Fractional Brownian Bridge Measures and Their Integration by Parts Formula

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1 Journal of Mahemaical Reearch wih Applicaion Jul., 218, Vol. 38, No. 4, pp DOI:1.377/j.in: Hp://jmre.lu.eu.cn Fracional Brownian Brige Meaure an Their Inegraion by Par Formula Xiaoxia SUN School of Mahemaic, Dongbei Univeriy of Finance an Economic, Liaoning 11625, P. R. China Abrac In hi paper, we focu on he characerizaion for fracional Brownian brige meaure. We give he inegraion by par formula for uch meaure by Bimu meho an heir pull back formula. Converely, we prove ha uch meaure can be eermine hrough heir inegraion by par formula. Keywor fracional Brownian brige meaure; inegraion by par formula; characerizaion MR21 Subjec Claificaion 6G15; 6G18; 6H3 1. Inroucion A fracional Brownian brige i a kin of Gauian brige. Similarly o a Brownian brige, a fracional Brownian brige ha i anicipaive an non-anicipaive repreenaion which are uie in [1. In hi paper, we aim o characerize a fracional Brownian brige meaure hrough i inegraion by par formula. Since inegraion by par formula for meaure are imporan in ochaic analyi, a lo of inereing work ha been one on hee fiel. The inegraion by par formula wa inveigae for Wiener meaure on he pah pace in [2 4. For Brownian brige meaure on he loop pace, [5 gave he inegraion by par formula on loop group, in which he vecor fiel i C 1 ; [6,7 eablihe he inegraion by par formula for uch meaure over Riemannian manifol wih Levi-Civia connecion. For fracional Wiener meaure, [8, 9 gave i inegraion by par formula uner ifferen inegral. [1 eablihe he inegraion by par formula for fracional Ornein-Uhlenbeck meaure. Converely, i i ignifican o conier he characerizaion of meaure hrough heir inegraion by par formula. I i prove ha Gauian meaure can be characerize hrough heir inegraion by par formula. [11 howe ha he inegraion by par formula can characerize abrac Wiener meaure. [12 prove ha Wiener meaure can be characerize by heir inegraion by par formula on he pah pace. [1 gave he characerizaion for fracional Ornein-Uhlenbeck meaure. 2. Preliminarie Receive Sepember 19, 216; Accepe April 27, 218 Suppore by he Founaion of Liaoning Eucaion Commiee Gran No. LN217QN43 an he Naional Naural Science Founaion of China Gran No are: xiaoxiaun@ufe.eu.cn

2 Fracional Brownian brige meaure an heir inegraion by par formula 419 For a coninuou Gauian proce G aring a an uch ha EG =, [, 1, i aociae brige proce i efine by X = G G 1, 1. If he Gauian proce G i a fracional Brownian moion, X i calle a fracional Brownian brige. We e he loop pace on R n a follow Ω = {ω C[, 1; R n ω = ω1 = }. Le Ω, F, F, P be a filere probabiliy pace, where P i fracional Brownian brige meaure uch ha coorinae proce X 1 = ω 1 i a fracional Brownian brige, F 1 i he P -complee naural filraion of X 1, an F = F 1 i he P -compleion of he Borel σ-algebra of Ω. equaion By [1, fracional Brownian brige X 1 aifie he following inegral X = B H X + where B H i a fracional Brownian moion wih H > 1 2 an k, = c H 1 2 H u H 1 2 u H 3 2 u, Ψ, = inπh H 1 2 H π in which c H = he fracional Brownian brige i where We e φ, = k1, k, Ψ, ux u k1,, a u2 u u H+ 1 2 u H+ 1 2 u u, 2.2 H2H1 B22H,H 1 2. By [1, Propoiion 18, he non-anicipaive repreenaion of X = B H { u 1 + Ψv, k1, v 2 v k1, ww2 v φ, B H, a Ψu, } u k1, k1, uk, uu. v2 v L 2 Ω; P = {F F : Ω R, F 2 := E P F < }. By [8, he iomorphim operaor K : L 2 Ω; P I H L2 Ω; P i efine by Kh = k, h, where h L 2 Ω; P an I H L2 Ω; P i H Höler lef fracional Riemann-Liouville inegral operaor. The invere operaor of K i enoe by K 1. By [8, he Cameron-Marin vecor fiel on Ω i H = {Kh h i aape proce, h L 2 Ω; P an Kh 1 = },

3 42 Xiaoxia SUN wih calar prouc Kh, Kg H = h, g L2 Ω;P = E P [ The irecional erivaive of F along Kh i h, g. D h F ω = lim F ω + δkh F ω, δ δ 1 if he limi exi in L 2 Ω, P. Denoe by FC Ω he e of all he mooh cylinrical funcion on Ω, i.e., FC Ω = {F F ω = fω 1,..., ω n, < 1 2 n 1, f C R n }. For F FC Ω, he irecional erivaive of F i D h F ω = n i F, Kh i R n, i=1 where i F = i fω 1,..., ω n i he graien wih repec o he i-h variable of f. graien DF : Ω H i eermine by DF, Kh H = D h F. The omain of D i enoe by DomD. 3. Inegraion by par formula By Bimu iea [13, we nee o conruc a R n -value proce β uch ha for any r ϵ, ϵ, he following inegral equaion X r = B H r ha oluion X r 1 aifying where B H r i efine by B H r = X r + X r 1 Ω for any r, The k1, k, Ψ, ux u r k1,, 3.1 u2 u r X r r= exi an r X r r= = Kh, 3.2 k, B r = k, B + r K 1 β u u, 3.3 in which B i a R n -value Brownian moion. Noe ha X 1 = X 1 an B 1 = B 1. The following lemma give he expreion of β uch ha he oluion X r 1 of 3.1 aifie 3.2. Lemma 3.1 If he oluion X r 1 of 3.1 aify 3.2, hen β = Kh + Kh + k1, k, Ψ, ukh u k1, u2 u.

4 Fracional Brownian brige meaure an heir inegraion by par formula 421 Proof Differeniaing 3.1 wih repec o r a r =, we obain r X r r= = r BH r r= r X r + r= Ψ, u r X ur k1, k, r= k1, u2 u. By 3.2 an 3.3, we ge which implie ha β = Kh + Thi complee he proof. r X r r= = Kh, Kh + r BH r r= = β, k1, k, Ψ, ukh u k1,. 3.4 u2 u We give he inegraion by par formula for fracional Brownian brige meaure. Theorem 3.2 For any T, 1, F DomD F T an Kh H, he inegraion by par formula for he fracional Brownian brige meaure P i E P [F K 1 β, B = E P [D h F, where B i a R n -value Brownian moion an β = Kh + Kh + Proof We e { ρ = exp r For H > 1 2, by 3.4, we have I follow ha K 1 β 2 2 K 1 β = h + Kh + h k1, k, Ψ, ukh u k1, u2 u. K 1 β, B r2 2 Kh 2 1 k1, u2 u + 4 By he efiniion of k in 2.2, we obain ha c H H 1 2 } K 1 β 2. k1, Ψ, ukh u k1, u2 u. Ψ, ukh u 2 k 2 1, k1, u2 u H 1 c H 2 k1, H H 1 H Since Kh i H-Höler coninuou an Kh 1 =, here i a conan C K uch ha Due o Kh C K 1 H h 2 Kh = c H u H u H 1 2 u H 3 2 h u,. 3.7

5 422 Xiaoxia SUN we have Kh = c H 1 2 H H 1 2 H 3 2 h. 3.8 Suppoe ha h i a boune aape proce. By , here i a conan C 1 uch ha Kh 2 lim 1 lim 1 k1, u2 u Kh 2 1 k1, + u2 u C 2 K 1 1 2H + 1 c 2 H 12H 2HH By 2.2, here i a conan C Ψ uch ha 2Kh Kh k1, u2 u 2C K 1 H c H 1 2 H H 1 2 H 3 2 C c H 12H 2HH Ψ, C Ψ 1 2 H 1 2 H 1 H By 3.8 an 3.1, i i eay o check ha here i a conan C 2 uch ha Ψ, uc H 1 2 H u H 1 2 u H 3 2 uh 2 k 2 1, k1, u2 u 2 C By 3.5, 3.9 an 3.11, ρ 1 i a uniformly inegrable maringale for any r ϵ, ϵ on Ω, F, F, P ue o Novikov crierion. Noe ha B r = B + r K 1 β u u. By Giraonv heorem, we conclue ha B r 1 i a Brownian moion for any r ϵ, ϵ uner ρ 1 P. Thu by [14, Theorem 2, B H r 1 i a fracional Brownian moion uner ρ 1 P. Since X = B H X r = B H r X + X r + Ψ, ux u k1, k, k1, u2 u, k1, k, Ψ, ux u r k1, u2 u, we conclue ha X r 1 an X 1 have he ame iribuion uner ρ 1 P an P, repecively, ha i, for any cylinrical funcion F = fx 1,..., X n, E ρ1 P [fx 1 r,..., X n r = E P [fx 1,..., X n. Differeniaing he above equaion wih repec o r, we obain r E P [ρ 1 fx 1 r,..., X n r [ r= = E P F Thu for any aape boune proce h, we ge E P [F K 1 β, B + E P [D h F =. K 1 β, B = E P [D h F. 3.12

6 Fracional Brownian brige meaure an heir inegraion by par formula 423 I i obviou ha K 1 β L 2 Ω; ν for h L 2 Ω; ν, hen 3.12 hol for any h L 2 Ω; ν. Moreover, ince D i a cloable operaor, inegraion by par formula 3.12 hol for any F DomD F T. The proof i complee. 4. Characerizaion of fracional Brownian brige Nex, we how ha a fracional Brownian brige meaure can be characerize hrough i inegraion by par formula. Suppoe ha Y i a emi-maringale an Y = Γ B + L, where Γ i a R n R n -value coninuou proce, B i a R n -value Brownian moion an L i a R n -value coninuou boune quaraic variaion proce. Theorem 4.1 Le Ω, F, F, µ be a probabiliy pace. If µ i a probabiliy meaure uch ha 1 Coorinae proce X aifie X = Y H X + k1, k, Ψ, ux u k1, u2 u, where Y H = k, Y ; 2 For any T, 1, F DomD F T an Kh H = {Kh h i aape proce, h L 2 Ω; µ an Kh 1 = }, i hol ha [F K 1 β, Y = [D h F, 4.1 where K 1 β = h + Kh + hen µ i a fracional Brownian brige meaure. k1, Ψ, ukh u k1, u2 u, Proof I uffice o prove ha Y i a Brownian moion. We eablih he proof in wo ep. 1 Le F = 1. By 4.1, we have which implie [F Coniering inegral equaion K 1 β = L, ha i I oluion i which implie Kh + Kh + Kh = KL K 1 β, Y =, K 1 β, L =. 4.2 Ψ, ukh u k1, k, k1, u2 u = KL. φ, KL, h = L K 1 φ, KL.

7 424 Xiaoxia SUN By he efiniion of iomorphim operaor K, i hol ha Hence, = = = φ, KL K { u 1 + Ψv, k1, v 2 v k1, v w2 w 2 u { u 1 + Ψv, k1, v 2 k, u v k1, v w2 w 2 { u u 1 + Ψv, k1, v 2 v k1, v w2 w 2 h = L 1 + Ψv, k1, v 2 v k1, w2 w 2 v Therefore, if we le h equal o 4.3, Eq. 3.2 i 1 + Ψu, } u k1, k1, uk, uu KL v2 v 1 + Ψu, } u k1, k1, ukl v2 u v 1 + Ψu, } u k1, k1, ukl v2 v. L, L =, 1 + Ψ, k1, k1, KL. 4.3 v2 v which yiel ha L = for any [, T. Due o he coninuiy of L in [, 1, we obain ha L = for any [, 1. 2 For an orhogonal bai {e i : i = 1,..., n} on R n, le F = Y T, e i. We give he erivaive of Y T, e i in wo way. Conier he following equaion X r = Y H r where Y H r i efine by X r + Y H r = Y H + rα, k1, k, Ψ, ux u r k1, u2 u, in which α i a R n -value aape proce. If he oluion aifie Kh = r Y r r=, we obain α = β = Kh + By he efiniion of Y H, we have which yiel ha Hence Y H r = Kh + k, Y r = Y r = Y + r k1, k, Ψ, ukh u k1,. 4.4 u2 u k, Y + r D h Y T, e i = r Y T r, e i r= = K 1 β. K 1 β u u, K 1 β, e i. 4.5

8 Fracional Brownian brige meaure an heir inegraion by par formula 425 Le F = Y T, e i. By 4.1, we have [D h Y T, e i = Γ e i, B K 1 β, Y = Γ Γ e i, K 1 β. 4.6 Noe ha Γ i a R n R n -value coninuou proce. Combining 4.5 an 4.6, we obain Γ Γ Ie i, K 1 β =. 4.7 By 4.4, we ge Γ Γ Ie i, K 1 β = Γ Γ k1, Ie i, h + Kh + Ψ, ukh u k1, u2 u = Γ Γ Ie i, h + P, Kh + Ψ, ukh u = Γ Γ Ie i, h + P, Kh + P, Ψ, ukh u, 4.8 where k1, P = k1, u2 u Γ Γ Ie i. The econ erm of 4.8 i P, Kh = P, c H 1 2 H u H 1 2 u H 1 2 uh u = c H P 1 2 H = c H P 1 2 H The hir erm of 4.8 i Ψ, up, Kh u u u = Ψ, up, c H u = c H 1 2 H u H 1 2 u H 3 2 = c H 1 2 H u H 1 2 u H 3 2 = c H P 1 2 H u H 1 2 u H 1 2 u, h u H 1 2 u H 1 2 u, h H u H 1 2 u H 3 2 h u u u Ψ, up u, h Ψ, up u, h u H 1 2 u H 3 2 Ψ, uu, h. 4.1

9 426 Xiaoxia SUN By , for any h, we obain Γ Γ Ie i + c H P 1 2 H Thu [Γ Γ Ie i + which implie ha u H 1 2 u H Ψ, uu, h =. c H P 1 2 H u H 1 2 u H Ψ, uu F =, Γ Γ Ie i + c H P 1 2 H u H 1 2 u H Ψ, uu F =. Le en o T. We ge Γ T Γ T Ie i =. Since Γ i coninuou in [, 1, Γ T Γ T = I for any T [, 1. Therefore, Y i a Brownian moion. The proof i complee. Acknowlegemen We hank he referee for heir ime an commen. Reference [1 D. GASBARRA, T. SOTTINEN, E. VALKEILA. Sochaic Analyi an Applicaion. Springer, Berlin, 27. [2 B. K. DRIVER. A Cameron-Marin ype quai-invariance heorem for Brownian moion on a compac Riemannian manifol. J. Func. Anal., 1992, 112: [3 Shizan FANG, P. MALLIAVIN. Sochaic analyi on he pah pace of a Riemannian manifol. I. Markovian ochaic calculu. J. Func. Anal., 1993, 1181: [4 E. P. HSU. Quai-invariance of he Wiener meaure on he pah pace over a compac Riemannian manifol. J. Func. Anal., 1995, 1342: [5 B. K. DRIVER. A Cameron-Marin ype quai-invariance heorem for pinne Brownian moion on a compac Riemannian manifol. Tran. Amer. Mah. Soc., 1994, 3421: [6 O. ENCHEV, D. W. STROOCK. Pinne Brownian moion an i perurbaion. Av. Mah., 1996, 1192: [7 E. P. HSU. Inegraion by par in loop pace. Mah. Ann., 1997, 392: [8 L. DECREUSEFOND, A. S. ÜSTÜNEL. Sochaic analyi of he fracional Brownian moion. Poenial Anal., 1999, 12: [9 T. E. DUNCAN, Yaozhong HU, B. PASIK-DUNAN. Sochaic calculu for fracional Brownian moion. I. Theory. SIAM J. Conrol Opim., 2, 382: [1 Xiaoxia SUN, Feng GUO. On inegraion by par formula an characerizaion of fracional Ornein- Uhlenbeck proce. Sai. Probab. Le., 215, 17: [11 H. SHIH. On Sein meho for infinie-imenional Gauian approximaion in abrac Wiener pace. J. Func. Anal., 211, 2615: [12 E. P. HSU. Characerizaion of Brownian Moion on Manifol Through Inegraion by Par. Singapore Univ. Pre, Singapore, 25. [13 J. M. BISMUT. Large Deviaion an he Malliavin Calculu. Birkhäuer, [14 D. NUALART, Y. OUKNINE. Regularizaion of ifferenial equaion by fracional noie. Sochaic Proce. Appl., 22, 121: