifractional Brownian motion and Tanaka formula (4), for any a>0, one can obtain(22)Xat=BatH,K−HK∫0atδBsH,KBs2HK−1ds=BatH,K−HK∫0tδBauH,Kau2HK−1adu=daHKBtH,K−HKa2HK∫0tδaHKBuH,Ku2HK−1du=aHKXt,where =d denotes that both stochastic processes have the same distributions. This proof is completed.
For stochastic processsignBtH,K, we first obtain the Wiener chaos expansion. Let In be the multiple Wiener integral of the stochastic process BH,K.Proposition 2.
For anyt≥0, one can obtain(23)signBtH,K=∑m=0∞b2m+1I2m+11,where(24)b2m+1=2−1m2m+12πt2m+1HKm!2m.Proof.
Forε>0, we denote(25)pεy=12πεe−y2/2ε,fεy=2∫−∞ypεzdz−1,y∈ℝ.
Then,(26)pt2HKy=12πtHKexp−y22t2HK,x∈ℝ,which is a density function of the bifractional Brownian motion BtH,K and fεBtH,K⟶signBtH,K in L2Ω as ε⟶0. By Stroock’s formula, one can obtain(27)fεBtH,K=∑m=0∞amεt∫0<s1<⋯<sn<tdBs1H,K,…,dBsnH,K,where(28)amεt=EDmfεBtH,K=2Epεm−1BtH,K=2−1m−1∂m−1∂zm−1EpεBtH,K−z|z=0=2−1m−1pε+t2HKm−10.
Asε⟶0, by taking the limit of (27) in the space L2Ω, one can obtain(29)signBtH,K=∑m=0∞ant∫0<s1<⋯<sn<tdBs1H,K,…,dBsnH,K,where amt=limε⟶0amεt=2−1m−1pt2HKn−10, which implies(30)amt=0,n=2k,2−1k2k!2πtnHk!2k,n=2k+1.
The proof is completed.
In this paper, the notationF≍G implies that there are two positive constants c1 and c2 such that(31)c1Gx≤Fx≤c2Gx,where C denotes a generic positive constant and F and G have the common domain.Proposition 3.
The random variablesignBtH,K belongs to the Sobolev–Watanabe space Dα,2 for any t≥0 and α<1/2.Proof.
By Stirling’s formula(32)limk⟶∞k!kk+1/2e−k=2π,we have(33)EI2m+1b2m+12=2m+1!b2m+1,b2k+1ℋ⊗2m+1=2m+1!4t2HK2m+12m+122πt2m+12HKm!2m2≍Cm−3/2.
The proof is completed.Proposition 4.
For anyt≥0, one has(34)∫0tsignBsH,KdBsH,K=∑m=1∞cmI2mh2m,where cm=−1m−12π2m−1m−1!2m−2−1 and(35)h2mB1,…,B2m=B1∨B2∨⋯∨B2m−2m−1HK.The above proposition is the chaos expansion of∫0tsignBtH,KdBtH,K and implies the following result, which can be proved by the method similar to Proposition 3.Proposition 5.
For anyα<1/2 and t≥0, the random variable ∫0tsignBsH,KdBsH,K belongs to the Sobolev–Watanabe space Dα,2. Now, we consider the stochastic process X:(36)ρn≔EXa+1−XaXn+1−Xn,where 0<a≤n.Definition 1.
We say a stochastic processXtt≥0 is long-range dependent (resp. short-range dependent) if for each a>0,(37)∑n≥aρn=∞,resp.∑n≥aρn<∞.Theorem 1.
The stochastic processX of (21) is short-range dependent. Before proving this theorem, a lemma given by Yan et al. [9] is stated.Lemma 1.
Let0≤r<s and 0<HK<1, one defines(38)ρr,s2≔s2HKr2HK−μ2,where μ=EBsH,KBrH,K. Then, we have(39)ρr,s2≍s−r2HKr2HK.Remark 1.
The proof of estimate (39) uses the following two inequalities:(40)1+xα≤1+2α−1xα,0≤x,α≤1,(41)u+v−1K≤uK+vK−1≤u+v−1K+uvK,0≤K≤1,where 0≤u,v≤1 and u+v≥1. It is not difficult to prove inequality (40), which is stronger than the well-known inequality(42)1+xα≤1+αxα≤1+xα,because 2α−1≤α for all 0≤α≤1.Proof of Theorem 1.
For0<a<k, one can obtain(43)ρk=E∫aa+1signBtH,KdBtH,K∫kk+1signBtH,KdBtH,K=∫aa+1∫kk+1ϕt,sEsignBsH,KsignBtH,Kdsdt+∫aa+1∫kk+1∫0s∫0tϕs,uϕt,vEδBsH,KδBtH,Kdudvdsdt=∫aa+1∫kk+1ϕt,sEsignBsH,KsignBtH,Kdsdt+4HK2∫aa+1∫kk+1ψ1s,tψ2s,tEδBsH,KδBtH,Kdsdt=∫aa+1∫kk+1ϕt,sEBsH,KBsH,KBtHKBtHKdsdt+4HK2∫aa+1∫nn+1ψ1s,tψ2s,tEδBsH,KδBtH,Kdsdt≍∫aa+1∫kk+1EBsH,KBsH,KBtHKBtHKϕt,sdsdt+4HK2∫aa+1∫kk+1EδBsH,KδBtH,K⋅s2HK−1+t−s2HK−1t2HK−1+t−s2HK−1dsdt≡ak+bk.
Now, we only need to estimateak and bk. For ak, by the orthogonal decomposition,(44)BtH,K=dθHs,tθHs,sBsH,K+βs,tN,where(45)βs,t2=ρs,t2s2HK,in which N∈N0,1 independent of BsH,K. Set(46)λs,t=θHs,tβs,ts2HK=1/2Kt2H+s2HK−t−s2HKsHKρs,t.
Since(47)1+x2HK−1−x2HK∼x2HK,x⟶0. By Lemma 1, we obtain(48)λs,t=θHs,tβs,ts2HK∼CHt−HK,as t⟶∞ and s∈0,1, which implies(49)EBsH,KBtH,KBsH,KBtH,K=EBsH,Kλs,tBsH,K+NB1H,Kλs,tBsH,K+N,=EB1H,KsH,Kλs,tB1H,K+NB1H,KsH,Kλs,tB1H,K+N=Ot−HK.
So, the termak behaves as k2HK−3ok−HK. Now, we evaluate the second term bn. For s<t, using Lemma 1, one can obtain(50)EδBsH,KδBtH,K=∫R2hy,zδyδzdydz=h0,0=12πρt,s≍s−HKt−s−HK,where hy,z is the density function of BsH,K,BtH,K. So,(51)bk=4HK22π∫aa+1∫kk+1s2HK−1+t−s2HK−1t2HK−1+t−s2HK−1stHK−1/41/2Kt2H+s2HK−t−s2HK21/2dsdt≍∫aa+1∫kk+1a+12HK−1+k+12HK−1k−a−12HK−1akHK−1/41/2Kk+12H+a+12HKK−k−a2HK21/2dsdt≍k3HK−3.
The proof is completed.


## 4. Case of Multidimension

We now consider thed-dimensional bifractional Brownian motion B=Bt1,…,Btdt≥0 with the index HK≥1/2, which implies the components Bi,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, Di and δi, with respect to each component Bi. Suppose that Di1,pℋ are the associated Sobolev spaces. Similarly, LHK,i1,p denotes the set of processes u in Di1,pℋ which satisfies(52)uLHK,i1,pp≔EuL1/HK0,Tp+EDiuL1/HK0,T2p<∞.Let(53)Rt=Bt=Bt12+⋯+Btd2be a bifractional Bessel process. In the following, we research the stochastic process:(54)Xt=∑i=1d∫0tBsiRsdBsi.The next theorem can be proved similar to Es-Sebaiy and Tudor [3].Theorem 2.
LetBt=Bt1,Bt2,…,Btd be a d-dimensional bifractional Brownian motion with 2HK>1 and f be a function of class C2ℝd. Then,(55)fBt=f0+∑i=1d∫0t∂f∂xiBsdBsi+HK∑i=1d∫0t∂2f∂xi2Bss2HK−1ds.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 thatR=Rtt≥0 is a bifractional Bessel process associated to the d-dimensional bifractional Brownian motion with index HK>1/2. For each i=1,…,d, one can obtain Bsi/Rss∈0,T∈LH,i1,1/HK and(56)Rt=∑i=1d∫0tBsiRsdBsi+HKd−1∫0tBs2HK−1Rsds.Proof.
Step1. We prove ∫0tBsi/RsdBsi are well defined which only proves Bsi/Rss∈0,T∈LHK,i1,1/HK for each i=1,…,d. Since Bsi/Rs≤1 for each i=1,…,d, one can obtain(57)E∫0TBsiRs1/HKds<∞,i=1,…,d.Together with the definition of the derivative operator and the self-similarity of the bifractional Brownian motion, one can obtain(58)E∫0T∫0TDsiBriRr1/HKdsdr=∫0TrE1Rr−Bri2Rr31/HKdr≤∫0TrERr−1/HKdr=TER1−1/HK=C∫0∞12πd/2e−u2/2ud−1−1/HKdu<∞,sinced−1−1/HK>−1. So, the integral ∫0tBsi/RsdBsi is well defined since Bsi/Rss∈0,T∈LH,K,i1,1/HK for each i=1,…,d.Step 2. We now prove (56). Note that f:ℝd⟶ℝ is defined by(59)fx=x12+⋯+xd2,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(60)Rt2=Bt12+⋯+Btd2,then one can apply the Itô formula (55), and we have(61)Rt2=2∑i=1d∫0tBsidBsi+HKdt2HK.
Set(62)gεy=38ε+34εy−18εεy2,y<ε,x,y≥ε.
For anyε>0, gεy∈C2ℝ, and limε⟶0gεy=y for any x≥0. Applying (55) to gεRt2, we obtain(63)gεRt2=38ε+∑i=1dIi,ε+IIε+IIIε,where(64)Ii,ε≔∫0t1Rs2<ε12ε3−Rs2ε+1Rs2≥ε1RsBsidBsi,IIε≔HKd−1∫0t1Rs2≥ε1RsB2HK−1ds,IIIε≔HK∫0t1Rs2<ε12ε3d−d+2Rs2εB2HK−1d.
Together with∫0ts2HK−1/Rs<∞ a.s. and the bounded convergence theorem, one can obtain(65)limε⟶0IIε=d−1HK∫0t1Rss2HK−1ds.
For the third term, by the substitutingu=ρ/sH,K and Fubini’s theorem, we can obtain(66)0≤EIIIε≤3HKd2ε∫0tPRs2<εs2HK−1ds≤3HKd2ε∫0tPBs12+Bs22<εs2HK−1ds=3HKd2ε∫0t∫02π∫0ερ2πs2HKe−ρ2/2Bs2HKdρdθs2HK−1ds≤3d2ε∫0ε∫ρtHK∞1ue−u2/2duρdρ,that is,(67)limε⟶0EIIIε=0.
Finally, we show that(68)limε⟶0Ii,ε=∫0tBsiRsdBsiin L1/HKΩ. We have(69)Aε≔E∫0T1−Rs2ε3−Rs2ε1/HKBsiRs1/HK1Rs2<εds≤∫0TPRs2<εds≤∫0TPBs12+Bs22<εds=∫0T∫02π∫0ερ12πB2HKe−ρ2/2B2HKdρdθds⟶0,as ε⟶0. On the other hand, one can obtain(70)Bε≔E∫0T∫0TDBiBriRr−Bri2ε3−Rr2ε1Rr2<ε1/HKdsdr=∫0TrE1Rr−Bri2Rr3−32ε+Rr2+2Bri22εε1/HK1Rr2<εdr≤∫0TrE1Rr+32ε+3Rr22εε1/HK1Rr2<εdr≤41HK∫0TER1−1/HK1R12<ε/r2HKdr.
The distribution ofB11,…,B1d in spherical coordinates yields(71)∫0TER1−1/HK1R1<ε/rHKdr≤Cd,H,K∫0T∫0ε/rHKe−u2/2lud−1−1/HKdudr,where Cd,H,K>0 is a constant which depends on H,K, and d. For any r∈0,T,(72)∫0ε/rHKe−u2/2lud−1−1/HKdu⟶0,ε⟶0,∫0ε/rHKe−u2/2lud−1−1/HKdu≤∫0∞e−u2/2lud−1−1/HKdu<∞.
We have(73)limε⟶0∫0TER1−1/HK1R1<ε/rHKdr=0,by the bounded convergence theorem, that is,(74)limε⟶0Bε=0.
This proves the desired convergence (68), and the proposition follows.Proposition 7.
Stochastic processX which is given by (54) is HK-self-similar.Proof.
Seta>0. Together with the HK-self-similarity property of the bifractional Brownian motion and (56), we can obtain(75)Xat=Rat−HKd−1∫0atBs2HK−1Rsds=Rat−HKd−1a2HK∫0tu2HK−1Raudu=daHKRt−HKd−1aH∫0tu2HK−1Rudu=aHKXt.
Forh∈ℋ⊗n, we denote(76)Ii1,…,ikh=∫0<s1,…,sk<ths1,…,skdBs1i1,…,dBsnin,1≤i1,…,ik≤d.Theorem 3.
Supposefj:ℝd⟶ℝ, j=1,2,…,d are with polynomial growth and smooth functions. Then, the stochastic process Zt=∑j=1d∫0tfjBsdBsj has the following chaos expansion:(77)Zt=∑j=1d∑k=1∞∑1≤i1,…,ii≤dIi1,…,ik,jigi1,…,injB1,…,Bk+1,where(78)gi1,…,ikjB1,…,Bk+1=−1nB1∨⋯∨Bi+1−kHK2πd/2×∫ℝd∂k∂yi1,…,∂yike−y2/2fiyB1∨⋯∨Bk+1HKdy.Proof.
For eachj=1,2,…,d, using Stroock’s formula, we can obtain(79)fjBs=∑k=0∞∑1≤i1,…,ik≤d1k!Ii1,…,infi1,…,inis10,s⊗n,where(80)fi1,…,ikjs=EDi1,…,DinfjBs=E∂nfj∂zi1,…,∂zikBs=12πlB2HKd/2∫ℝd∂nfi∂zi1,…,∂zinze−z2/2s2HKdz=l−nB−nH2πd/2∫ℝd∂kfi∂zi1,…,∂zinyBH,Ke−y|2/2dy=−1nB−nH2πd/2∫ℝdfjyBH,K∂n∂zi1,…,∂zine−y|2/2dy.
So,(81)Zt=∑j=1d∫0tfjBsdBsj,=∑i=1d∑k=0∞∑1≤i1,…,in≤dIi1,…,ik,isymmfi1,…,ikis10,s⊗n10,ts=∑i=1d∑k=0∞∑1≤i1,…,in≤dIi1,…,in,ifi1,…,ikiB1∨⋯∨Bk+1∏j=1k+110,tBj.
This completes the proof.
Letfjx=xj/x12+⋯+xd2; then, fjtx=fjx. So, for such fj, one can obtain(82)gi1,…,ikjB1,…,Bk+1=−1kl−kB1∨⋯∨Bk+1−kHK2πd/2×∫ℝd∂k∂yi1⋯∂yine−y2/2fjydy.
Then,Btj/Rt can be denoted by(83)BtjRt=∑k=0∞∑1≤i1,…,in≤d−1kt−kHK2πd/2×∫ℝd∂n∂yi1,…,∂yine−y2/2fjydy∫0<s1<⋯<sk<tdBs1i1,…,dBskin,and the chaos expansion of ∫0tBBi/RsdBsi is(84)∫0tBsiRsdBsi=∑n=1∞∑1≤j1,…,jk≤d−1k2πd/2∫ℝd∂k∂yj1,…,∂yjne−|y|22fiydy×∫0<s1,…,sn,sk+1<ts1∨⋯∨sk+1−kHKdBs1j1,…,dBskjndBsk+1i.
The theorem is proved.Theorem 4.
The stochastic processX is short-range dependent.Proof.
Let(85)ρk=E∑i=1d∫01BsiBsdBsi∑i=1d∫kk+1BsiBsdBsi.
For everyK≥1, by the formula, we can decompose ρk as(86)ρk=∑i,j=1d∫01∫kk+1EBsiBtjBsBtϕt,sdsdt+∑i,j=1d∫01∫kk+1∫0t∫0Bϕs,uϕt,vEDvjBsiBsDuiBtjBtdudvdsdt≡ρk,1+ρk,2.
Forρk,1, one can use the decomposition(87)Bt=dθHs,tθHs,sBs+βs,tN,where(88)βs,t2=ρs,t2s2HK,where N is independent of Bs and is denoted as ad-dimensional standard normal random variable. By Lemma 13 in [11],(89)EBs,BtBsBt=EBs,λs,tBs+NBsλs,tBs+N=EB1,sH,Kλs,tB1+NBssH,Kλs,tB1+N=sH,Kλs,tEB12N2−B1,N2BsN3+Ot−HK.
Thus,(90)EBs,BtBsBt≍sHKt−HK,which implies that the term ρk,1 behaves as kH−3.
Forρk,2, one has(91)ρk,2=HK2∑i,j=1d∫01∫kk+1EδijBs−BsiBsjBsH,K3δijBt−BtiBtj|Bt|3ψ1s,tψ2s,tdsdt=HK2∫01∫kk+1EdBsBt−|Bs|2Bs3Bt−|Bt|2BsBt3+Bs,Bt2Bs3Bt3ψ1s,tψ2s,tdsdt=HK2∫01∫nn+1EBs,Bt2Bs3Bt3+d−2BsBtψ1s,tψ2s,tdsdt.
Since(92)EBs,Bt2Bs3Bt3+d−2BsBt,behaves as Mt−HK as t⟶∞, where(93)M=EB1,N2B13N3+d−2B1N>0.
We see that the termρk,2 also behaves as kHK−3, and the theorem follows.


## 5. The Local Times of∫0tsignBsH,KdBsH,K

Now, we consider the local times of the stochastic processX=Xt,t≥0 defined by(94)Xt=∫0tsignBsH,KdBsH,K.Lemma 2.
Let1/2<HK<1. Then, for all t≥0, we have(95)signBtH,KDtXt≥0,a.s.Proof.
Using the Itô formula, one can obtain(96)Xt2=2∫0tXssignBsH,KdBsH,K+2∫0tsignBsH,KDsXsds,σt≡EXt2=2E∫0tsignBsH,KDsXsds=∫0t∫0sEsignBsH,KsignBsH,Kϕs,ududs,and note that the function t⟶σt is increasing since(97)EsignBsH,KsignBuH,K=∑m=1∞42m!1/2Kt2H+s2HK−t−s2HK2m+12m+122πk!2k2su2m+1.
So,(98)E∫t0tsignBsH,KDsXsds≥0for all t≥t0≥0. Now, let us prove signBtH,KDsXt≥0, a.s. We only need to show that(99)∫t0tsignBsH,KDsXsds,0≤t≤Tis nondecreasing. Let(100)Vt0t=∫tt0tsignBsH,KDsXsds,0≤t0≤t,Φx=1x<0,Φε∈C2R,I=ΦVt0t,Iε=ΦεVt0t,where Φεx=0 for x>ε and Φεx=1 for x<ε. Thus, one can obtain(101)IDsIε=0,I⋅Vt0t≤0.
So, for allt≥t0≥0, one can obtain(102)EIVt0t=E∫t0tIsignBsH,KDsXsds=limε⟶0E∫t0tIIεsignBsH,KDsXsds≥E∫t0tIsignBsH,KDsXsds≥0,which implies that I=0 and Vt0t≥0. Therefore,(103)∫t0tsignBsH,KDsXsds,0≤t≤Tis nondecreasing. This completes the proof.Theorem 5.
Let the stochastic processX=Xt,t≥0 be defined by(104)Xt=∫0tsignBsH,KdBsH,K,and let Φ:R+⟶R be a convex function with polynomial growth. Then, there is a continuous increasing process AΦ which satisfies(105)ΦXt=Φ0+∫0tD−XssignBsH,KdBsH,K+12AtΦ,where D−Φ denotes the left-hand derivative of Φ.Proof.
IfΦ∈C2, then this is the Itô formula, and(106)AtΦ=∫0tΦ″XssignBsH,KDsXsds,together with Lemma 2, implies that the stochastic process AΦ is increasing.
Now, letΦ∉C2. For x∈R and ε>0, one sets(107)pεx=12πεe−1/2εx2,Φεx=∫Rpεx−yΦydy.
It is easy to see thatΦε∈C2 and has polynomial growth. So, for all ε>0, there exists a continuous increasing process AΦε such that(108)ΦεXt=Φε0+∫0tΦε′XssignBsH,KdBsH,K+12AtΦε,AtΦε=∫0tΦε″XssignBsH,KDsXsds=∫RΦε″x∫0tδXs−xsignBsH,KDsXsdsdx.
Note that(109)limε↓0Φεx=Φx,limε↓0Φε′x=D−Φx,and one can obtain as ε⟶0(110)∫0tΦε′XssignBsH,KdBsH,K⟶∫0tD−XssignBsH,KdBsH,K,in probability. So, AΦε converges to a stochastic process AΦ which, as a limit of increasing stochastic processes, is itself an increasing stochastic process and(111)ΦXt=Φ0+∫0tD−XssignBsH,KdBsH,K+12AtΦ,where AΦ can be chosen to be a.s. continuous. The proof is completed.Corollary 1.
For the processXt=∫0tsignBsH,KdBsH,K and all x∈R, there exists a local time ℒxX such that(112)Xt−x=x+∫0tsignXs−xdXs+ℒtxX.Proof.
Note that the left derivative of the functionΦy=y−x+ is equal to 1x,∞y. By Theorem 5, one can obtain(113)Xt−x+=−x∨0+∫0t1Xs>xsignBsH,KdBsH,K+12A+,where A+ is a continuous increasing stochastic process. Similarly, there exists a continuous increasing stochastic process A− which satisfies(114)Xt−x−=x∨0+∫0t1Xs>xsignBsH,KdBsH,K+12A−.
Therefore, one can obtain(115)Xt=∫0tsignBsH,KdBsH,K+12A++A−,which implies that A+=A− a.s. and we set ℒxxX=At+. This completes the proof.
Combining this corollary with Es-Sebaiy and Tudor [3], we can obtain the following results.Corollary 2.
Suppose thatℒxX is the local time of the process X and ℒxBH,K is the weighted local time of the bifractional Brownian motion BH,K defined by(116)ℒtxBH,K=2HK∫0tδBH,K−xs2HK−1ds.Then, we have(117)ℒtxX−ℒtxBH,K=Xt−x−BH,K−x+2∫0t1Xs≤xsignBsH,K−xdBsH,K.Proof.
By the Tanaka formula, one can obtain(118)Xt−x−BH,K−x=ℒtxX−ℒtxBH,K+∫0tsignXs−xsignBsH,K−xdBsH,K−∫0tsignBsH,K−xdBsH,K=ℒtxX−ℒtxBH,K+∫0tsignXs−x−1signBsH,K−xdBsH,K=ℒtxX−ℒtxBH,K−2∫0t1Xs≤xsignBsH,K−xdBsH,K,which implies that (117) holds.Corollary 3.
For anyt≥0 and x∈R, we have(119)ℒtxX=∫0tδXs−xsignBsH,KDsXsds.
Moreover, letΦ:R+⟶R be a convex function with polynomial growth; one can obtain the following Itô–Tanaka formula:(120)ΦXt=Φ0+∫0tD−ΦXssignBsH,KdBsH,K+12∫RℒtxXμΦdx,where D−Φ denotes the left derivative of Φ and signed measure μΦ which is defined by(121)μΦa,b=D−Φb−D−Φa,a<b,a,b∈R.
Finally, one can prove that the local time of the process(122)∑i=1d∫0tBsiRsdBsi,d≥2exists by the same method and can obtain the similar results.


## 6. Conclusions

This paper presents theorems and propositions associated with respect to the stochastic processRt=Bt12+⋯+Btd2, where B=Bt1,…,Btdt≥0 is a d-dimensional bifractional Brownian motion and 2HK≥1. Since there is no Lévy’s characterization theorem for a general bifractional Brownian motion, to prove whether a stochastic process(123)Xt≔∑i=1a∫0tBsiRsdBsi,d≥1is a bifractional Brownian motion or not is difficult. Theorems 1 and 4 prove Xt is short-range dependent in one-dimensional case and multidimensional case, respectively. Theorem 5 considers the local times of the stochastic process of Xt in one-dimensional case. Theorem 2 gives the the following chaos expansion of the stochastic process Zt=∑i=1d∫0tfiBsdBsi. Moreover, significance results associated with the above theorem are given.



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*Source: 7037602-2020-10-19.xml*