Communiction on Stochtic Anlyi Vol. 7, No. 2 213 245-253 Seril Publiction www.erilpubliction.com LINEA STOCHASTIC DIFFEENTIAL EQUATIONS WITH ANTICIPATING INITIAL CONDITIONS NAJESS KHALIFA, HUI-HSIUNG KUO, HABIB OUEDIANE, AND BENEDYKT SZOZDA Abtrct. In thi pper we ue the new tochtic integrl introduced by Ayed nd Kuo 1 nd the reult obtined by Kuo, Se-Tng nd Szozd 1 to find olution to drift-free liner tochtic differentil eqution with nticipting initil condition. Our olution i bed on well-known reult from clicl Itô theory nd nticiptive Itô formul reult from 1. We lo how tht the olution obtined by our method i conitent with the olution obtined by the method of Mllivin clculu, ee e.g., 3. 1. Introduction The im of the preent pper i to etblih olution to liner tochtic differentil eqution with n nticipting initil condition of certin form, nmely dx t = α t X t db t + β t X t dt t, b 1.1 X = pb b B. In the ce with X = x, it i well-known fct tht the unique olution i given by X t = x exp α db + β 1 2 α2 d. 1.2 For detil, ee for exmple, 6, Section 11.1. The ignificnce of our reult ly in the fct tht the olution X t of Eqution 1.1 i n nticipting tochtic proce nd it cnnot be obtined by the clicl tool from the Itô theory of tochtic integrtion. Inted, we ue the integrl of dpted nd intntly independent procee introduced by Ayed nd Kuo 1, 2 nd further developed by Kuo, Se- Tng nd Szozd 8, 9, 1. In contrt to reult obtined by Eunge 4 nd Buckdhn nd Nulrt 3, our reult do not rely on white noie nlyi or Mllivin clculu nd re nchored in bic probbility theory. The reminder of thi pper i orgnized follow. In Section 2 we recll ll the necery definition nd previou reult ued in the ret of the pper. Section 3 contin imple exmple tht illutrte our method nd Section 4 preent our min reult, Theorem 4.1. We conclude with ome exmple in Section 5. eceived 212-12-2; Communicted by the editor. 2 Mthemtic Subject Clifiction. Primry 6H5; Secondry 6H2. Key word nd phre. Brownin motion, Itô integrl, dpted tochtic procee, tochtic integrl, intntly independent tochtic procee, liner tochtic differentil eqution, nticipting tochtic differentil eqution. 245
246 N. KHALIFA, H.-H. KUO, H. OUEDIANE, AND B. SZOZDA 2. Preliminry Definition In thi ection we et the nottion nd recll everl definition ued in the reminder of thi work. We denote by C k the pce of ll function f : tht re k time continuouly differentible, nd by C the pce of function whoe derivtive of ll order exit nd re continuou. The pce of ll mooth function whoe Mclurin erie converge for ll x i denoted by M, tht i M = f C fx = k= f k x x k for ll x k! where f k x tnd for the k-th derivtive of fx. We denote by S the Schwrtz cl of rpidly decreing function, tht i S = f C up x n f m x <, for ll m, n N. 2.1 x It i well known fct tht S i cloed under the Fourier trnform, which we define ˆfζ = fxe 2πixζ dx, with the invere Fourier trnform given by fx = ˆfζe 2πixζ dζ. In thi etting, we hve the following property of the Fourier trnform d dx fx ζ = 2πiζ ˆfx. 2.2 Let Ω, F, P be complete probbility pce, B t be tndrd Brownin motion on Ω, F, P nd F t t, be right-continuou, complete filtrtion uch tht: 1 for ech t,, the rndom vrible B t i F t -meurble; 2 for ll nd t uch tht < t, the rndom vrible B t B i independent of F. Following Ayed nd Kuo 1, we y tht tochtic proce X t i intntly independent with repect to F t t, if for ech t,, the rndom vrible X t i independent of F t. ecll tht if f t i dpted nd ϕ t i intntly independent with repect to F t, the Itô integrl of the product of f nd ϕ i defined the limit b n f t ϕ t db t = lim f ti 1 ϕ ti B ti B ti 1, 2.3 n i= whenever the limit exit in probbility. Note tht if ϕ 1, then the integrl defined in Eqution 2.3 reduce to the ordinry Itô integrl for dpted procee. Thi kind of integrl w firt introduced by Ayed nd Kuo 1, 2 nd tudied further by Kuo, Se-Tng nd Szozd 8, 9, 1. Following the nottion of 6, we denote by L 2 d Ω, b the pce of ll b dpted tochtic procee X t uch tht E X2 t db t <. It i well-known fct tht the Itô integrl i well-defined for procee from L 2 d Ω, b. A in the Itô theory of tochtic integrtion, the key tool ued in thi work will be the Itô formul. We tte below one of the reult of Kuo, Se-Tng,
LINEA SDES WITH ANTICIPATING INITIAL CONDITIONS 247 nd Szozd 1 where the uthor provide everl formul of thi type. Multidimenionl verion nd further generliztion of Itô formul together with n nticiption verion of the Girnov theorem cn be found in 7. Theorem 2.1 1, Corollry 6.2. Suppoe tht θt, x, y = τtfxϕy, where τ C 1, f C 2, nd ϕ M. Let X t = where α, β L 2 d Ω, b. Then α db + θt, X t, B b B = θ, X, B b B + + 1 2 + + β d, θ x, X, B b B dx 2 θ x 2, X, B b B dx 2 2 θ x y, X, B b B dx db θ t, X, B b B d. 2.4 Equivlently, we cn write the Eqution 2.4 in differentil form dθt, X t, B b B = θ x t, X t, B b B dx t + 1 2 θ 2 x 2, X t, B b B dx t 2 + 2 θ x y t, X t, B b B dx t db t + θ t t, X t, B b B d. 2.5 3. A Motivtionl Exmple In thi ection, we preent n exmple tht illutrte the method for obtining olution of Eqution 1.1. We begin with the implet poible ce of Eqution 1.1, tht i we et α 1, β nd px = x, nd retrict our conidertion to the intervl, 1. Thu we wih to find olution to dx t = X t db t, t, 1 3.1 X = B 1. The nturl gue for the olution of Eqution 3.1 i obtined by putting B 1 for x in Eqution 1.2 to obtin X t = B 1 exp B t 1 2 t. 3.2 Uing the Itô formul, it i ey to how tht the proce X t i not olution of Eqution 3.1, but it i olution of dx t = X t db t + e B t 1 2 t dt, 3.3 which i obviouly different from Eqution 3.1. The filure of thi pproch come the fct tht we do not ccount for the new fctor in the eqution, nmely
248 N. KHALIFA, H.-H. KUO, H. OUEDIANE, AND B. SZOZDA B 1. To ccount for B 1 in Eqution 3.1, we cn introduce correction term to X t tht will counterct the dt term ppering in Eqution 3.3. Now, we will ue the following n ntz for the olution of Eqution 3.1 X t = B 1 ξt exp B t 1 2 t, 3.4 where ξt i determinitic function. The reon for thi prticulr choice i imple. We ee tht the difference between Eqution 3.3 nd 3.1 i the term expb t 1 2t dt, nd to counterct thi, we need to introduce nother dt-term with the oppoite ign. Looking t the Itô formul in Theorem 2.1, we ee tht we hve to introduce correction fctor tht depend only on t. We ue the Itô formul from Theorem 2.1 with θt, x, y = y ξte x 1 2 t, nd to obtin θ t = ξ te x 1 2 t 1 2 y ξtex 1 2 t, θ x = y ξte x 1 2 t, θ xx = y ξte x 1 2 t, θ xy = e x 1 2 t, dθt, B t, B 1 = B 1 ξt e B t 1 2 dbt + 1 2 B 1 ξt e B t 1 2 dt + e B t 1 2 t dt ξ te B t 1 2 t + 1 2 B 1 ξte B t 1 t 2 dt = B 1 ξt e Bt 1 2 dbt + e Bt 1 2 t ξ te Bt 1 t 2 dt. So for X t = θt, B t, B 1 to be the olution of Eqution 3.1, function ξt h to tify the following ordinry differentil eqution ξ t = 1, t, 1 3.5 ξ =. Thu, with ξt = t, proce X t given in Eqution 3.4 i olution to tochtic differentil eqution 3.1, tht i X t = B 1 t exp B t 1 2 t 3.6 olve Eqution 3.1. We point out tht the olution in Eqution 3.6 coincide with the one tht cn be obtined by method of Buckdhn nd Nulrt 3, where in Propoition 3.2 uthor tte tht the unique olution of Eqution 3.1 h the form X t = gt, x exp B t 1 2 t, x=b1 where g olve prtil differentil eqution g t t, x = g x t, x, t, 1 g, x = x. In our ce, gt, x = x t.
LINEA SDES WITH ANTICIPATING INITIAL CONDITIONS 249 4. Generl Ce Theorem 4.1 give the olution to Eqution 1.1 for certin cl of coefficient α t nd β t, nd initil condition px with x = B b B. The proof of thi theorem ue the ide of correction term introduced in the previou ection. Theorem 4.1. Suppoe tht α L 2, b nd β L 2 d Ω, b. Suppoe lo tht p M S. Then the tochtic differentil eqution dx t = α t X t db t + β t X t dt, t, b 4.1 X = pb b B, h unique olution given by where nd X t = pb b B ξt, B b B Z t, 4.2 ξt, y = α p y Z t = exp α db + α u du d, 4.3 β 1 2 α2 d. emrk 4.2. Before we proceed with proof of Theorem 4.1, let u remrk tht if =, α t α nd β t β, tht i the coefficient re contnt nd evolution trt t, we cn gin pply the reult of Propoition 3.2 of 3. In our nottion, the bove mentioned propoition tte tht the olution to Eqution 4.1 h the form X t = gt, B 1 exp αb t + β 12 α2 t, 4.4 where gt, x i the olution of the following prtil differentil eqution g t t, x = αg x t, x t, b g, x = px. 4.5 Hence in order to how tht our olution nd the one given by Eqution 4.4 coincide, it i enough to how tht gt, x = px ξt, x olve Eqution 4.5. Note tht in the ce of contnt coefficient, gt, x = px αt. Now it i mtter of imple computtion to check tht g olve Eqution 4.5. Proof. The uniquene of olution follow from linerity of Eqution 4.1 nd tndrd rgument. To prove the exitence of olution, firt oberve tht Z t i olution of tochtic differentil eqution given by dz t = α t Z t db t + β t Z t dt, t, b Z = 1. Conider pbb dx t = d B ξt, B b B Z t = d pb b B Z t d ξt, Bb B Z t,
25 N. KHALIFA, H.-H. KUO, H. OUEDIANE, AND B. SZOZDA where ξt, y = ξ n ty n, for ll t, y. 4.6 n= Note tht ince the function zξ n ty n tifie the umption of the Theorem 2.1, we cn write dz t ξt, B b B = d ξ n tb b B n = = Z t n= d Z t ξ n tb b B n n= n= ξ n tb b B n dz t + Z t ξ tb b B n dt + Z t ξ n tnb b B n 1 dz t db t = ξt, B b B dz t + Z t ξ t t, B b B dt + ξ y t, B b B dz t db t. 4.7 Uing Theorem 2.1 nd Eqution 4.7 we obtin dx t = pb b B dz t + p B b B dz t db t ξ t t, B b B Z t dt + ξt, B b B dz t + ξ y t, B b B dz t db t = pb b B ξt, B b B dz t + p B b B dz t db t ξ t t, B b B Z t dt ξ y t, B b B dz t db t. So for X t to be olution of Eqution 4.1, we need p B b B dz t db t ξ t t, B b B Z t dt ξ y t, B b B dz t db t = 4.8 for ll t, b. Note tht dz t db t = α t Z t db t + β t Z t dt db t Putting together Eqution 4.8 nd 4.9 yield = α t Z t dt. 4.9 p B b B α t Z t dt ξ t t, B b B Z t dt ξ y t, B b B α t Z t dt =, or equivlently, p B b B α t ξ t t, B b B ξ y t, B b B α t Xt dt =.
LINEA SDES WITH ANTICIPATING INITIAL CONDITIONS 251 Hence it i enough to find ξt, y uch tht p yα t ξ ξ t t, y y t, yα t =, t, b ξ, y =. 4.1 Thu the problem of finding olution to the tochtic differentil eqution 4.1 h been reduced to tht of finding olution to the determinitic prtil differentil eqution 4.1. In order to olve Eqution 4.1, we pply the Fourier trnform to both ide of Eqution 4.1, to obtin p ζα t t ξt, ζ 2πiζ ξt, ζα t =. 4.11 Note tht Eqution 4.11 i n ordinry differentil eqution in t, with n integrting fctor exp 2πiζ α d. Hence Eqution 4.11 i equivlent to t ξt, ζ exp 2πiζ α d = p t ζα t exp 2πiζ α d. 4.12 Integrtion with repect to t of both ide of Eqution 4.12 yield ξt, ζ exp 2πiζ α d = p ζ α exp 2πiζ α u du d+ĉζ, 4.13 for ome function Ĉζ S. Thu, the Fourier trnform of function ξt, y, tht i olution of Eqution 4.1, i given by ξt, ζ = p ζ α exp 2πiζ α u du d + Ĉζ exp 2πiζ α d. 4.14 Now, we pply the invere Fourier trnform to get ξt, y = p ζ α exp 2πiζ α u du d exp 2πiyζ dζ = = + + Ĉζ exp α 2πiζ p ζ exp πiζ Ĉζ exp α p y πiζ y α d exp 2πiyζ dζ y α u du d + C α u du dζ d α u du dζ y α d. Uing the initil condition from Eqution 4.1, we ee tht Cy. Hence X t in Eqution 4.2 i olution of Eqution 4.1.
252 N. KHALIFA, H.-H. KUO, H. OUEDIANE, AND B. SZOZDA emrk 4.3. Although very tediou, it i trightforwrd to check tht function ξt, y in Eqution 4.3 cn be expreed in the form of Eqution 4.6. 5. Exmple Below we give everl exmple of tochtic differentil eqution with either determinitic or nticipting initil condition. It i intereting to compre the olution to ee how nticipting initil condition ffect the olution. Exmple 5.1 Adpted. Eqution dx t = X t db t + X t dt X = x h olution given by X t = x exp B t + 1 2 t. Exmple 5.2 Anticipting, compre with Exmple 5.1. Eqution dx t = X t db t + X t dt X = B 1 h olution given by X t = B 1 t exp B t + 1 2 t. Exmple 5.3 Anticipting, compre with Exmple 5.1. Eqution dx t = X t db t + X t dt X = e B1 h olution given by X t = e B1 t exp B t 1 2 t Exmple 5.4 Adpted. Eqution dx t = α t X t db t + β t X t dt X = x h olution given by X t = x exp α db + β 1 2 α2 d. Exmple 5.5 Anticipting, compre with Exmple 5.4. Eqution dx t = α t X t db t + β t X t dt X = B 1 h olution given by X t = B 1 α d exp α db + β 1 2 α2 d.
LINEA SDES WITH ANTICIPATING INITIAL CONDITIONS 253 Acknowledgment. Hui-Hiung Kuo i grteful for the upport of Fulbright Lecturing/eerch Grnt 11-6681, April 1 to June 3, 212 t the Univerity of Tuni El Mnr, Tunii. Benedykt Szozd cknowledge the upport from The T.N. Thiele Centre For Applied Mthemtic In Nturl Science nd from CEATES DNF78, funded by the Dnih Ntionl eerch Foundtion. eference 1. Ayed, W. nd Kuo, H.-H.: An extenion of the Itô integrl, Communiction on Stochtic Anlyi 2, no. 3 28 323 333. 2. Ayed, W. nd Kuo, H.-H.: An extenion of the Itô integrl: towrd generl theory of tochtic integrtion, Theory of Stochtic Procee 1632, no. 1 21 1 11. 3. Buckdhn,. nd Nulrt, D.: Liner tochtic differentil eqution nd Wick product, Probbility Theory nd elted Field 99, no. 4 1994 51 526. 4. Eunge, J.: A cl of nticipting liner tochtic differentil eqution, Communiction on Stochtic Anlyi 3, no. 1 29 155 164. 5. Itô, K.: Stochtic integrl, Proceeding of the Imperil Acdemy 2, no. 8 1944 519 524. 6. Kuo, H.-H.: Introduction to Stochtic Integrtion, Univeritext, Springer, 26. 7. Kuo, H.-H., Peng, Y., nd Szozd, B.: Itô formul nd Girnov theorem for new tochtic integrl, preprint 213 8. Kuo, H.-H., Se-Tng, A., nd Szozd, B.: A tochtic integrl for dpted nd intntly independent tochtic procee, in Advnce in Sttitic, Probbility nd Acturil Science Vol. I, Stochtic Procee, Finnce nd Control: A Fetchrift in Honour of obert J. Elliott ed.: Cohen, S., Mdn, D., Siu, T. nd Yng, H., World Scientific, 212, 53 71. 9. Kuo, H.-H., Se-Tng, A., nd Szozd, B.: An iometry formul for new tochtic integrl, In Proceeding of Interntionl Conference on Quntum Probbility nd elted Topic, My 29 June 4, 211, Levico, Itly, QP PQ: Quntum Probbility nd White Noie Anlyi 29 213 222 232. 1. Kuo, H.-H., Se-Tng, A., nd Szozd, B.: The Itô formul for new tochtic integrl, Communiction on Stochtic Anlyi 6, no. 4 212 64 614. Nrje Khlif: Deprtment of Mthemtic, Univerity of Tuni El Mnr, 16 Tuni, Tunii E-mil ddre: nrjekhlif@yhoo.fr Hui-Hiung Kuo: Deprtment of Mthemtic, Louiin Stte Univerity, Bton ouge, LA 783, USA E-mil ddre: kuo@mth.lu.edu UL: http://www.mth.lu.edu/ kuo Hbib Ouerdine: Deprtment of Mthemtic, Univerity of Tuni El Mnr, 16 Tuni, Tunii E-mil ddre: hbib.ouerdine@ft.rnu.tn Benedykt Szozd: The T.N. Thiele Centre for Mthemtic in Nturl Science, Deprtment of Mthemticl Science & CEATES, School of Economic nd Mngement, Univerity of Arhu, Arhu, Denmrk E-mil ddre: zozd@imf.u.dk