Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H > 1 2

Transcription

1 Sochasic delay differenial equaions driven by fracional Brownian moion wih Hurs parameer H > 1 2 by Marco Ferrane 1, and Carles Rovira 2 1 Diparimeno di Maemaica Pura ed Applicaa, Universià degli Sudi di Padova, via Belzoni 7, I Padova, Ialy marco.ferrane@unipd.i, hp:// ferrane/index.hml 2 Facula de Maemàiques, Universia de Barcelona, Gran Via 585, 87-Barcelona, Spain carles.rovira@ub.edu Absrac We consider he Cauchy problem for a sochasic delay differenial equaion driven by a fracional Brownian moion wih Hurs parameer H > 1 2. We prove an exisence and uniqueness resul for his problem, when he coefficiens have enough regulariy. Furhermore, if he diffusion coefficien is bounded away from zero and he coefficiens are smooh funcions wih bounded derivaives of any order, we prove ha he law of he soluion admis a smooh densiy wih respec o Lebesgue s measure on R. Running ile: SDDE wih fbm. Keywords: fracional Brownian moion, sochasic delay differenial equaion. AMS subjec classificaions: 6H5, 6H7, 6H1, 34K5.

2 1 Inroducion A general heory for he sochasic differenial equaions (SDE) driven by a fracional Brownian moion (fbm) is no ye esablished and jus a few resuls have been proved (see e.g. Nualar and Rascanu (22), Nualar and Ouknine (22, 23), and Couin and Qian (2)) using differen approach. Acually, he same definiion of he sochasic inegraion wih respec o fbm is no ye complely esablished and differen approach have been proposed in he las years (see, among ohers, Alòs and Nualar (23), Carmona and Couin (2) and Couin and Qian (22)). Due o he iniial sage of he general heory, i could appear a leas srange ha one plans o consider he class of he sochasic delay differenial equaions (SDDE). Indeed, his equaions provide in general an infinie dimensional problem, much more difficul o be solved compared o he cusomary SDE s. Neverheless, hey include also problems ha are easier o be solved compared o he SDE s, bu ha are sill of grea ineres in he applicaions, for example o finance (see he recen papers of Arriojas el al. (23) and, in a more complicae case, of Hobson and Rogers (1998)). In he presen paper we shall consider he Cauchy problem for a SDDE dx() = b(x())d + σ(x( r))db(), [, T ] X(s) = φ(s) s [ r, ], where φ C([ r, ]) and he noise process {B(), } represen a fracional Brownian moion (fbm) wih Hurs parameer H > 1. As a soluion 2 o his problem, we shall define a process {X(), [ r, T ]} saisfying X() = φ() + b(x(s))ds + X(s) = φ(s) s [ r, ], σ(x(s r))db(s), [, T ] where φ C([ r, ]). The sochasic inegral, ha appear in he previous equaion, will be he Sraonovich inegral wih respec o he fbm, recenly developed by some auhors (see Definiion 1 of he nex secion). Neverheless, all he resuls presened in his paper can also be obained dealing wih Skorohod inegrals. To find a soluion, we shall firs solve he equaion wihin he inerval [, r]; hen, we use his soluion process as he iniial daa o solve he 2

3 equaion wihin he inerval [r, 2r], and so on. This procedure allow us o consruc a soluion sep by sep, providing a any sage is uniqueness and is regulariy. Wih he same approach, we will be able o prove, under he cusomary assumpion of non degeneracy of he diffusion coefficien, ha he law of he soluion a any ime, admis a densiy wih respec o Lebesgue s measure on R. Le us give now some remarks abou he way one can ge hese resuls. We have used he classical echniques of sochasic calculus combined wih some special properies of fbm. The delay allows us o avoid some of he usual problems ha appear working wih fbm and o use classical mehods as Picard s ieraions. Our paper is divided as follows: in he nex secion we will inroduce he basic noaions concerning he fracional Brownian moion and we recall some resuls aken basically from Nualar (23) and Alòs and Nualar (23). Secion 3 is devoed o sae he exisence and uniqueness of our sochasic delay differenial equaion driven by fbm. Finally, in secion 4, we obain he smoohness of he densiy of such soluion. 2 Fracional Brownian moion Le us sar wih some basic facs abou he fracional Brownian moion (fbm) and he sochasic calculus ha can be developed wih respec o his process. 1 Fix a parameer < H < 1. The fracional Brownian moion of Hurs 2 parameer H is a cenered Gaussian process B = {B(), [, T ]} wih he covariance funcion R (, s) = 1 ( s 2H + 2H s 2H). (1) 2 Le us assume ha B is defined in a complee probabiliy space (Ω, F, P ). One can show (see, for insance, Alòs and Nualar (23)) ha R (, s) = s where K(, s) is he kernel defined by K(, r)k(s, r)dr, (2) K(, s) = c H s 1 2 H (r s) H 3 2 r H 1 2 dr, s 3

4 [ for s <, where c H = H(2H 1) B(2 2H,H 1 2 ) ] 1/2 and B(α, β) is he Bea funcion. We assume ha K(, s) = if s >. I is worh o noice ha equaion (2) implies ha R is nonnegaive definie and, herefore, here exiss a Gaussian process wih his covariance. Le us denoe by E he se of sep funcions on [, T ]. Le H be he Hilber space defined as he closure of E wih respec o he scalar produc 1[,], 1 [,s] = R(, s). One can show ha R(, s) = α H where α H = H(2H 1). This implies ϕ, ψ H = α H T T H s r u 2H 2 dudr, r u 2H 2 ϕ(r)ψ(u)dudr (3) for all ϕ and ψ in E. The mapping 1 [,] B() can be exended o an isomery beween H and he firs chaos H 1 associaed wih B, and we denoe his isomery by ϕ B(ϕ). The elemens of H may no be funcions bu disribuions of negaive order. Due o his reason, i is convenien o inroduce he Banach space H of measurable funcions ϕ on [, T ] saisfying ϕ 2 H := α H T T ϕ(r) ϕ(u) r u 2H 2 drdu <. (4) One can prove (Pipiras and Taqqu, 2) ha he space H equipped wih he inner produc ϕ, ψ H is no complee and i is isomeric o a subspace of H, ha we will idenify wih H. The coninuous embedding L 1 H ([, T ]) H has been proved in Mémin e al. (21). 2.1 Malliavin calculus and sochasic inegrals for he fbm In order o consruc a sochasic calculus of variaions wih respec o he Gaussian process B, we shall follow he general approach inroduced, for insance, in Nualar (1995) (see also Nualar el al., 23). Le us recall he 4

5 definiion of he derivaive and divergence operaors and some basic facs of his sochasic calculus of variaions, aken mainly from Alòs and Nualar (23). Le S be he se of smooh and cylindrical random variables of he form F = f(b(φ 1 ),..., B(φ n )), (5) where n 1, f Cb (R n ) (f and all is parial derivaives are bounded), and φ i H. The derivaive operaor D of a smooh and cylindrical random variable F of he form (5) is defined as he H-valued random variable DF = n j=1 f x j (B(φ 1 ),..., B(φ n ))φ j. The derivaive operaor D is hen a closable operaor from L p (Ω) ino L p (Ω; H) for any p 1. For any k 1 se D k he ieraion of he derivaive operaor. For any p 1 he Sobolev space D k,p is he closure of S wih respec o he norm ( k F p k,p = E F p + E D i F ). p H i Proceeding as before, given a Hilber space V we denoe by D 1,p (V ) he corresponding Sobolev space of V -valued random variables. The divergence operaor δ is he adjoin of he derivaive operaor, defined by means of he dualiy relaionship i=1 E(F δ(u)) = E DF, u H, (6) where u is a random variable in L 2 (Ω; H). We say ha u belongs o he domain of he operaor δ, denoed by Dom δif he mapping F E DF, u H is coninuous in L 2 (Ω). A basic resul says ha he space D 1,2 (H) is included in Dom δ. Two basic properies of he divergence operaor will follow: i) For any u D 1,2 (H) Eδ(u) 2 = E u 2 H + E Du, (Du) H H, (7) where (Du) is he adjoin of (Du) in he Hilber space H H. 5

6 ii) For any u D 2,2 (H), δ(u) belongs o D 1,2 and for any h in H Dδ(u), h H = δ ( Du, h H ) + u, h H. (8) Le us now consider he space H H H H of measurable funcions ϕ on [, T ] 2 such ha ϕ 2 H H : = α 2 H [,T ] 4 ϕ(r, s) ϕ(r, s ) r r 2H 2 s s 2H 2 drdsdr ds <. Le us denoe by D 1,2 ( H ) he space of processes u such ha E u 2 H + E Du 2 H H <. (9) Then D 1,2 ( H ) is included in D 1,2 (H), and for a process u in D 1,2 ( H ) we can wrie E u 2 H = α H u(s)u(r) r s 2H 2 drds (1) [,T ] 2 and E Du, (Du) H H (11) = αh 2 D r u(s)d r u(s ) r s 2H 2 r s 2H 2 drdr dsds. [,T ] 4 The elemens of D 1,2 ( H ) are sochasic processes and we will make use of he inegral noaion δ(u) = T u()δb(), and we will call his inegral he Skorohod inegral wih respec o he fbm. Moreover, if u D 1,2 ( H ) one can also define an indefinie inegral process given by X = u(s)δb(s). Le us define now a Sraonovich ype inegral wih respec o B. By convenion we pu B() = if / [, T ]. Following he approach by Russo and Vallois (1993) we can give he following definiion: Definiion 1 Le u = {u(), [, T ]} be a sochasic process wih inegrable rajecories. The Sraonovich inegral of u wih respec o B is defined as he limi in probabiliy as ε ends o zero of T (2ε) 1 u(s)(b(s + ε) B(s ε))ds, provided his limi exiss, and i is denoed by T u()db(). 6

7 I has been shown in Alòs and Nualar (23) ha a process u D 1,2 ( H ), such ha T T a.s. is Sraonovich inegrable and T u(s)db(s) = T D s u() s 2H 2 dsd < u(s)δb(s) + α H T T D s u() s 2H 2 dsd. (12) On he oher hand, if he process u has a.s. λ-hölder coninuous rajecories wih λ > 1 H, hen he Sraonovich inegral T u(s)db(s) exiss and coincides wih he pah-wise Riemann-Sieljes inegral. When ph > 1, can also be inroduced he space L 1,p H D 1,2 ( H ) such ha [ T u p,1 = of processes u ( T ( T ) phds )] E( u(s) p )ds + E D r u(s) 1 1 p H dr I is also known (Nualar, 23) ha ( E [,T ] u(s)db(s) p) C u p p,1, where he consan C > depends on p, H and T. 2.2 Some useful resuls <. In he nex secion, we shall need some addiional resuls. They provide sufficien condiions for processes Z = {Z(), [, T ]} o belong o he space D 1,2 ( H ). Le us recall once more ha hese resuls holds in he case H > 1. 2 Lemma 2 Le Z = {Z(), [, T ]} be a sochasic process such ha for any [, T ], Z() D 1,2 and s E( Z(s) 2 ) c 1 and E( D r Z(s) 2 ) c 2. r,s Then he sochasic process Z belongs o D 1,2 ( H ) and E Z 2 H + E DZ 2 H H < c H,T (c 1 + c 2 ). 7

8 Proof. The proof follows he compuaions given in Nualar (23). For insance, we can compue E DZ 2 H H ( ) = E αh 2 D r Z(s) D r Z(s ) r s 2H 2 s r 2H 2 drdsdr ds [,T ] ( 4 ) E αh 2 D r Z(s) 2 r s 2H 2 s r 2H 2 drdsdr ds [,T ] ( ) 4 T αh 2 2H 1 2 H 1 E( D r Z(s) 2 )drds 2 [,T ] ( ) 2 T αh 2 2H 1 2 H 1 T 2 E( D r Z(s) 2 ) 2 r,s = c H,T c 2. Lemma 3 Le Z = {Z(), [, T ]} be a sochasic process such ha for any [, T ], Z() D 1,2 and s E( Z(s) 2 ) c 1 and E( D r Z(s) 2 ) c 2. r,s Then given r > and f a deerminisic coninuous funcion, he sochasic process V = {V (), [, T ]} defined as { Z( r), si > r, V () = f(), si < r, belongs o D 1,2 ( H ). Proof. I is clear ha V belongs o D 1,2 (H) and ha { Ds Z( r), si > r, D s V () =, si < r. Then i is enough o apply Lemma 2. Given s = (s 1,..., s k ) [, T ] k ; we denoe by s he lengh of s, ha means k. For a random variable Y D k,p and s [, T ] k, we denoe by 8

9 Ds k Y he ieraive derivaive D sk D sk 1 D s1 Y. Le f C, b (R), he space of coninuous funcions defined on R infiniely differeniable wih bounded derivaives. Se Γ s (f; Y ) = s m=1 f (m) (Y ) m i=1 D p i p i Y, where he second sum exends o all pariions p 1,, p m of lengh m of s. We also need he following lemma ha is an exension of he resul proved in Rovira and Sanz (1996). Lemma 4 Le {F n, n 1} be a sequence of random variables in D k,p, k 1, p 2. Assume here exiss F D k 1,p such ha {D k 1 F n, n 1} converges o D k 1 F in L p (Ω; [, T ] (k 1) ) as n goes o infiniy and, moreover, he sequence {D k F n, n 1} is bounded in L p (Ω; [, T ] (k 1) ). Then, F D k,p. 3 Exisence and uniqueness Le B = {B(), [, T ]} be a one dimensional fracional Brownian moion (fbm) wih Hurs parameer H > 1 2. Le us define he following sochasic delay differenial equaion (SDDE) driven by a fbm X() = φ() + b(x(s))ds + σ(x(s r))db(s), [, T ] (13) where r > and φ C([ r, ]). For simpliciy le us assume T = Mr. The sochasic inegral in (13) has o be inended as he Sraonovich ype inegral defined before (see Definiion 1). We shall assume ha b and σ are real funcions ha saisfy he following condiions: Hypoheses (H) : b and σ are M-imes differeniable funcions wih bounded derivaives up o order M. Moreover, σ is bounded and b() c 1 for some consan c 1. Theorem 5 Under Hypoheses (H), he SDDE (13) admis a unique soluion X on [, T ]. 9

10 The proof of his heorem is based in he following lemmas and proposiions. Lemma 6 Le M = {M(), [, T ]} be a quadraic inegrable sochasic process. Assume ha b is a Lipschiz funcion defined on R, such ha b() c 1 for some consan c 1. Then, fixed T 1 T, he sochasic inegral equaion X() = x + b(x(s))ds + M(), [, T 1 ], (14) X() = if > T 1, admis a unique soluion X on [, T ]. Proof. In order o prove he exisence and uniqueness, we can prove ha he classical Picard-Lindelöf ieraions converge o a soluion of (14). Consider { X (n+1) () = x + b(x(n) (s))ds + M(), X () (15) () = x + M(), for [, T 1 ], and X (n) () = for any T 1 and all n. Noice ha we only need o deal wih [, T 1 ]. We have E( X (1) () X () () 2 ) K 2 E( uniformly in. For a generic n we hus have (x 2 + M(s) 2 )ds) + K 2 K 2 E( X (n+1) () X (n) () 2 ) K 2 E( X (n) (s) X (n 1) (s) 2 )ds s1 sn 1 K2 n 1 E( X (1) (s) X () (s) 2 )ds n... ds 1 K Kn 1 2 (16) n! From his we can easy prove ha he Picard-Lindelöf ieraions converge in L 2 (Ω) on [, T 1 ] o a soluion for equaion (14). A similar argumen give he uniqueness. Le us inroduce some new noaion. Fixed m 1 and p 2, we will say ha a sochasic process Z = {Z(), [, T ]} saisfies condiion (D (,m,p) ) if Z() belongs o D m,p for any [, T ] and E( Z() p ) c 1,N,p and 1 u, u =k E( D k u Z() p ) c 2,N,k,p,

11 for any k m and for some consans c 1,N,p, c 2,N,k,p. Noice ha if Z saisfies condiion (D (,m,p) ), hen lemma 2 yields ha Z belongs o D 1,2 ( H ). Proposiion 7 Le M = {M(), [, T ]} be a sochasic process saisfying condiion (D (,m,p) ). Assume ha b has bounded derivaives up o order m and ha b() c 1 for some consan c 1. Then, fixed T 1 T, he sochasic inegral equaion X() = x + b(x(s))ds + M(), [, T 1 ], (17) X() = if > T 1, admis a unique soluion X on [, T ]. Moreover he sochasic process X = {X(), [, T ]} saisfies condiion (D (,m,p) ). Proof. The exisence of a unique soluion has been proved in Lemma 6, where we have proved ha he Picard-Lindelöf ieraions converge o a soluion of (17). Some easy compuaions give us ha so E( X() p ) K p (E( E( X() p ) K p (1 + b(x(s))ds p ) + E( M() p ) + x p ), and finally we ge using a Gronwall s lemma ha ) E( X(s) p )ds E( X() p ) < c 1,N,p <. In order o check ha for all k m, X() belongs o D k,p for all [, T ] and E( D ux() k p ) c 2,N,k,p, u, u =k we will use he Picard-Lindelöf ieraions defined in (15). We consider he hypohesis of inducion, for k m, (Ĥk): (a) for all n, X (n) () D k,p for all. (b) D k 1 X (n) () converges o D k 1 X() in L p (Ω, [, T ] k 1 ) when n ends o. 11

12 (c) n u, u =k E( D k u X (n) () p ) K p,k <. Noice ha hypohesis (Ĥk) implies ha X() D k,p. Observe ha we only need o sudy he case [, T 1 ], since when > T 1 all he resuls are obvious. Sep 1. We prove (Ĥ1), ha is, he case k = 1. Since X (n) () converges, when n ends o o X() in L p (Ω) we know ha (b) is rue. In order o prove (a) and (c), we will use anoher inducion argumen o check for all n he hypohesis ( H n ) (i) X (n) () D 1,p for all [, T ]. (ii) u E( Du X (n) () p ) K n,p,1 <. From de definiion of X () i is clear ha X () () D 1,p for all and ha for [, T 1 ] D u X () () = D u M(). So, our hypohesis of inducion ( H ) has been proved. Assume now ha he hypohesis of inducion ( H n ) is rue. Then from he definiion of X (n+1) i follows ha X (n+1) () D 1,p and for any [, T 1 ] D u X (n+1) () = b (X (n) (s))d u X (n) (s)ds + D u M(). Moreover, using ha M saisfies condiion (D,m,p ) we have E( Du X (n+1) () p ) u ( K p u K n+1,p <. From here, ( H n+1 ) can be easily proved. Finally, since we have he relaionship u E( Du X (n+1) () p ) K p E( Du X (n) (s) p )ds + E( D u M() p ) u E( Du X (n) (s) p )ds + K p, u ) 12

13 ieraing n imes his inequaliy we ge u E( Du X (n+1) () p ) (K p ) 2 So we have ha n (K p ) k= s k+1 k k! K p exp(k p ). n (Ĥ1) has been proved. E( Du X (n 1) (v) p )dvds + Kp 2 + K p u E( Du X (n) () p ) K p exp(k p T ) <. u Noice now ha applying lemma 4 we have ha, for all [, T ], X() D 1,p. Moreover we can obain ha D u X() = and we easily have ha b (X(s))D u X(s)ds + D u M(), E( D u X() p ) <. u Sep 2. Le us assume ha (Ĥi), i k m 1 is rue. We wan o check (Ĥk+1). We will prove (a) doing anoher inducion over n similar o he inducion done in Sep 1. Le us consider, for all n, he hypohesis ( H n ) (i) X (n) () D k+1,p for all [, T ]. (ii) u, u =k+1 E( D k+1 u X (n) () p ) K n,p,1 <. Since for all, M() D k+1,p, from he definiion of X () i is clear ha, for all, X () () D k+1,p and ha for u, u = k + 1 u X () () = Du k+1 M(), D k+1 13

14 for any [, T 1 ]. Then ( H ) is rue. Assuming now ha is rue unil n, from he definiion of X (n+1) i follows ha, for all, X (n+1) () D k+1,p and for u, u = k + 1 D k+1 u X (n+1) () = Γ u (b; X (n) (s))ds + Du k+1 M(), for any [, T 1 ]. The proof of (b) can be obained easily from he expressions of D k ux (n) () and D k ux(). Finally o prove (c) se u (b; X (n) (s)) = Γ u (b; X (n) (s)) b (k+1) (X (n) (s))du k+1 X (n) (s). Noice ha using by he hypohesis of inducion u, u =k+1 E( u (b; X (n) (s)) p ) K p. s Then D k+1 u X (n+1) () = u (b; X (n) (s))ds +D k+1 u M() + b (k+1) (X (n) (s))du k+1 X (n) (s)ds. Repeaing he same calculaions we did in he proof of (Ĥ1) we can finish he proof of (Ĥk+1). Noice now ha applying again lemma 4 we have ha, for all, X() D k+1,p and we easily have ha u, u =k+1 E( D k+1 u X() p ) K p. The following Proposiion sudies he behavior of he sochasic inegral. Proposiion 8 Le Y = {Y (), [, T ]} be a sochasic process saisfying condiion (D,m+1,p ). Then he sochasic Sraonovich inegral M() := Y (s)db(s), [, T ], (18) is well defined and he sochasic process M = {M(), [, T ]} saisfies condiion (D,m,p ). 14

15 Proof. Clearly, Y is Sraonovich inegrable, bu he Sraonovich inegral and he divergence operaor do no coincide and we have ( T ) Y (s)db(s) = δ(y 1 [,] ) + α H D v Y (s) s v 2H 2 dv ds. Then for insance ( p) E( M() p ) = E Y (s)db(s) K p E ( δ(y 1[,] ) ( p ) ( T + Kp E α H T ( T K p Y p p,1 + K pc H D v Y (s) p s,v K p <. So we have ha E ( M() p ) K p. ) D v Y (s) s v 2H 2 dv ) s v 2H 2 dv ds p (19) p) ds On he oher hand, using an inducion argumen i is easy o check ha for any k m, M() D k,p for any and D k (u 1,...,u k )M() = k i=1 D k 1 (u 1,...,û i,...,u k ) Y (u i)1 [,] (u i ) + δ(d k uy 1 [,] ), where (u 1,..., û i,..., u k ) denoes he poin u wihou he componen u i. Repeaing he argumen we have used in (19) we obain ha for all k m u, u =k E( D k u M() p ) K p. Proof of Theorem. To prove ha equaion (13) admis a unique soluion on [, T ], we shall firs prove he resul on [, r]. Then by inducion, we shall prove ha if equaion (13) admis a unique soluion on [, Nr], hen we can exend his soluion o he inerval [, (N + 1)r] and ha his exension is unique. Acually our hypohesis of inducion, for N M, is he following: 15

16 (H N ) The equaion X() = φ() + b(x(s))ds + σ(x(s r))db(s), [, Nr], X() = if > Nr, has an unique soluion. Moreover for all p 2, X() saisfies condiion (D (, M N,p) ). Noice ha for simpliciy we omi he dependence on N of he soluion X. Noice also ha in each sep we loose one degree of regulariy. Check (H 1 ). Le [, r]; equaion (13) can be wrien in he following easy form X() = φ() + b(x(s))ds + σ(φ(s r))db(s). (2) Le us define he process M() := σ(φ(s r))1 {<r} db(s), for [, T ]. Since φ is deerminisic, i is immediae o see ha σ(φ( r)) D 1,2 ( H ) and ha Dσ(φ(s r)) =. For his reason he sochasic inegral in (2) is well defined and coincides wih he divergence operaor, due o (12). Again since φ is a deerminisic coninuous funcion we have ha for all k 1, p 2, M() D k,p and and and D u M() = σ(φ(u r))1 {u<<r}, D k M() =, when k 2. Then, for all k 1, p 2 we have ha E( M() p ) σ(φ(. r)) p p,1 c 3,1,p u, u =k E( D k u M() p ) c 4,1,k,p, (21) 16

17 for some consans c 3,1,p, c 4,1,k,p. So, we have proved ha M saisfies condiion (D (,k,p) ) for any k 1. From Proposiion 7 we have ha here exis a unique soluion X and ha his soluion saisfies condiion (D (, M 1,p) ). Inducion Assume ha (H N ) is rue unil N wih N < M. We wan o check (H N+1 ). Consider he sochasic process {Z(), [, T ]} defined as ϕ( r), si r, Z() = X( r), si r < (N + 1)r,, si > (N + 1)r, where here X is he soluion obained in (H N ). Se now Y () = σ(z()). Then our problem became for [, (N + 1)r] X() = φ() + Le us define he process M() := b(x(s))ds + Y (s)1 {<(N+1)r} db(s), [, T ]. Y (s)db(s). (22) and prove ha he Sraonovich inegral is well defined. To his aim, we shall need o prove ha (as poined ou in he previous chaper): 1. Y D 1,2 ( H ); 2. T T D uy (s) s u 2H 2 dsdu < These wo condiions can be obained from Lemma 3 and he following facs: σ is a bounded funcion wih bounded derivaives, Z is in D 1,2 ( H ), u E( D u Z() p ) c 2,1,p, and D u Y () = σ (Z())D u Z(). On he oher hand, using ha he sochasic process Z saisfies condiion (D (, M N,p) ) and ha σ has derivaives up o order M, i is clear ha for any [, T ], Y () D k,p for all k M N and D u Y () = σ (Z())D u Z() 17

18 and sill more generally, for k M N D k uy () = Γ u (σ, Z()). Furhermore, Y will also saisfy condiion (D (, M N,p) ). Applying Proposiion 8 we ge ha M saisfies condiion (D (, M N 1,p) ). Finally, using Proposiion 7, as we did in Sep 1, we finish he proof of his Theorem. Remark 9 Noice ha in he proof of he exisence and uniqueness of soluion o our sochasic differenial equaion we need o sudy he Malliavin derivaives unil order M of our soluion, in order o define he sochasic inegrals appearing in our Picard ieraions. To do his sudy, we assume ha he coefficiens σ and b have bounded derivaives up o order M, ha are no he usual assumpions in his case when he sochasic differenial equaion is driven by a sandard Brownian moion. As a by-produc of our mehod, we can prove easily ha if he coefficiens σ and b have bounded derivaives of any order, he soluion X() belongs o D. Moreover assuming he nondegeneracy on σ we can also have he smoohness of he densiy. 4 Regulariy of he densiy Le us consider now anoher se of hypoheses (Ĥ). Hypoheses (Ĥ) : b and σ are real funcions wih bounded derivaives of any order. Moreover, σ is bounded and b() c 1 for some consan c 1. Theorem 1 Assume Hypoheses (Ĥ). Then if here exiss a posiive consan c such ha σ(x) > c for all x, for any [, T ] he soluion of he SDDE (13) X() has an infiniely differeniable densiy wih respec o Lebesgue s measure on R. Proof. Fixed [, T ] and using he Malliavin s crierion for he exisence of a smooh densiy we have o check wo hings: 1. X() D, 2. ( T D ux() 2 du) 1 p 1 Lp (Ω). 18

19 Following he same seps of Theorem 5, we can prove ha for any [, T ], X() D. In order o prove he second condiion i is enough o check ha for any p 1 here exiss ε > such ha for all ε ε. From equaion (13) we can wrie X() = φ() + P ( T D u X() 2 du ε ) ε p, So, for any u r we obain D u X() = b(x(s))ds + σ(x(s r))δb(s) ( T +α H D v σ(x(s r)) s v 2H 2 dv ) ds. + u b (X(s))D u X(s)ds + σ(x(u r)) u+r +α H σ (X(s r))d u X(s r)δb(s) u+r and when u ( r, ) we have D u X() = u ( s r D u D v σ(x(s r)) s v 2H 2 dv ) ds, b (X(s))D u X(s)ds + σ(x(u r)). Then, using a Gronwall s inequaliy we have ha E ( D u X(s) q) K q. u ( r,),s ( r,) On he oher hand, we can wrie P ( T D u X() 2 du ε ) P ( ε α u p 1,ε + p 2,ε, b (X(s))D u X(s)ds + σ(x(u r)) 2 du ε ) 19

20 wih p 1,ε = P ( ε α u u ( ε α,) p 2,ε = P ( u ( ε α,) b (X(s))D u X(s)ds + σ(x(u r)) 2 du ε, ε α b (X(s))D u X(s) ds ε β) ε α b (X(s))D u X(s) ds > ε β). Since σ(x) > c for all x, when α < 1 we clearly have ha p 1,ε =. On he oher hand, using Chebyshev s inequaliy, for any q > 1 p 2,ε 1 E( εβq u ( ε α,) ε α b (X(s))D u X(s) ds q) ε (α β)q KE ( D u X(s) q). u ( ε α,),s ( ε α,) So, choosing β < α < 1 he proof is complee. Acknowledgemens Marco Ferrane was parially pored by he gran COFIN of MIUR and a fellowship gran of he Cenre de Recerca Maemáica (Bellaerra, Barcelona), insiuion ha he auhor wish o hank for is warm hospialiy. Carles Rovira was parially pored by he DGES gran BFM References Alòs, E. and Nualar, D. (23) Sochasic inegraion wih respec o he fracional Brownian moion. Sochasics and Sochasics Rep. 75, Arriojas, M., Hu, Y., Mohammed, S.A.D. and Pap, Y. (23) A Delayed Black and Scholes Formula. Preprin. Carmona, P. and Couin, L. (2) Sochasic inegraion wih respec o fracional Brownian moion. C.R. Acad. Sci. Paris Ser. I. Mah. 33,

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