Fractional Ornstein-Uhlenbeck Bridge

Transcription

1 WDS'1 Proceeding of Conribued Paper, Par I, 21 26, 21. ISBN MATFYZPRESS Fracional Ornein-Uhlenbeck Bridge J. Janák Charle Univeriy, Faculy of Mahemaic and Phyic, Prague, Czech Republic. Abrac. In hi work we udy he fracional Ornein-Uhlenbeck bridge. Fir, we recall he noion of he fracional Brownian moion (fbm and inroduce a linear ochaic differenial equaion which i driven by (fbm inead of andard Brownian moion. We ummarize he reul on exience and uniquene of oluion o hee equaion ha are called he fracional Ornein-Uhlenbeck procee. We inroduce a concep of he Gauian bridge and we derive i repreenaion, which we ue for obaining he formula for fracional Ornein- Uhlenbeck bridge. The reul are applied o one pecial example. In he la par of he paper we menion a nonanicipaive repreenaion of he bridge. Inroducion Definiion 1. A Gauian proce (B H (, on a probabiliy pace (Ω, F, P wih value in R i called he fracional Brownian moion (fbm wih Hur parameer H (,1, if hi proce aifie EB H ( = for any R +, cov (B H (,B H ( = 1 2 (2H + 2H 2H for any, R + and (B H (, ha coninuou pah P a.., where cov (X,Y denoe he covariance of random variable X and Y. From he definiion i follow ha B H ( = P a.. and (fbm wih H = 1 2 i he andard Brownian moion. For H 1 2 he proce BH doe no have he emimaringale propery, o i wa neceary o develop an inegraion heory for (fbm. The conrucion of he ochaic inegral wih repec o (fbm can be found in [Decreuefond, Üunel, 1999] and for he cae H > 1 2 in [Janák, 29]. The proof of Theorem 1, 2, 3 and 4 are aed in [Janák, 29]. The proof of Theorem 5 a well a many idea abou Gauian bridge come from [Benh, Nunno, Lindrøm, Økendal, 25]. The infinie dimenional cae of Ornein- Uhlenbeck bridge and i applicaion in a Hilber pace are udied in [Goldy, Malowki, 28], bu he fracional analogy i no dicovered ye. The formulae for he repreenaion of a general Gauian bridge and he fracional Ornein-Uhlenbeck bridge are origin reul. Linear ochaic equaion Conider he following equaion dx = A(X d + b(d + Φ(dB H, X = x, (1 where, A : [, R n n i a coninuou funcion, b : [, R n, Φ : [, R n m are locally bounded Borel funcion, x R n and B H i an R m -valued (fbm wih Hur parameer H > 1 2. The oluion of he equaion (1 on a probabiliy pace (Ω, F, P, where R m -valued (fbm B H i defined, i a coninuou proce (X,, ha aifie X = x + A(X d + b(d + Φ(B H, P a.. (2 21

2 In Theorem 1 we will give a cloed formula for he oluion, which we call fracional Ornein-Uhlenbeck proce. In Theorem 2, we will how wha he diribuion of he oluion i. Theorem 1. Le U(,, T, be a fundamenal yem of he differenial equaion Ẋ = A(X, hen X = U(, x + U(,rb(rdr + U(,rΦ(rdBr H (3 i he P a.. unique oluion of he equaion (1. Theorem 2. The proce defined by (2 ha a coninuou modificaion, i Gauian, wih expecaion EX = U(, x + U(,rb(rdr (4 and covariance Q, = for,, where φ(u = H(2H 1 u 2H 2. Gauian bridge U(,vΦ(vΦ(u T U(,u T φ(u vdudv, (5 We ar wih a moivaion of he definiion of Gauian bridge. Le X be a coninuou Gauian proce uch ha X = and E(X =. Fix T > and define he o called X-bridge U T, by he formula U T, = X T X T. (6 The proce U T, i Gauian, becaue i i an affine ranformaion of Gauian random vecor. I i a bridge in he ene U T, = U T, T =. If X i a andard Brownian moion, i i known ha he probabiliy law of he proce U T, defined in (6 i he ame a he condiional law of andard Brownian moion Law((X T X T = = Law((U T, T. (7 We would like o ue he previou equaion for he definiion of a general Gauian bridge, bu we are condiioning by an even ha ha a zero meaure. We need o ake a regular verion of hi condiional probabiliy. Since he proce X ake value in R n, ha i a Polih pace, uch regular verion clearly exi. In fac, here are infinie many verion of he condiional probabiliy. We will chooe he coninuou one, which give u he following Theorem 3. Theorem 3. Le Y : (Ω, F, P (R n, B(R n be a fixed real vecor, X be a coninuou ochaic proce on [,T] wih value in R n and le exi a regular verion of he condiional probabiliy L : y R n Law(X, [,T] Y = y, (8 ha i coninuou in y in he ene of weak convergence of meaure on C([,T], where y upp Law(Y = {y R n : ε >, P( Y y < ε > }. Then he law of L(y i unique. Definiion 2. Le X be a coninuou R n -valued Gauian proce uch ha X = x. Fix T > and θ R n. A coninuou Gauian proce i Gauian bridge from (,x o (T,θ, if Law( aifie condiion (8 from he Theorem 3 for Y = X T and y = θ. 22

3 Remark 1. Theorem 3 give u he uniquene of Law(. If upp Law(X T = R n and Law(X T,ϑ i coninuou a he poin ϑ = θ in he diribuion on C([,T]; R n, hen condiion (8 i aified (ee [Janák, 29]. Since we know how he condiional diribuion in Gauian diribuion look like (ee [Mandelbaum, 1984], we are able o obain he repreenaion for Gauian bridge. Theorem 4. Le X be a coninuou Gauian proce wih expecaion funcion µ(. Denoe Q T, = cov(x T,X and Q = cov(x,x = var X and aume ha Q T i poiive definie and up T Tr Q <. Then he Gauian bridge from (,µ( o (T,θ ha he following repreenaion Moreover cov ( E( = X Q,T Q 1 T (X T θ. (9 = µ( Q,T Q 1 T (µ(t θ (1, = Q, Q,T Q 1 T Q T,. (11 By Theorem 4, if Q T i regular, we ge he following repreenaion for he fracional Ornein-Uhlenbeck bridge = X Q,T Q 1 T (X T θ = U(, x + U(,rb(rdr + U(,rΦ(rdBr H ( U(T,vΦ(vΦ(u T U(,u T φ(u vdudv ( 1 U(T,vΦ(vΦ(u T U(T,u T φ(u vdudv ( U(T, x + U(T,rb(rdr + U(T,rΦ(rdBr H θ. The expecaion and covariance can be wrien in he following form E( = µ( Q,T Q 1 (µ(t θ T = U(, x + U(,rb(rdr ( U(T,vΦ(vΦ(u T U(,u T φ(u vdudv ( U(T,vΦ(vΦ(u T U(T,u T φ(u vdudv ( U(T, x + U(T,rb(rdr θ, 1 23

4 cov(, = Q, Q,T Q 1 T Q T, = U(,vΦ(vΦ(u T U(,u T φ(u vdudv ( U(T,vΦ(vΦ(u T U(,u T φ(u vdudv ( U(T,vΦ(vΦ(u T U(T,u T φ(u vdudv ( U(,vΦ(vΦ(u T U(T,u T φ(u vdudv. Unforunaely, here i nohing we can do abou implifying hee expreion in he general cae, bu i i poible o do o in ome pecial example. Aume he calar cae (n = m = 1 in equaion (1 and le A = and Φ = 1 wih a aring poin x =. The oluion of hi equaion i obviouly he fracional Brownian bridge aring a he poin. Uing he previou formulae we ge he repreenaion for one-dimenional fracional Brownian bridge cov (B T,θ B T,θ = B H T 2H + 2H T 2H 2T 2H (B H T θ E(B T,θ,B T,θ = T 2H + 2H T 2H 2T 2H θ = 2H + 2H 2H 2 (T 2H + 2H T 2H (T 2H + 2H T 2H 4T 2H. 1 A non-anicipaive repreenaion of a Gauian bridge Fix T > and ξ,θ R n. Le W = (W(, T be a andard Brownian moion on a probabiliy pace (Ω, F, P uch ha W( = ξ. We can repreen he Brownian bridge by he following way dy T,θ Y T,θ = dw + θ Y T,θ T d, = ξ + (θ ξ T + (T Y T,θ = ξ (12 dw T (13 W T,θ = W T (W T θ. (14 The repreenaion (13 i a oluion of he differenial equaion (12. So equaion (12 and (13 define ame proce Y T,θ. The procee Y T,θ and W T,θ have he ame probabiliy law, ince hey have he ame covariance and expecaion. From Theorem 4 i follow ha he proce W T,θ i he Brownian bridge and due o Definiion 2 he proce Y T,θ a well. The procee Y T,θ and W T,θ are differen, becaue he proce Y T,θ i adaped o he filraion of W, while he proce W T,θ i no. To conruc W T,θ from (14 we need informaion abou he random variable W T. The repreenaion of he bridge in (14 i called anicipaive. The repreenaion of he bridge in (12 and (13 are called non-anicipaive. They are quie difficul o obain for general Gauian bridge (conruced from he Gauian proce X. We will how one poible mehod for which we need he following aumpion. 24

5 Aumpion (A1. There exi a Volerra kernel k and a coninuou Gauian maringale M wih ricly increaing bracke M uch ha X admi he repreenaion X = Le K exend he relaion K : 1 [, k(, linearly. We have f(dx = g(dm = k(,dm. (15 K[f](dM, K 1 [g](dx, for any g L 2 ([,T],d M and uch funcion f ha are in he preimage of L 2 ([,T],d M under K. Aumpion (A2. For any T he equaion K[f] = 1 [, ha a oluion f K 1 (L 2 ([,T],d M. Aumpion (A3. For any T he equaion K[g] = 1 [, k(t, ha a oluion g L 2 ([,T],d M. Denoe Ψ T (, := K 1 [1 [, K[1 [,T ]](. (16 Theorem 5. Le X aify aumpion (A1, (A2 and (A3. Then he bridge aifie he inegral equaion ( = X + θ X T,θ Ψ T (,udxu T,θ k(t, k(, k(t,u2 d M u d M. (17 Moreover, he bridge admi he non-anicipaive repreenaion where = θ R(T, R(T,T + X ϕ T (,dx, (18 ϕ T (, = ( u (1 + Ψ T (v,k(t,v 2 ( v k(t,w2 d M w 2 d M v 1 + Ψ T (u, u k(t,v2 d M v k(t,uk(,ud M u. (19 A calar fracional Brownian moion wih Hur parameer H (, 1 aifie aumpion (A1, (A2 and (A3, o now we are able o find he repreenaion for he fracional Brownian 25

6 bridge even for H < 1 2. The kernel Ψ T wa compued in [Pipira, Taqqu, 21] and ha he form Ψ T (, = in π(h H ( 1 2 H u H+1 2(u H+1 2 du. π u A for he Volerra kernel k(, noe ha for H (,1 we have where k(, = c H ( ( H 1 2 ( H 1 2 ( H 1 12 H u H 3 2 (u H 1 2 du, 2 c H = Γ(p = 2HΓ( 3 2 H Γ(H Γ(2 2H, e x x p 1 dx, p >. The repreenaion for he fracional Brownian bridge follow from Theorem 5 wih M = W and d M = d. Acknowledgmen. Thi work wa uppored by he GA UK gran no and by he GA ČR gran no. P 21/1/752. Reference F. E. Benh, G. D. Nunno, T. Lindrøm, B. Økendal, T. Zhang, Sochaic Analyi and Applicaion, Proceeding of he econd Abel ympoium, 25 L. Decreuefond, A. S. Üunel, Sochaic analyi of he fracional brownian moion, Poenial Anal., 1, , 1999 B. Goldy, B. Malowki, Ornein-Uhlenbeck bridge and applicaion o Markov emigroup, Sochaic proce. Appl., 118, , 28 J. Janák, Ornein-Uhlenbeck bridge, Diploma hei, 29 A. Mandelbaum, Linear eimaor and meaurable linear ranformaion on a Hilber pace, Z. Wahrch. Verw. Gebiee 65, , 1984 V. Pipira, M. Taqqu, Are clae of deerminiic inegrand for fracional Brownian moion on an inerval complee?, Bernoulli 7, , 21 26