FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES

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1 FRACTIONAL ORNSTEIN-ULENBECK PROCESSES Prick Cheridio Deprmen of Mhemics, ET Zürich C-89 Zürich, Swizerlnd ideyuki Kwguchi Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn Mkoo Mejim Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn Absrc The clssicl sionry Ornsein-Uhlenbeck process cn be obined in wo differen wys. On he one hnd, i is sionry soluion of he Lngevin equion wih Brownin moion noise. On he oher hnd, i cn be obined from Brownin moion by he so clled Lmperi rnsformion. We show h he Lngevin equion wih frcionl Brownin moion noise lso hs sionry soluion nd h he decy of is uo-covrince funcion is like h of power funcion. Conrry o h, he sionry process obined from frcionl Brownin moion by he Lmperi rnsformion hs n uo-covrince funcion h decys eponenilly. Keywords Frcionl Brownin moion, Lngevin equion, long-rnge dependence, selfsimilr processes, Lmperi rnsformion AMS subjec clssificion primry: 6; secondry: 6G5, 6G8, 45F5 Inroducion Le Ω, F, P be probbiliy spce. Definiion. A frcionl Brownin moion wih urs prmeer,, is n lmos surely coninuous, cenered Gussin process B R wih Cov B, B s = + s s,, s R.. The firs uhor would like o hnk he Deprmen of Mhemics, Keio Universiy, for hving mde possible his sy in Yokohm. Curren ddress: Risk Mngemen Sysems Deprmen, Sumiomo Misui Bnking Corporion, -3-, Mrunouchi, Chiyod, Tokyo -5, Jpn; kwguchi hideyuki@dn.smbc.co.jp

2 For n in-deph inroducion o frcionl Brownin moions we refer he reder o Secion 7. of Smorodnisky nd Tqqu 994 or Chper 4 of Embrechs nd Mejim. I is cler h for ll,, B = lmos surely. Moreover, i cn be deduced from. h for ll,, B R hs sionry incremens nd is -selfsimilr, h is, for every c >, B c R d = c B R, where d = denoes equliy of ll finie-dimensionl disribuions. B R is wo-sided Brownin moion. In priculr, i hs independen incremens. For,,, B is no semimringle nd i cn be derived from. h for ll h R nd < < s, Cov Bh+ B h, B h+s+ h+s B = Cov B, Bs+ Bs = n= n n! n k s n, k= in priculr, for every N =,,..., for ll h R nd >, Cov Bh+ B h, B h+s+ B h+s N n n = k s n + Os N, s s.. n! n= k= This shows h for,, n= Cov B, Bn+ B n =, phenomenon referred o s long-rnge dependence or long memory of he incremens process Bn+ B n n=. The clssicl Ornsein-Uhlenbeck process wih prmeers λ > nd σ > sring R, is he unique srong soluion of he Lngevin equion wih Brownin moion noise X = ξ λ X s ds + σb,,.3 wih iniil condiion ξ =. I is given by he lmos surely coninuous Gussin Mrkov process Y, := e λ + σ e λu db u,. The unique srong soluion of.3 wih iniil condiion ξ = σ e λu db u, is given by he resricion o non-negive s of he sionry, lmos surely coninuous, cenered Gussin Mrkov process Y := σ e λ u db u, R.

3 I cn esily be checked h Cov Y, Y +s = σ λ e λ s,, s R. This implies h Y R is ergodic. Moreover, for ll R, Y Y, = e λ Y, s, lmos surely. From his i cn be derived h if Y is sionry process h solves.3 wih ny iniil condiion ξ L d Ω, hen Y = Y. Now le α >. Then, Z := e λ B, R, αe λ is lso sionry, lmos surely coninuous, cenered Gussin process, nd Cov Z, Z +s = αe λ s,, s R. ence, for α = σ λ, Y R = d Z R. I is shown in Lmperi 96 h for every >, sochsic process X is - selfsimilr if nd only if for ll λ, α >, he process X = e λ X α ep λ, R,.4 is sionry. We cll.4 he Lmperi rnsformion from selfsimilr processes o sionry processes nd X R he Lmperi rnsform of X. For =, frcionl Brownin moion cn be represened s follows: B = η, R, where η is sndrd norml rndom vrible. equion, X = ξ λ cn ph-wise be reduced o he ordinry differenil equions, For every iniil condiion ξ L Ω, he X s ds + σb,,.5 X ω = λx ω + σηω, ω Ω, wih iniil condiions which hve he unique soluions X ω = ξω, ω Ω, Y,ξ ω := e λ { ξω σ λ ηω } + σ λ ηω,, ω Ω. Equion.5 hs only sionry soluion for he iniil condiion ξ = σ λη. I is given by Y := σ λ η,, 3

4 which, for ll, equls he Lmperi rnsform Z := e λ B α epλ = αη, R, if α = σ λ. This leds us o he quesion wheher for,,, he Lngevin equion wih noise process σb hs sionry soluion, if is disribuion is unique nd if i is equl in some sense o he Lmperi rnsform Z := e λ B α ep λ, R, for n ppropriely chosen α >. The srucure of he pper is s follows. In Secion we show h for ll,, he Lngevin equion wih frcionl Brownin moion noise hs for ll iniil condiions ξ L Ω, unique srong soluion Y,ξ. Moreover, here eiss sionry, lmos surely coninuous, cenered Gussin process Y R such h Y solves he Lngevin equion wih frcionl Brownin moion noise, nd every oher sionry soluion is equl o Y in disribuion. The decy of he uo-covrince funcion of Y R is for ll,, similr o h of he incremens of B R see.. In priculr, Y R is ergodic, nd for,, i ehibis long-rnge dependence. In Secion 3 we show h for ll, he uo-covrince funcion of Z R decys eponenilly, which implies h for,,, Y R cnno hve he sme disribuion s Z R. Frcionl Ornsein-Uhlenbeck processes Le λ, σ > nd ξ L Ω. Since he Lngevin equion, X = ξ λ X s ds + N,, only involves n inegrl wih respec o, i cn be solved ph-wise for much more generl noise processes N hn Brownin moion. For emple, i follows from Proposiion A. h for ech, nd for every [,, e λu db u, >, eiss s ph-wise Riemnn-Sieljes inegrl, which is lmos surely coninuous in, nd Y,ξ := e ξ λ + σ e λu dbu,, is he unique lmos surely coninuous process h solves he equion, X = ξ λ X s ds + σb,.. 4

5 In priculr, he resricion o posiive s of he lmos surely coninuous process Y := σ solves. wih iniil condiion ξ = Y e λ u db u, R,. I is cler h Y R is Gussin process, nd i follows immediely from he sionriy of he incremens of frcionl Brownin moion h i is sionry. Furhermore, s in he Brownin moion cse, for every ξ L Ω, Y Y,ξ = e λ Y ξ, s, lmos surely, which implies h every sionry soluion of. hs he sme disribuion s Y cll Y,ξ sionry frcionl Ornsein-Uhlenbeck process. frcionl Ornsein-Uhlenbeck process wih iniil condiion ξ nd Y. We R In Pipirs nd Tqqu i is shown h for, nd wo rel-vlued mesurble funcions { } f, g f : fu fv u v dudv <, he wo inegrls fudb u, gudb u cn in consisen wy be defined s limis of inegrls of elemenry funcions, nd [ E fudbu gudbu = fugv u v dudv. For,, he kernel u v cnno be inegred over he digonl. owever, if f nd g re regulr enough nd he inersecion of heir suppors is of Lebesgue mesure zero, he sme holds rue. We will only need his resul for he cse where f nd g re given by fu = gu = e λu nd heir suppors re disjoin inervls. owever, he following lemm cn esily be generlized. Lemm. Le,,, λ > nd < b c < d <. Then [ b d b d E e λu dbu e λv dbv = e λu e λv v u dv du. c Proof. We firs ssume b = = c. By Proposiion A. we ge [ d E e λu dbu e λv dbv [ d = E e λ B λ e λu Bu du e λd B d λ e λv Bv dv = eλ e λd [ + d d + d λeλ e λv [ + v v dv λeλd e λu [ u + d d u du + d λ e λv e λu [ u + v v u du dv. 5 c

6 Afer pril inegrion wih respec o u, his becomes e λd e λu [ u d u du d +λ e λv e λu [ u v u du dv, which, by pril inegrion wih respec o v, is equl o e λu d e λv v u dv du. Now we ssume b = < c. I follows from bove h [ d [ d c E e λu dbu e λv dbv = E e λu dbu e λv dbv e λu dbu e λv dbv c [ d c = e λu e λv v u dv du e λu e λv v u dv du = e λu d c e λv v u dv du. For generl < b c < d <, he process B = B+b B b frcionl Brownin moion. Therefore, [ b E e λu dbu = = b b [ e λv dbv = E e λw+b d B w c b d b e λw+b e λ+b w d d c b d e λu e λv v u dv du, c d b c b dw, R, is gin e λ+b d B nd he proof is complee. Lemm. Le β <. Then for ech N =,,,..., N n e e y y β dy = β + β k β n + O β N, s, n= k= nd e e y y β dy = β + N n n β k β n + O β N, s, n= k= where n= mens. 6

7 Proof. We hve nd e e y y β dy = e e β + β e y y β dy =... = β + β β + ββ β ββ... β N + β N +e ββ... β N e y y β N dy, e e y y β N dy e e y β N dy = β N, which proves he firs sserion. On he oher hnd, nd e e y y β dy = e e β e β e y y β dy =... = β β β N ββ... β N + β N e e { β N ββ... β N + } e N ββ... β N e e y y β N dy e e y dy + e y y β N dy, e y β N dy e β N +. This proves he second pr of he lemm. Theorem.3 Le,, nd N =,,.... Then for fied R nd s, Cov Y, Y+s N n = λ n k s n + Os N. Proof. By Lemm., = E Cov Y [ σ = e λs E σ n=, Y+s = Cov Y [ σ e λu db u σ s e λu db u σ +σ e λs, Ys k= e λs v dbv λ e λv db v s e λu e λv v u dv du λ 7

8 by he chnge of vribles: w = λu, = λv λs = σ e λs e w e w d dw + Oe λs λ by he chnge of vribles: y = w, z = + w σ { λs y = e λs y e z dz dy λ y λs y } + y e z dz dy + Oe λs = = λs σ y e λs λ { λs e y y dy + λs σ λs {e λs λ e λs y y dy e y y dy + e λs The proof cn now be concluded by pplying Lemm.. Theorem.3 shows h for,,, he decy of is very similr o he decy of λs Cov Y, Y+s, for s, Cov B h+ B h, B h+s+ B h+s, for s } e y y dy + Oe λs } e y y dy + Oe λs. see.. In priculr, Y R is ergodic, nd for,, i ehibis long-rnge dependence. Remrk.4 Le s R. For ll,, he funcions f = { } e λ nd g = { s} e λ belong o he inner produc spce Λ defined on pge 89 of Pipirs nd Tqqu. ence, for ll, s R, Cov Y, Y+s is equl o σ e λs Γ + sinπ f, = σ g Λ π e is λ d.. + Therefore, he epression given in he he semen of Theorem.3 is n sympoic epnsion of he righ hnd side in. s s. The ne corollry shows h for he soluion Y, of. wih deerminisic iniil vlue Y, = R, Cov Y,, Y, +s, for s, decys like power funcion of he order s well. Corollry.5 Le,,, R nd N =,,.... Then for fied nd s, Cov Y,, Y, +s = N n { σ λ n k s n e λ + s n} + Os N. n= k= 8

9 Proof. Cov [ = E σ [ = E σ [ = E σ Y, e λs E, Y, +s e λ u db u σ e λ u db u σ +s +s e λ u dbu e λ [ σ e λ u dbu σ e λ+s v dbv e λ+s v dbv e λs e λ v dbv +s e λu dbu e λ+s v dbv e λ v dbv = Cov Y, Y+s e λ Cov Y, Y+s + Oe λs. Now, he corollry follows from Theorem.3. 3 The Lmperi rnsform of frcionl Brownin moion Le λ > nd α >. For ech,, we se Z := e λ B α ep λ, R. Theorem 3. Le, nd, s R. Then Cov { Z, Z+s α = e λ s + n n n= e λ n s }. 3. Proof. Wihou loss of generliy we cn ssume h s. Then, Cov Z, Z+s = e λ λ+s α e { = α eλs = α eλs = α { e λ+s + e λ e λ +s e λ } + e λs e λ s } { } + e λs e λ s n n n= { } e λs + e n λ n s, n n= which proves he heorem. I follows from Theorem 3. h for every N =,,..., for ech, nd ll R, Cov { Z, Z+s α = e λ s + N } e n λ n s n n= N+ + O e λ s, 9

10 s s. This shows h for ll,, he uo-covrince funcion of Z R decys eponenilly. I follows h for,,, Z R cnno hve he sme disribuion s Y R. For,, he leding erm in 3. for s, is α e λ s, wheres for,, i is α e λ s. Noe h for,, he leding erm of Cov Z, Z+s for s, is posiive, wheres he leding erm of Cov Y, Y+s for s, is negive see Theorem.3. Appendi: The Lngevin equion Lngevin 98 pioneered he following pproch o he movemen of free pricle immersed in liquid: e described he pricle s velociy v by he equion of moion dv d = f F v + m m A. where m > is he mss of he pricle, f > fricion coefficien nd F flucuing force resuling from impcs of he molecules of he surrounding medium. Uhlenbeck nd Ornsein 93 imposed probbiliy hypoheses on F nd hen derived h for v = R, v is e λ, for λ = f m nd σ = fkt m, normlly disribued wih men e λ nd vrince σ λ where k is he Bolzmnn consn nd T he emperure. Doob 94 noiced h if v is rndom vrible which is independen of F nd normlly disribued wih men zero nd vrince σ λ, hen he soluion v of A. is sionry nd {>} v λ ln,, is Brownin moion, from which he concluded h every soluion of A. hs lmos surely coninuous phs which re nowhere differenible. To void he embrrssing siuion h he equion A. involves he derivive of v bu leds o soluions v h do no hve derivive, he gve rigorous mening o sochsic differenil equions of he form dx = λx d + dn, A. for he cse h N is Lévy process nd showed h for ll R, he equion A. wih iniil condiion X = R, hs he unique soluion X = e λ + e λu dn u,. In he modern heory of sochsic differenil equions see e.g. Proer 99 he equion A. wih iniil condiion X = ξ L Ω is undersood s he inegrl equion X = ξ λ X s ds + N,, A.3

11 nd i cn be shown h he unique srong soluion of A.3 is given by X ξ := e λ ξ + e λu dn u,, whenever N is semimringle wih respec o he filrion genered by N nd ξ. Proposiion A. Le B R be frcionl Brownin moion wih urs prmeer, nd ξ L Ω. Le < nd λ, σ >. Then for lmos ll ω Ω, we hve he following: For ll >, e λu db u ω eiss s Riemnn-Sieljes inegrl nd is equl o b The funcion e λ B ω e λ B ω λ e λu db u ω, >, is coninuous in. c The unique coninuous funcion y h solves he equion, is given by y = ξω λ y = e λ {ξω + σ In priculr, he unique coninuous soluion of he equion, B u ωe λu du. ysds + σb ω,. A.4 y = σ e λu dbu ω λ } e λu dbu ω,. A.5 ysds + σb ω,, is given by y = σ e λ u dbu ω,. Proof. I cn esily be checked h B s := {s<} s B s + {s>} s B s, s R, is gin frcionl Brownin moion. I follows from he Kolmogorov-Čensov heorem see e.g. Theorem..8 of Krzs nd Shreve 99 h here eiss mesurble null se

12 N Ω, such h for every ω Ω \ N, Bs ω nd B s ω re coninuous in s, nd for ll β <, B s ω lim s s β =. This implies h for ll γ >, ence, for ll >, Bs ω lim s s γ =. B u ωe λu du eiss s Riemnn inegrl, which, by Theorem. of Wheeden nd Zygmund 977, implies h he Riemnn-Sieljes inegrl eiss oo nd is equl o e λu db u ω e λ B ω e λ B ω λ This proves. b follows from nd he fc h he funcion e λ B ω λ B u ωe λu du. B u ωe λu du, >, is coninuous in. A coninuous funcion y solves A.4 if nd only if he funcion z = solves he liner differenil equion: Since he unique soluion of A.6 is given by ysds,, z = λz + ξω + σb ω, z =. A.6 z = e λ e λu ξω + σbu ω du,, he unique coninuous funcion y h solves A.4 is given by λe λ e λu ξω + σbu ω du + ξω + σb ω,, which, by, is equl o A.5. This shows c.

13 Remrk A. Equion A.3 cn be solved ph-wise for ll sochsic processes N h hve lmos ll phs in { } L loc R + := h : R + R : h is mesurble nd, hs ds <, nd even when he consn λ is replced by sochsic process wih lmos ll phs in { } L loc R + := g : R + R : g is mesurble nd, sup gs <. s Indeed, if h L loc R + nd g L loc R +, hen i cn esily be checked h he funcion f := h + is in L loc R + nd solves he inegrl equion f = On he oher hnd, if f L loc R + is soluion of A.8, hen f f = gser s gudu hsds,, A.7 gsfsds + h,. A.8 nd i follows from vrin of Gronwll s lemm h [ gs f f ds,, f f =,. ence, A.7 is he only funcion in L loc R + h solves A.8. If he funcions g nd h re boh in L loc R + nd coninuous on R + \ C, where C is of Lebesgue mesure zero, hen i cn be deduced from Theorems 5.54 nd. of Wheeden nd Zygmund 977 h f cn be wrien s follows: R f = e gudu h + e R s gudu dhs,, where e R s gudu dhs is Riemnn-Sieljes inegrl. Noe h lmos ll phs of semimringle re righ-coninuous nd hve lef limis, in priculr, hey re in L loc R + nd hve mos counbly mny disconinuiies. References Doob, J.L. 94, The Brownin movemen nd sochsic equions, Ann. of Mh. 43, Embrechs, P. nd Mejim, M., Selfsimilr Processes, Princeon Series in Applied Mhemics, Princeon Universiy Press. 3

14 Krzs, I. nd Shreve, S.E. 99, Brownin Moion nd Sochsic Clculus, Grdue Tes in Mhemics, Springer-Verlg, New York. Lmperi, J.W. 96, Semi-sble sochsic processes, Trns. Amer. Mh. Soc. 4, Lngevin, P. 98, Sur l héorie du mouvemen brownien, C.R. Acd. Sci. Pris 46, Pipirs, V. nd Tqqu, M., Inegrion quesions reled o frcionl Brownin moion, Prob. Th. Rel. Fields 8, -9. Proer, P. 99, Sochsic Inegrion nd Differenil Equions, Springer-Verlg, Berlin. Smorodnisky, G. nd Tqqu, M.S. 994, Sble Non-Gussin Rndom Processes, Chpmn & ll, New York. Uhlenbeck, G.E. nd Ornsein, L.S. 93, On he heory of he Brownin moion, Physicl Review 36, Wheeden, R.L. nd Zygmund, A. 977, Mesure nd Inegrl, Mrcel Dekker, New York- Bsel. 4

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