U T,0. t = X t t T X T. (1)

Transcription

1 Gauian bridge Dario Gabarra 1, ommi Soinen 2, and Eko Valkeila 3 1 Deparmen of Mahemaic and Saiic, PO Box 68, 14 Univeriy of Helinki,Finland dariogabarra@rnihelinkifi 2 Deparmen of Mahemaic and Saiic, PO Box 68, 14 Univeriy of Helinki, Finland, ommioinen@helinkifi 3 Iniue of Mahemaic, PO Box 11, 215 Helinki Univeriy of echnology, Finland, ekovalkeila@hufi Summary We conider Gauian bridge; in paricular heir dynamic repreenaion We prove a Giranov heorem beween he law of Gauian bridge and he original Gauian proce, which hold wih naural aumpion Wih ome addiional condiion we obain dynamical repreenaion for a Gauian bridge We dicu briefly he iniial enlargemen of filraion in hi conex Mahemaic Subjec Claificaion 2: 6G15; 6G18, 6G25, 6G44 Key word and phrae: Gauian procee; Brownian bridge; pinned Gauian procee; ied down Gauian procee; enlargemen of filraion; fracional Brownian moion; fracional Brownian bridge 1 Inroducion Moivaion Le X be a coninuou Gauian proce uch ha X and EX Fix > and define he bridge of X U, by U, X X 1 I i clear ha he proce U, i a Gauian proce Moreover, i i a bridge in he ene ha U, U, If X i a andard Brownian moion, hen i i known ha he law of he proce U, defined by 1 i he ame a he condiional law of he andard Brownian moion: P Law X X P Law U,

2 2 Dario Gabarra, ommi Soinen, and Eko Valkeila I i well-known ha in he cae of andard Brownian moion he bridge proce U, ha a repreenaion a oluion o he differenial equaion 2 We refer o he nex ubecion for more informaion on Brownian bridge We udy he properie of he bridge proce U, of X in he cae of arbirary Gauian proce X We define he bridge proce uing he condiional law of X I urn ou ha i i quie eay o obain he analog of 1 for he arbirary Gauian proce X; ee Propoiion 4 for he exac reul If he Gauian proce X i a maringale, hen i i quie eay o o decribe he bridge proce U, a a oluion o a differenial equaion analogou o 2 Bu if he proce X i a fracional Brownian moion, hen he correponding differenial equaion conain Volerra operaor Repreenaion for he Brownian bridge Fix > and le W W [, ] be a andard Brownian moion on a probabiliy pace Ω, F, P aring from W ξ Le, θ be a condiioning hen he noaion W,θ mean ha he proce W i condiioned o be θ a ime ha i W,θ i a bridge from, ξ o, θ For he Brownian bridge W,θ from, ξ o, θ one find in he lieraure he following hree equivalen definiion dy,θ Y,θ dw + θ Y,θ d, ξ + θ ξ + W,θ θ + W W Y,θ ξ, 2 dw, 3 4 he repreenaion 3 i ju he oluion of he ochaic or pahwie differenial equaion 2 So, he equaion 2 and 3 define he ame proce Y,θ he equaion 4, however, doe no define he ame proce a he equaion 2 and 3 he equaliy beween repreenaion 2 3 and 4 i only an equaliy in law: LawY,θ ; P LawW,θ ; P ha he procee Y,θ and W,θ are differen i obviou from he fac ha he proce Y,θ i adaped o he filraion of W while he proce W,θ i no Indeed, o conruc W,θ by uing 4 we need informaion of he random variable W he fac ha he wo procee Y,θ and W,θ have he ame law i alo obviou, ince hey have he ame covariance and expecaion I i alo worh noicing ha if he Brownian bridge Y,θ i given by he equaion 3 hen he original Brownian moion W may be recovered from he bridge W,θ by uing he equaion 2 In paricular, hi mean ha in hi cae he filraion of hi Brownian bridge i he ame a he filraion of he Brownian moion: F Y,θ F W

3 Gauian bridge 3 he non-adaped repreenaion 4 come from he orhogonal decompoiion of Gauian variable Indeed, he condiional law of proce W [, ] given he variable W i Gauian wih EW W W ξ + ξ, CovW, W W he econd-order rucure of he Brownian bridge i eaily calculaed from he repreenaion 4: Cov E W,θ W,θ, W,θ Giranov heorem and Brownian bridge ξ + θ ξ, 5 6 We know ha Brownian bridge i defined only up o diribuion Pu P,θ : LawW,θ ; P We have ha P,θ P W θ, where P i he law of he Brownian moion W Conider now he rericion of he meaure P,θ and P on he igma-algebra F W : denoe he rericion by P and P,θ We know ha P,θ P for all [,, bu, of coure, P θ P From 2 we ge, by Giranov heorem, ha dp,θ θ Y,θ exp dp dw 1 θ Y,θ 2 d 2 hi i a key obervaion for he non-anicipaive repreenaion in he general cae Non-anicipaive and anicipaive repreenaion Le now X X [, ] be a Gauian proce on Ω, F, P wih X ξ We wan o underand wha i he correponding bridge X,θ from, ξ o, θ If one merely replace he Brownian moion W wih he proce X in repreenaion 2 4 hen he X-bridge obained from he fir wo repreenaion of coure coincide However, he bridge obained from he la one doe no coincide wih he fir wo one he following example, communicaed o u by M Lifhi, elaborae hi poin Example 1 Le f n n 1 be a equence of mooh iomorphim of [, ] ono ielf ake X n, : W fn and e

4 4 Dario Gabarra, ommi Soinen, and Eko Valkeila hen X 1,,θ n, : θ + X n, X n,, X 2,,θ n, Cov n,1, : Cov : θ + X 1,,θ n,, Xn, 1,,θ dx n, f n + f n f n, Cov n,2, : Cov X 2,,θ n,, Xn, 2,,θ df n u u 2 he covariance Cov n,1 and Cov n,2 are no he ame in general Indeed, le f n 1 {1} hen for all, < 1 we have ha a n, Cov n,1, while Cov n,2, Srucure of he paper We will udy Gauian bridge Afer he definiion of Gauian bridge we obain he anicipaive repreenaion of he Gauian bridge, which i a generaliaion of he repreenaion 4 Nex we give he deniy beween he bridge meaure P,θ and he original meaure P and give an abrac verion of he non-anicipaive repreenaion 3 in he general eup In he ecion hree we udy bridge of Gauian maringale, and hi par i an eay generaliaion of he Brownian bridge In he nex ecion we udy bridge of cerain pecial Gauian procee: Wiener predicable proce, Volerra proce and fracional Brownian moion We end he paper by giving he connecion o he enlargemen of filraion heory, where he enlargemen i an iniial enlargemen wih he final value of he Gauian proce X 2 Gauian bridge in general 21 Definiion of he X-bridge he fac ha for Brownian moion he Brownian bridge in unique up o law only ugge he following definiion in he cae of an arbirary Gauian proce Definiion 2 Le X be a Gauian ochaic proce wih X ξ hen he Gauian proce X,θ i an X-bridge from, ξ o, θ if Law X,θ ; P Law X; P,θ, 1 where he meaure P,θ on Ω, F i defined by P,θ P X θ 2

5 Gauian bridge 5 Remark 3 he definiion above aume ha he proce X,θ exi in he original pace Ω, F, P Alo, we have 1 PX θ X θ P,θ X θ P X,θ θ, a we hould Noe ha in 2 we condiion on a e of zero meaure However, we can define 2 a a regular condiional diribuion in he Polih pace of coninuou funcion on [, ] ee Shiryaev [9, pp ] In wha follow we denoe by µ and R he mean and covariance of X, repecively 22 Anicipaive repreenaion he anicipaive repreenaion correponding o 4 i eaily obained from he orhogonal decompoiion of X wih repec o X Indeed, LawX X i Gauian wih EX X X µ R, R, + µ, CovX, X X R, R, R, R, hu, we have an anicipaive repreenaion for any Gauian bridge Propoiion 4 Le X be a Gauian proce wih mean µ and covariance R hen he X-bridge X,θ from, µ o, θ admi a repreenaion R, θ R, + X, X,θ R, θ R, + X R, R, X 3 Moreover, Cov E X,θ X,θ θ µ R, + µ, 4 R,, X,θ R, R, R, 5 R, Example 5 Le X be a cenered fracional Brownian moion he bridge proce Z : X X i a H- elf imilar proce, bu i i no a fracional Brownian bridge in he ene of definiion 2 he correc fracional Brownian bridge in he ene of he definiion 2 i X,θ X 2H + 2H 2H 2 2H X

6 6 Dario Gabarra, ommi Soinen, and Eko Valkeila X-bridge and drif Le a W i a Brownian moion wih drif a R, ie W : a W a i a andard Brownian moion aring from ξ hen from 4 i eay o ee ha he Brownian bridge i invarian under hi drif: a W,θ W,θ Conider now a general cenered Gauian proce X, and le µ be a deerminiic funcion wih µ Define µ X by µ X : X + µ ranform µ X o µ X,θ by 3 hen µ X i a Gauian proce wih he ame covariance R a X and wih mean µ When doe µ X,θ define he ame bridge a X,θ in he ene of Definiion 2? From 3 i follow ha an invarian mean funcion µ mu aify he equaion µ R, µ R, So, uch an invarian mean µ may depend on he ime of he condiioning Indeed, µ µ ar, for ome a R In paricular, we ee ha µ i independen of if and only if R, f for ome funcion f Bu hi mean ha X ha independen incremen, or in oher word ha X EX i a maringale X-bridge and elf-imilariy he Brownian moion W aring from W i 1/2-elf-imilar Ie Law W [, ] ; P Law 1/2 W τ ; P τ [,1] Conequenly, we have for he Brownian bridge he caling propery Law W,θ ; P Law 1/2 W ; P [, ] 1,θ 1/2 τ τ [,1] From 3 i i eay o ee ha if he proce X X [, ] i H-elfimilar, ie Law X [, ] ; P Law H X τ τ [,1] ; P hen he correponding bridge aifie he caling propery Law X,θ ; P Law H X [, ] 1,θ H τ τ [,1] ; P

7 Gauian bridge 7 So, we may repreen he bridge X,θ a X,θ where τ / [, 1] H 1,θ X R1, τ θ R1, 1 + H X τ R1, τ R1, 1 H X 1, 23 Deniy beween he bridge meaure P,θ and P When we look for analogie for he non-anicipaive, or dynamic, repreenaion 3 and he correponding differenial equaion 2, hen he main idea i o work wih he predicion maringale of X and o ue he Giranov heorem We inroduce ome noaion Le X,θ and P,θ be a in 1 and 2 Le ˆX ˆX [, ] be he predicion maringale of X Ie ˆX : E X F X For he incremenal bracke of he Gauian maringale ˆX we ue he horhand noaion ˆX, : ˆX ˆX : ˆX, ˆX ˆX, ˆX Noe ha ince ˆX i a Gauian maringale i ha independen incremen, and conequenly i bracke ˆX i deerminiic Denoe P : P F X and P,θ : P,θ F X Le α denoe he regular condiional law of X given he informaion F X and le α α be he law of X So, if p denoe he Gauian deniy pθ; µ, σ 2 1 2πσ e 1 2 θ µ σ 2, i i eay enough o ee ha α dθ p θ; ˆX, ˆX, dθ, α dθ p θ; µ, ˆX dθ Now, by uing Baye rule we have ha

8 8 Dario Gabarra, ommi Soinen, and Eko Valkeila dp,θ dp dα θ dα p θ; ˆX, ˆX, p θ; µ, ˆX ˆX ˆX exp 1 2 θ ˆX 2, ˆX θ µ 2, ˆX 6 Since we wan o ue he Giranov heorem laer we need o aume ha he predicion maringale ˆX i coninuou Anoher way of aing hi aumpion i he following: A he hiory of X i coninuou, ie F X F+; X here F < F and F + u> F u Alo, in order for he calculaion above o make ene we need he o aume ha P,θ P for all < Or, ince he boh meaure are Gauian, we may a well aume ha: A1 P P,θ for all < From equaion 6 we ee ha aumpion A1 ay ha ˆX < ˆX for all < So, anoher way of aing aumpion A1 i ha he value of X canno be prediced for cerain by uing he informaion F X only Indeed, ˆX, Var ˆX i he predicion error of ˆX Le u noe ha in general he meaure P and P,θ are of coure ingular, ince X,θ i degenerae a In wha follow β,θ i a non-anicipaive funcional acing on Gauian predicion maringale m : β,θ m : θ m m, he following propoiion i he key ool in finding a non-anicipaive repreenaion Propoiion 6 Le X be a Gauian proce on Ω, F, P aifying he aumpion A and A1 hen he bridge meaure P,θ on Ω, F may be repreened a dp,θ L,θ dp, where L,θ exp β,θ ˆX d ˆX 1 2 β,θ ˆX 2 d ˆX

9 Gauian bridge 9 Proof he claim follow from equaion 6 Indeed, ju ue Iô formula wih he maringale ˆX o he funcion g, x : 1 θ x 2 2 ˆX,, and here you have i 24 Non-anicipaive repreenaion In order o come back from he predicion maringale level o he acual proce we ill need one aumpion A2 he non-anicipaive linear mapping F ending he pah of he Gauian proce X o he pah of i predicion maringale ˆX i injecive he aumpion A2 ay imply ha he proce X may be recovered from ˆX by X F 1 ˆX Alo, noe ha he aumpion A2 implie ha he predicion filraion and he original filraion are he ame: F X F ˆX Le m be a Gauian maringale We denoe by S,θ m he unique oluion of he differenial equaion wih iniial condiion m,θ ζ, ie m,θ dm,θ dm + β,θ m,θ d m,θ 7 S,θ m ζ + θ ζ m,θ m,θ + m,θ, dm m,θ, In order o ee ha S,θ m i indeed he oluion o 7 one ju ue he inegraion by par I i alo worh noicing ha claical heory of differeniaion applie here: he differenial equaion 7 may be underood pahwie Finally, noe ha by he Giranov heorem he bracke of he Gauian maringale m and m,θ coincide: m m,θ Le u now abue he noaion lighly and e hen we have he decompoiion where S m : S, m S,θ m θk m + S m, K m : m m and S i independen of θ he following heorem i he analogy of he non-anicipaive repreenaion 3

10 1 Dario Gabarra, ommi Soinen, and Eko Valkeila heorem 7 Le X be a Gauian proce wih mean µ and covariance R aifying A, A1 and A2 hen he bridge X,θ from, µ o, θ admi he non-anicipaive repreenaion X,θ F 1 S,θF X 8 θ F 1 K F X + F 1 S F X 9 R, θ R, + X, 1 by Moreover, he original proce X may be recovered from he bridge X,θ X F 1 S 1,θ F X,θ 11 Proof Le u fir prove he equaion 8 1 By he equaion 3 we already know he conribuion coming from θ Indeed, we mu have F 1 K R, F X θ R, So, we may aume ha θ and conider he correponding bridge X, Now, we map X o i predicion maringale F X hen S F X i he oluion of he ochaic differenial equaion 7 wih m F X and he iniial condiion ζ µ Conequenly, he Giranov heorem and Propoiion 6 ell u ha Law S F X ; P Law F X ; P,θ 12 So, he claim 8 follow imply by recovering he proce X by uing he map F 1 on boh ide of he equaion 12 he equaion 11 i now obviou, ince S,θ i inverible Indeed, S 1,θ F X,θ S 1,θ hi finihe he proof ˆX,θ ˆX,θ + β ˆX,θ d ˆX,θ Remark 8 For he differenial equaion 2 have he following formal analogy X,θ X + F 1 θ F X,θ u d F X F X u ;,u In he following ecion we conider ome pecial Gauian bridge and give he omewha abrac heorem 7, and highly abrac Remark 8, more concree form In paricular, we conider cae where he operaor F and F 1 may be repreened a Wiener inegral

11 Gauian bridge 11 3 Bridge of Gauian maringale he cae of Gauian maringale i exremely imple Indeed, he analogy o he Brownian cae i complee Propoiion 9 Le M be a coninuou Gauian maringale wih ricly increaing bracke M and M ξ hen he M-bridge M,θ admi he repreenaion dm,θ M,θ M,θ Moreover, we have dm + θ M,θ d M M, M,θ ξ, 1, ξ + θ ξ M + M M, θ M + M M M M M dm M,, 2 3 CovM,θ EM,θ ξ + θ ξ M M,, M,θ M M M M Proof Since M i coninuou and M i ricly increaing he aumpion A and A1 are aified he aumpion A2 i rivial in hi cae Now, he oluion of 1 i 2 and hi i ju he equaion 8 where F i he ideniy operaor Repreenaion 3 a well a he mean and covariance funcion come from he repreenaion 3 Indeed, for Gauian maringale we have R, M Remark 1 Acually, one can deduce he reul of Propoiion 9 wihou uing he Baye Iô Giranov machinery inroduced in Secion 2 Indeed, he reul follow quie eaily from equaion 2 4 and he repreenaion of he Gauian maringale M a he ime-changed Brownian moion W M 4 Bridge of Wiener predicable procee Le u fir conider abrac Wiener-inegraion wih repec o Gauian procee he linear pace H of a Gauian proce X i he cloed Gauian ubpace of L 2 Ω, F, P generaed by he random variable X, For he predicion maringale of X i i well known ha ˆX H Le E denoe he pace of elemenary funcion over [, ] equipped wih he inner produc generaed by he covariance of X : 1[,, 1 [,u : R, u,

12 12 Dario Gabarra, ommi Soinen, and Eko Valkeila Le Λ be he compleion of E in he inner produc, Now he mapping I : 1 [, X exend o an iomery beween Λ and H We call hi exenion he abrac Wiener inegral Ala, he pace Λ i no in general a pace of funcion or more preciely a pace of equivalence clae of funcion However, we can find a ubpace of i whoe elemen may be idenified a equivalence clae of funcion Viz he pace Λ which coni of uch funcion f ha up π i, j π f i 1 f j 1 1[i 1, i, 1 [j 1, j < Here he upremum i aken over all pariion π of he inerval [, ] he reaon o ake a upremum ined of leing he meh of he pariion go o zero i ha he, -norm of a funcion may increae when muliplied by an indicaor funcion For deail of hi phenomenon in he cae of fracional Brownian moion ee Bender and Ellio [1] If f Λ hen we wrie f dx : I [f] 1 So, he Wiener inegral 1 of a funcion f Λ i defined a a, -limi of imple funcion Noe ha if hen Λ Λ and I [f] I [ f1[, ] for f Λ Since he operaor F i linear and non-anicipaive we have ˆX I [p, ] for ome p, Λ We aume now ha hi predicion kernel p, i acually a funcion in Λ : A3 here exi a Volerra kernel p uch ha p, Λ for all and m may be repreened a he Wiener inegral ˆX p, dx 2 Repreenaion 2 ugge ha, if we are lucky enough, he invere operaor F 1 may be repreened a a Wiener inegral wih repec o ˆX hi i he meaning of he nex aumpion we make A4 here exi a Volerra kernel p uch ha he original Gauian proce X may be reconruced from he predicion maringale m a a Wiener inegral X p, d ˆX 3

13 Gauian bridge 13 Remark 11 he Wiener inegral in A4 may underood a an abrac Wiener inegral or, a well, a he ochaic inegral wih repec o he maringale m Indeed, in hi cae Λ i he funcion pace L 2 [, ], d ˆX Alo, aumpion A4 give u an alernaive way of defining he Wiener inegral 2 Indeed, le he operaor P be he linear exenion of he map 1 [, p, hen he aumpion A3 may be reaed a: he operaor P ha he indicaor funcion 1 [,,, ], in i image In hi cae we may define Wiener inegral wih repec o X a f dx : P [f] d ˆX for uch f ha P [f] L2 [, ], d ˆX Moreover, in hi cae p, P 1 [ 1 [, ] Indeed, hi i he approach aken in he nex ecion Remark 12 Obviouly A4 implie A2 Alo, we have implicily aumed ha X i cenred wih X However, adding a mean funcion o X caue no difficulie Indeed, le m be he predicion maringale of he cenred proce X µ and le p and p be he kernel aociaed o hi cenred proce hen ˆ X ˆX µ, X ˆX p, d ˆX + µ, p, dx µ + µ Remark 13 he relaion 3 ay ha he covariance R of X may be wrien a R, p, up, u d ˆX u 4 So, p i a quare roo of R Noe, however, ha in general a decompoiion like 4 i by no mean unique, even if he meaure i given hi mean ha from an equaion like 4 we canno deduce he kernel p even if we knew he meaure d ˆX induced by he bracke ˆX We have he following analogue of repreenaion 2 and 3 Propoiion 14 Le X be a Gauian proce wih covariance R aifying A, A1, A3 and A4 hen he bridge X,θ aifie he inegral equaion

14 14 Dario Gabarra, ommi Soinen, and Eko Valkeila { } X,θ p X + θ p, u dxu,θ, ˆX d ˆX 5, Moreover X,θ admi he non-anicipaive repreenaion where φ, R, θ R, + X X,θ u φ, dx, 6 p v, ˆX 2 d ˆX v p u, ˆX,v p, u d ˆX u,u Remark 15 Noe ha unlike he equaion 2 and 1 he equaion 5 i no of differenial form Indeed, i i clear by now ha he differenial connecion i characeriic o he maringale cae Proof [Propoiion 14] Conider he predicion maringale ˆX Uing he relaion 7 ie wih 3 yield d X,θ X +,θ,θ ˆX d ˆX θ ˆX + ˆX d ˆX, { θ },θ p ˆX, ˆX d ˆX 7, he inegral equaion 5 follow now from 7 and 2 Le u now derive he non-anicipaive repreenaion 6 Inering he,θ oluion ˆX S,θ ˆX o he equaion 7 we obain X,θ X + p, ˆX, θ X + ˆX + { θ ˆX, ˆX + ˆX, p, d ˆX d ˆX u ˆX,u p, d ˆX : X + θf + Φ ˆX d ˆX u ˆX,u } d ˆX Noe now ha X ˆX p, d ˆX,

15 which implie ha p, 1 [, Conequenly, by 4 p, d ˆX and, ince ˆX R,, we have f Gauian bridge 15 p, p, d ˆX R,, R, R, a fac ha we acually knew already by 4 Now we wan o expre Φ ˆX in erm of X We proceed by inegraing by par: d ˆX u ˆX,u ˆX ˆX, ˆX u ˆX 2,u d ˆX u 8 Uing he aumpion A4 o 8 and changing he order of inegraion we obain d ˆX u 1 ˆX,u ˆX p, u dx u, hu, Φ ˆX p, u ˆX, p u, ˆX,u φ, dx 1 ˆX 2,u p, u ˆX, u u hi prove he decompoiion 6 u u p u, v dx v d ˆX u p v, u ˆX 2 d ˆX v dx u,v p v, u ˆX 2 d ˆX v dx up, d ˆX,v p v, ˆX 2 d ˆX v p, u d ˆX u dx,v 5 Bridge of Volerra procee he reul of he previou ecion i ill raher implici Indeed, we have no explici relaion beween he covariance R of X and he bracke ˆX of

16 16 Dario Gabarra, ommi Soinen, and Eko Valkeila he predicion maringale m Moreover, in general here i no imple way of finding, or even inuring he exience, of he kernel p and p In hi ecion we conider a model where hee connecion are clear, alhough he formula urn ou o be raher complicaed A5 here exi a Volerra kernel k and a coninuou Gauian maringale M wih ricly increaing bracke M uch ha X admi a repreenaion X k, dm 1 Remark 16 Since M i coninuou, M i alo coninuou Alo, if M i no ricly increaing on an inerval [a, b], ay, hen nohing happen on ha inerval Conequenly, we could ju remove i Remark 17 he connecion beween he covariance R and he kernel k i R, k, uk, u d M u 2 Moreover, if R admi he repreenaion 2 wih ome meaure d M, hen X admi he repreenaion 1 Now we define he Wiener inegral wih repec o X by uing he way decribed in Remark 11 Le K exend he relaion K : 1 [, k, linearly So, we have f dx g dm K[f] dm, 3 K 1 [g] dx for any g L 2 [, ], d M and uch funcion f ha are in he preimage of L 2 [, ], d M under K We need o have he invere K 1 defined for a large enough cla of funcion hu, we aume A6 For any he equaion ha a oluion in f A7 For any he equaion ha a oluion in g Kf 1 [, Kg 1 [, k, By he aumpion A6, we have a revere repreenaion o 1 Indeed,

17 where we have denoed M Gauian bridge 17 k, dx, 4 Since M i a maringale we have k, : K 1 [ 1 [, ] dm k, dm By aumpion A6 we have k, for < d M -almo everywhere and a M i ricly increaing alo d-almo everywhere So, we may wrie hu, X dm d ˆX k, k, k, d ˆX and we have he aumpion A4 aified wih p, k, k, Conequenly, he aumpion A2 i alo aified Alo, he aumpion A6 implie he aumpion A1, ince d ˆX k, 2 d M Indeed, hi implie ha ˆX i ricly increaing For he kernel p we find he repreenaion by uing he aumpion A7 a follow: ˆX X + k, dm K [ 1 [, ] dm K 1 [ 1 [, K [ 1 [, ]] dx {K 1 K [ 1 [, ] + K 1 [ 1 [, K [ 1 [, ]] } dx where we have denoed Ψ, dx,

18 18 Dario Gabarra, ommi Soinen, and Eko Valkeila So, we have found ha Ψ, : K 1 [ 1 [, K [ 1 [, ]] 5 d ˆX k, 2 d M, p, 1 [, + Ψ,, p, k, k, and we may rewrie Propoiion 14 a follow Propoiion 18 Le X aify aumpion A5, A6 and A7 hen he bridge X,θ aifie he inegral equaion X,θ X { + θ X,θ } Ψ, u dxu,θ k, k, k, d M 6 u2 d M u Moreover, he bridge X,θ admi he non-anicipaive repreenaion where R, θ R, + X X,θ ϕ, dx, 7 ϕ, { u 1 + Ψ v, k, v 2 v k, 2 d M v w2 d M w 1 + Ψ u, u k, v2 d M v }k, uk, u d M u 6 Fracional Brownian bridge he fracional Brownian moion Z i a cenred aionary incremen Gauian proce wih variance E Z 2 2H for ome H, 1 Anoher way of charaeriing he fracional Brownian moion if o ay ha i i he unique up o muliplicaive conan cenred H-elf-imilar Gauian proce wih aionary incremen In order o repreen he fracional Brownian moion a a Volerra proce we fir recall ome preliminarie of fracional calculu For deail we refer o Samko e al [8] Le f be a funcion over he inerval [, 1] and α > hen I α ± [f] : 1 1 f Γ α 1 α d ±

19 Gauian bridge 19 are he Riemann Liouville fracional inegral of order α For α, 1, D α ± [f] : ±1 Γ 1 α d d 1 f α d ± are he Riemann Liouville fracional derivaive of order α; I ± and D ± are ideniy operaor If one ignore he rouble concerning divergen inegral and formally change he order of differeniaion and inegraion one obain I α ± D α ± We hall ake he above a he definiion for fracional inegral of negaive order and ue he obviou unified noaion Now, he fracional Brownian moion i a Volerra proce aifying aumpion A5 and A6 Indeed, le K be a weighed fracional inegral or differenial operaor where K [f] : c H 1 2 H I H 1 2 c H [ H 1 2 f ], 2HH 1 2 Γ H BH 1 2, 2 2H and Γ and B are he gamma and bea funcion hen we have he relaion 1 for fracional Brownian moion: Z K [ 1 [, ] dw, 1 where W i he andard Brownian moion hu, he fracional Brownian moion aifie he aumpion A5 he operaor K aifie he aumpion A6 Indeed, K 1 [f] 1 ] 1 2 H I 1 2 H [ H 1 2 f c H he kernel Ψ ha been calculaed eg in Pipira and aqqu [7], heorem 71 Indeed, for any H, 1 we have Ψ, in πh H 1 2 H u H+ 1 2 u H+ 1 2 π u A for he kernel p noe ha for H, 1 we have k, K [ ] 1 [, { H 1 c 2 H H 1 2 H H du u H 3 2 u H 1 2 du },

20 2 Dario Gabarra, ommi Soinen, and Eko Valkeila where c H 2HΓ 3 2 H Γ H Γ 2 2H If H > 1/2 hen we have a lighly impler expreion, viz k, c HH H u H 1 2 u H 3 2 du For he derivaion of hee formula ee Norro e al [6] and Jo [5] he repreenaion for he fracional Brownian bridge follow now by plugging in our Ψ and k o he formula 6 and 7 in Propoiion 18 wih M W and d M d Unforunaely, i eem ha here i really nohing we can do o implify he reuling formula excep ome rivial ue of he H-elf-imilariy, even in he cae H > 1/2 So, we do no boher o wrie he equaion 6 and 7 again here 7 Enlargemen of filraion poin of view Le u denoe by F X be i naural coninuou filraion of he Gauian proce X Seing in our condiioning imply θ : X we may inerpre he bridge meaure P,X,X a iniial enlargemen of he filraion F X by he random variable X Le F X,X be hi enlarged filraion We have formally Ω, F, F X, P,X,X Ω, F, F X,X, P For he Brownian moion W we have he following: wih repec o he meaure P θ he Browian moion ha he repreenaion W W θ + β,θ W d W θ θ W + d hi mean ha wih repec o he filraion F W,W Brownian moion W ha he repreenaion and meaure P W W FW,W + W W d, where Law W FW,W P LawW P, W FW,W i a P, F W,W Brownian moion, bu W i a P, F W,W emimaringale Similarly, if we have an arbirary gauian proce uch ha he proce ha a Volerra repreenaion 3 X p, d ˆX

21 Gauian bridge 21 we can ue he enlargemen of filraion reul o give a emimaringle repreenaion for he maringale ˆX wih repec o P, F X,X : ˆX ˆX FX,X X + ˆX ˆX d ˆX, 2, where he P, F X,X - gauian maringale ˆX FX,X ha he ame law a ˆX ee [2, 3] for more deail We can now ue 3, 6 and 2 o obain he following repreenaion for he proce X wih φ, X X FX,X + X R, R, u Acknowledgemen φ, dx, 3 p v, ˆX 2 d ˆX v p u, ˆX,v p, u d ˆX u,u Soinen i graeful for parial uppor from EU-IHP nework DYNSOCH We hank M Lifhi for he example 1 Reference 1 C Bender, R Ellio, R On he Clark-Ocone heorem for fracional Brownian moion wih Hur parameer bigger han a half, Soch Soch Rep D Gabarra, E Valkeila, Iniial enlargemen: a Bayeian approach, heory Soch Proce D Gabarra, E Valkeila, L Vorikova, Enlargemen of filraion and addiional informaion in pricing model: a Bayeian approach In: From Sochaic Analyi o Mahemaical Finance YKabanov, RLiper, and J Soyanov, ed Springer 26, Jeulin, Semi-maringale e Groiemen d une Filraion, Lec Noe Mah 1118, Springer, Berlin C Jo, ranformaion formula for fracional Brownian moion Aricle in pre in Sochaic Procee and heir applicaion, 26 6 I Norro, E Valkeila, J Viramo, An elemenary approach o a Giranov formula and oher analyical reul on fracional Brownian moion, Bernoulli V Pipira, M aqqu, Are clae of deerminiic inegrand for fracional Brownian moion on an inerval complee?, Bernoulli SG Samko, AA Kilba, OI Marichev, Fracional inegral and derivaive heory and applicaion, Gordon and Breach Science Publiher, Yverdon AN Shiryaev, Probabiliy Springer, Berlin, 1984