Malliavin calculus for fractional delay equations
aa r X i v : . [ m a t h . P R ] D ec MALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONSJORGE A. LEÓN AND SAMY TINDELAbstra t. In this paper we study the existen e of a unique solution to a general lass ofYoung delay di(cid:27)erential equations driven by a Hölder ontinuous fun tion with parametergreater that / via the Young integration setting. Then some estimates of the solutionare obtained, whi h allow to show that the solution of a delay di(cid:27)erential equation drivenby a fra tional Brownian motion (fBm) with Hurst parameter H > / has a C ∞ -density.To this purpose, we use Malliavin al ulus based on the Fré het di(cid:27)erentiability in thedire tions of the reprodu ing kernel Hilbert spa e asso iated with fBm.1. Introdu tionThe re ent progresses in the analysis of di(cid:27)erential equations driven by a fra tionalBrownian motion, using either the omplete formalism of the rough path analysis [3, 10,18℄, or the simpler Young integration setting [25, 33℄, allow to study some of the basi properties of the pro esses de(cid:28)ned as solutions to rough or fra tional equations. Thisglobal program has already been started as far as moments estimates [13℄, large deviations[16℄, or properties of the law [2, 21℄ are on erned. It is also natural to onsider someof the natural generalizations of di(cid:27)usion pro esses, arising in physi al appli ations, andsee if these equations have a ounterpart in the fra tional Brownian setting. Some partialdevelopments in this dire tion on ern pathwise type PDEs, su h as heat [7, 11, 12, 30℄,wave [28℄ or Navier-Stokes [4℄ equations, as well as Volterra type systems [5, 6℄. As weshall see, the urrent paper is part of this se ond kind of proje t, and we shall deal withsto hasti delay equations driven by a fra tional Brownian motion with Hurst parameter H > / .Indeed, we shall onsider in this arti le an equation of the form: dy t = f ( Z yt ) dB t + b ( Z yt ) dt, t ∈ [0 , T ] , (1)where B is a d -dimensional fra tional Brownian motion with Hurst parameter H > / , f : C γ ([ − h, R n ) → R n × d and b : C γ ([ − h, R n ) → R n satisfy some suitable regularity onditions, C γ designates the spa e of γ -Hölder ontinuous fun tions of one variable (seeSe tion 2.1 below) and Z yt : [ − h, → R n is de(cid:28)ned by Z yt ( s ) = y t + s . In the previ-ous equation, we also assume that an initial ondition ξ ∈ C γ is given on the interval [ − h, . Noti e that equation (1) is a slight extension of the typi al delay equation whi hDate: November 5, 2018.2000 Mathemati s Subje t Classi(cid:28) ation. 60H10, 60H05, 60H07.Key words and phrases. Delay equation, Young integration, fra tional Brownian motion, Malliavin al ulus.J.A. León is partially supported by the CONACyT grant 98998. S. Tindel is partially supported bythe ANR grant ECRU. 1 JORGE A. LEÓN AND SAMY TINDELis obtained for some fun tions f and b of the following form: f : C γ ([ − h, R n ) → R n × d , with f ( Z yt ) = σ (cid:18)Z − h y t + θ ν ( dθ ) (cid:19) , (2)for a regular enough fun tion σ , and a (cid:28)nite measure ν on [ − h, . This spe ial ase ofinterest will be treated in detail in the sequel. Our onsiderations also in lude a fun tion f de(cid:28)ned by f ( Z yt ) = σ ( Z yt ( − u ) , . . . , Z yt ( − u k )) for a given k ≥ , ≤ u < . . . < u k ≤ h and a smooth enough fun tion σ : R n × k → R n × d .The kind of delay sto hasti di(cid:27)erential system des ribed by (1) is widely studied whendriven by a standard Brownian motion (see [20℄ for a ni e survey), but the results inthe fra tional Brownian ase are s ar e: we are only aware of [8℄ for the ase H > / and f ( Z y ) = σ ( Z y ( − r )) , ≤ r ≤ h , and the further investigation [9℄ whi h establishesa ontinuity result in terms of the delay r . As far as the rough ase is on erned, anexisten e and uniqueness result is given in [22℄ for a Hurst parameter H > / , and [31℄extends this result to H > / . The urrent arti le an be thus seen as a step in the studyof pro esses de(cid:28)ned as the solution to fra tional delay di(cid:27)erential systems, and we shallinvestigate the behavior of the density of the R n -valued random variable y t for a (cid:28)xed t ∈ (0 , T ] , where y is the solution to (1). More spe i(cid:28) ally, we shall prove the followingtheorem, whi h an be seen as the main result of the arti le:Theorem 1.1. Consider an equation of the form (1) for an initial ondition ξ lying inthe spa e C γ ([ − h, R n ) . Assume b ≡ , and that f is of the form (2) for a given (cid:28)nitemeasure ν on [ − h, and σ : R n → R n × d a four times di(cid:27)erentiable bounded fun tion withbounded derivatives, satisfying the non-degenera y ondition σ ( η ) σ ( η ) ∗ ≥ ε Id R n , for all η , η ∈ R n . Suppose moreover that
H > H , where H = (7 + √ / ≈ . . Let t ∈ (0 , T ] be an arbitrary time, and y be the unique solution to (1) in C κ ([0 , T ]; R n ) , for a given / < κ < H . Then the law of y t is absolutely ontinuous with respe t to Lebesguemeasure in R n , and its density is a C ∞ -fun tion.Noti e that this kind of result, whi h has its own interest as a natural step in the studyof pro esses de(cid:28)ned by delay systems, is also a useful result when one wants to evaluatethe onvergen e of approximation s hemes in the fra tional Brownian ontext. We planto report on this possibility in a subsequent ommuni ation. The reader may also wonderabout our restri tion H > H above. It will be ome lear from Remark 3.15 that thisassumption is due to the fa t that we onsider a delay whi h depends ontinuously onthe past. For a dis rete type delay of the form σ ( y t , y t − r , . . . , y t − r q ) , with q ≥ and r < · · · < r q ≤ h , we shall see at Remark 4.7 that one an show the smoothness of thedensity up to H > / , as for ordinary di(cid:27)erential equations. Finally, the ase b ≡ hasbeen onsidered here for sake of simpli ity, but the extension of our result to a non trivialdrift is just a matter of easy additional omputations.Let us say a few words about the strategy we shall follow in order to get our Theorem 1.1.First of all, as mentioned before, there are not too many results about delay systemsgoverned by a fra tional Brownian motion. In parti ular, equation (1) has never been onsidered (to the best of our knowledge) with su h a general delay dependen e. Weshall thus (cid:28)rst show how to de(cid:28)ne and solve this di(cid:27)erential system, by means of a slightALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 3variation of the Young integration theory ( alled algebrai integration), introdu ed in [10℄and also explained in [21℄. This setting allows to solve equations like (1) in Hölder spa esthanks to ontra tion arguments, in a rather lassi al way, whi h will be explained atSe tion 3.1. In fa t, observe that our resolution will be entirely pathwise, and we shalldeal with a general equation of the form dy t = f ( Z yt ) dx t + b ( Z yt ) dt, t ∈ [0 , T ] , (3)for a given path x ∈ C γ ([0 , T ]; R d ) with γ > / , where the integral with respe t to x hasto be understood in the Young sense [32℄. Furthermore, in equations like (3), the driftterm b ( Z y ) is usually harmless, but indu es some umbersome notations. Thus, for sakeof simpli ity, we shall rather deal in the sequel with a redu ed delay equation of the type: y t = a + Z t f ( Z ys ) dx s , t ∈ [0 , T ] . On e this last equation is properly de(cid:28)ned and solved, the di(cid:27)erentiability of the solution y t in the Malliavin al ulus sense will be obtained in a pathwise manner, similarly to the ase treated in [26℄. Finally, the smoothness Theorem 1.1 will be obtained mainly bybounding the moments of the Malliavin derivatives of y . This will be a hieved thanks toa areful analysis and some a priori estimates for equation (1).Here is how our arti le is stru tured: Se tion 2 is devoted to re all some basi fa tsabout Young integration. We solve, estimate and di(cid:27)erentiate a general lass of delayequations driven by a Hölder noise at Se tion 3. Then at Se tion 4 we apply those generalresults to fBm and prove our main Theorem 1.1.2. Algebrai Young integrationThe Young integration an be introdu ed in several ways ( onvergen e of Riemann sums,fra tional al ulus setting [33℄). We have hosen here to follow the algebrai approa hintrodu ed in [10℄ and developed e.g. in [12, 21℄, sin e this formalism will help us later inour analysis.2.1. In rements. Let us begin with the basi algebrai stru tures whi h will allow us tode(cid:28)ne a pathwise integral with respe t to irregular fun tions: (cid:28)rst of all, for an arbitraryreal number T > , a topologi al ve tor spa e V and an integer k ≥ we denote by C k ( V ) (or by C k ([0 , T ]; V ) ) the set of ontinuous fun tions g : [0 , T ] k → V su h that g t ··· t k = 0 whenever t i = t i +1 for some i ≤ k − . Su h a fun tion will be alled a ( k − -in rement,and we will set C ∗ ( V ) = ∪ k ≥ C k ( V ) . An important elementary operator is δ , whi h isde(cid:28)ned as follows on C k ( V ) : δ : C k ( V ) → C k +1 ( V ) , ( δg ) t ··· t k +1 = k +1 X i =1 ( − k − i g t ··· ˆ t i ··· t k +1 , (4)where ˆ t i means that this parti ular argument is omitted. A fundamental property of δ ,whi h is easily veri(cid:28)ed, is that δδ = 0 , where δδ is onsidered as an operator from C k ( V ) to C k +2 ( V ) . We will denote ZC k ( V ) = C k ( V ) ∩ Ker δ and BC k ( V ) = C k ( V ) ∩ Im δ .Some simple examples of a tions of δ , whi h will be the ones we will really use through-out the paper, are obtained by letting g ∈ C ( V ) and h ∈ C ( V ) . Then, for any JORGE A. LEÓN AND SAMY TINDEL s, u, t ∈ [0 , T ] , we have ( δg ) st = g t − g s , and ( δh ) sut = h st − h su − h ut . (5)Furthermore, it is easily he ked that ZC k ( V ) = BC k ( V ) for any k ≥ . In parti ular, thefollowing basi property holds:Lemma 2.1. Let k ≥ and h ∈ ZC k +1 ( V ) . Then there exists a (non unique) f ∈ C k ( V ) su h that h = δf .Observe that Lemma 2.1 implies that all the elements h ∈ C ( V ) su h that δh = 0 an be written as h = δf for some (non unique) f ∈ C ( V ) . Thus we get a heuristi interpretation of δ | C ( V ) : it measures how mu h a given 1-in rement is far from being anexa t in rement of a fun tion, i.e., a (cid:28)nite di(cid:27)eren e.Remark 2.2. Here is a (cid:28)rst elementary but important link between these algebrai stru -tures and integration theory: let f and g be two smooth real valued fun tion on [0 , T ] .De(cid:28)ne then I ∈ C ( V ) by I st = Z ts df v Z vs dg w , for s, t ∈ [0 , T ] . Then, some trivial omputations show that ( δI ) sut = [ g u − g s ][ f t − f u ] = ( δf ) ut ( δg ) su . This is a helpful property of the operator δ : it transforms iterated integrals into produ tsof in rements, and we will be able to take advantage of both regularities of f and g inthese produ ts of the form δf δg .For sake of simpli ity, let us spe ialize now our setting to the ase V = R m for anarbitrary m ≥ . Noti e that our future dis ussions will mainly rely on k -in rements with k ≤ , for whi h we will use some analyti al assumptions. Namely, we measure the sizeof these in rements by Hölder norms de(cid:28)ned in the following way: for ≤ a < a ≤ T and f ∈ C ([ a , a ]; V ) , let k f k µ, [ a ,a ] = sup r,t ∈ [ a ,a ] | f rt || t − r | µ , and C µ ([ a , a ]; V ) = (cid:8) f ∈ C ( V ); k f k µ, [ a ,a ] < ∞ (cid:9) . Obviously, the usual Hölder spa es C µ ([ a , a ]; V ) will be determined in the following way:for a ontinuous fun tion g ∈ C ([ a , a ]; V ) , we simply set k g k µ, [ a ,a ] = k δg k µ, [ a ,a ] , (6)and we will say that g ∈ C µ ([ a , a ]; V ) i(cid:27) k g k µ, [ a ,a ] is (cid:28)nite. Noti e that k · k µ, [ a ,a ] isonly a semi-norm on C µ ([ a , a ]; V ) , but we will generally work on spa es of the type C µv,a ,a ( V ) = (cid:8) g : [ a , a ] → V ; g a = v, k g k µ, [ a ,a ] < ∞ (cid:9) , (7)for a given v ∈ V , or C µ̺,a ,a ( R d ) := { ζ ∈ C µ ([ a − h, a ]; R d ); ζ = ̺ on [ a − h, a ] } , (8)where ≤ a < a and ̺ ∈ C µ ([ a − h, a ]; R d ) . These last two spa es are omplete metri spa es with the distan e d µ ( g, f ) = k g − f k µ . More spe i(cid:28) ally, the metri we shall useon the spa e C µ̺,a ,a ( R d ) is: d µ,a ,a ( g, f ) , k g − f k µ, [ a − h,a ] . ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 5In some ases we will only write C µk ( V ) instead of C µk ([ a , a ]; V ) when this does not leadto an ambiguity in the domain of de(cid:28)nition of the fun tions under onsideration. For h ∈ C ([ a , a ]; V ) set in the same way k h k γ,ρ, [ a ,a ] = sup s,u,t ∈ [ a ,a ] | h sut || u − s | γ | t − u | ρ (9) k h k µ, [ a ,a ] = inf (X i k h i k ρ i ,µ − ρ i ; h = X i h i , < ρ i < µ ) , where the last in(cid:28)mum is taken over all sequen es { h i ∈ C ( V ) } su h that h = P i h i and for all hoi es of the numbers ρ i ∈ (0 , µ ) . Then k · k µ is easily seen to be a norm on C ([ a , a ]; V ) , and we set C µ ([ a , a ]; V ) := { h ∈ C ([ a , a ]; V ); k h k µ < ∞} . Eventually, let C ([ a , a ]; V ) = ∪ µ> C µ ([ a , a ]; V ) , and remark that the same kind ofnorms an be onsidered on the spa es ZC ([ a , a ]; V ) , leading to the de(cid:28)nition of somespa es ZC µ ([ a , a ]; V ) and ZC ([ a , a ]; V ) .With these notations in mind, the ru ial point in our approa h to pathwise integrationof irregular pro esses is that, under mild smoothness onditions, the operator δ an beinverted. This inverse is alled Λ , and is de(cid:28)ned in the following proposition, whose proof an be found in [10℄.Proposition 2.3. Let ≤ a < a ≤ T . Then there exists a unique linear map Λ : ZC ([ a , a ]; V ) → C ([ a , a ]; V ) su h that δ Λ = Id ZC ([ a ,a ]; V ) . In other words, for any h ∈ C ([ a , a ]; V ) su h that δh = 0 there exists a unique g =Λ( h ) ∈ C ([ a , a ]; V ) su h that δg = h . Furthermore, for any µ > , the map Λ is ontinuous from ZC µ ([ a , a ]; V ) to C µ ([ a , a ]; V ) and we have k Λ h k µ, [ a ,a ] ≤ µ − k h k µ, [ a ,a ] , h ∈ ZC µ ([ a , a ]; V ) . (10)Moreover, the operator Λ an be related to the limit of some Riemann sums, whi hgives a se ond link (after Remark 2.2) between the previous algebrai developments andsome kind of generalized integration.Corollary 2.4. For any 1-in rement g ∈ C ( V ) su h that δg ∈ C , set δf = ( Id − Λ δ ) g .Then ( δf ) st = lim | Π st |→ n − X i =0 g t i t i +1 , where the limit is over any partition Π st = { t = s, . . . , t n = t } of [ s, t ] , whose mesh tendsto zero. Thus, the 1-in rement δf is the inde(cid:28)nite integral of the 1-in rement g . JORGE A. LEÓN AND SAMY TINDEL2.2. Young integration. In this se tion, we will de(cid:28)ne a generalized integral R ts f u dg u fora C κ ([0 , T ]; R n × d ) -fun tion f , and a C γ ([0 , T ]; R d ) -fun tion g , with κ + γ > , by means ofthe algebrai tools introdu ed at Se tion 2.1. To this purpose, we will (cid:28)rst assume that f and g are smooth fun tions, in whi h ase the integral of f with respe t to g an be de(cid:28)nedin the Riemann sense, and then we will express this integral in terms of the operator Λ .This will lead to a natural extension of the notion of integral, whi h oin ides with theusual Young integral. In the sequel, in order to avoid some umbersome notations, wewill sometimes write J st ( f dg ) instead of R ts f u dg u .Let us onsider then for the moment two smooth fun tions f and g de(cid:28)ned on [0 , T ] .One an write, thanks to some elementary algebrai manipulations, that: J st ( f dg ) ≡ Z ts f u dg u = f s ( δg ) st + Z ts ( δf ) su dg u = f s ( δg ) st + J st ( δf dg ) . (11)Let us analyze now the term J ( δf dg ) , whi h is an element of C ( R n ) . Invoking Remark2.2, it is easily seen that, for s, u, t ∈ [0 , T ] , h sut ≡ [ δ ( J ( δf dg ))] sut = ( δf ) su ( δg ) ut . The in rement h is thus an element of C ( R n ) satisfying δh = 0 (re all that δδ = 0 ). Letus estimate now the regularity of h : if f ∈ C κ ([0 , T ]; R n × d ) and g ∈ C γ ([0 , T ]; R d ) , fromthe de(cid:28)nition (9), it is readily he ked that h ∈ C γ + κ ( R n ) . Hen e h ∈ ZC γ + κ ( R n ) , and if κ + γ > (whi h is the ase if f and g are regular), Proposition 2.3 yields that J ( δf dg ) an also be expressed as J ( δf dg ) = Λ( h ) = Λ ( δf δg ) , and thus, plugging this identity into (11), we get: J st ( f dg ) = f s ( δg ) st + Λ st ( δf δg ) . (12)Now we an see that the right hand side of the last equality is rigorously de(cid:28)ned whenever f ∈ C κ ([0 , T ]; R n × d ) , g ∈ C γ ([0 , T ]; R d ) , and this is the de(cid:28)nition we will use in order toextend the notion of integral:Theorem 2.5. Let f ∈ C κ ([0 , T ]; R n × d ) and g ∈ C γ ([0 , T ]; R d ) , with κ + γ > . Set J st ( f dg ) = f s ( δg ) st + Λ st ( δf δg ) . (13)Then(1) Whenever f and g are smooth fun tion, J st ( f dg ) oin ides with the usual Riemannintegral.(2) The generalized integral J ( f dg ) satis(cid:28)es: |J st ( f dg ) | ≤ k f k ∞ k g k γ | t − s | γ + c γ,κ k f k κ k g k γ | t − s | γ + κ , for a onstant c γ,κ whose exa t value is (2 γ + κ − − .(3) We have J st ( f dg ) = lim | Π st |→ n − X i =0 f t i δg t i t i +1 , where the limit is over any partition Π st = { t = s, . . . , t n = t } of [ s, t ] , whosemesh tends to zero. In parti ular, J st ( f dg ) oin ides with the Young integral asde(cid:28)ned in [32℄.ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 7Proof. The (cid:28)rst laim is just what we proved at equation (12). The se ond assertionfollows dire tly from the de(cid:28)nition (13) and the inequality (10) on erning the operator Λ . Finally, our third property is a dire t onsequen e of Corollary 2.4 and the fa t that δ ( f δg ) = − δf δg , whi h means that J ( f dg ) = [ Id − Λ δ ] ( f δg ) . (cid:3) A Fubini type theorem for Young's integral will be needed in the last se tion of thispaper. Its proof below is a good example of the importan e of Proposition 2.3 andTheorem 2.5.Proposition 2.6. Assume that γ > λ > / . Let f and g be two fun tions in C γ ([0 , T ] : R ) and h : { ( t, s ) ∈ [0 , T ] ; 0 ≤ s ≤ t ≤ T } → R a fun tion su h that h ( · , t ) (resp. h ( t, · ) )belongs to C λ ([ t, T ]; R ) (resp. C λ ([0 , t ]; R ) ) uniformly in t ∈ [0 , T ] , and k h ( r , · ) − h ( r , · ) k λ, [0 ,r ∧ r ] ≤ C | r − r | λ . (14)Then Z ts Z rs h ( r, u ) dg u df r = Z ts Z tu h ( r, u ) df r dg u , ≤ s ≤ t ≤ T. (15)Proof. Fix s, t ∈ [0 , T ] , with s < t , and divide the proof in several steps.Step 1. Here we see that R ts R rs h ( r, u ) dg u df r is well-de(cid:28)ned. Note that we only need toshow that R · s h ( · , u ) dg u belongs to C λ ([ s, T ]; R ) due to Theorem 2.5.Let r , r ∈ [ s, t ] , r < r , then Theorem 2.5.(2) gives (cid:12)(cid:12)(cid:12)(cid:12)Z r s h ( r , u ) dg u − Z r s h ( r , u ) dg u (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z r s ( h ( r , u ) − h ( r , u )) dg u (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z r r h ( r , u ) dg u (cid:12)(cid:12)(cid:12)(cid:12) ≤ k g k γ (cid:0) k h ( r , · ) − h ( r , · ) k ∞ , [0 ,r ] ( r − s ) γ + c γ,λ k h ( r , · ) − h ( r , · ) k λ, [0 ,r ] ( r − s ) γ + λ (cid:1) + k g k γ (cid:0) k h ( r , · ) k ∞ , [0 ,r ] ( r − r ) γ + c γ,λ k h ( r , · ) k λ, [0 ,r ] ( r − r ) γ + λ (cid:1) . Hen e (14) implies our laim. The de(cid:28)nition of R ts R tu h ( r, u ) df r dg u follows along the samelines.Step 2. Let Π st = { t = s, . . . , t n = t } be a partition of the interval [ s, t ] . Then, a ordingto Proposition 2.5, for any v ∈ [0 , t ) we have Z vs h ( t, u ) dg u = lim | Π st |→ n − X i =0 h ( t, t i ) ( δg ) t i ∧ v,t i +1 ∧ v . (16)Our assumption (14) allows now to take limits in the equation above, so that we obtain,for any ≤ s < t ≤ T , q st := Z ts h ( t, u ) dg u = lim | Π st |→ n − X i =0 h ( t, t i ) δg t i ,t i +1 := q st . (17)In order to see that the relation above holds in C λ ([0 , T ]; R ) , it is now enough to he kthat both q and q in (17) are elements of C λ ([0 , T ]; R ) . JORGE A. LEÓN AND SAMY TINDELHowever, the fa t that q ∈ C λ ([0 , T ]; R ) an be proved along the same lines as in Step 1.The assertion q ∈ C λ ([0 , T ]; R ) an be proved by observing that the limit de(cid:28)ning q st donot depend on the sequen e of partitions under onsideration. In parti ular, onsider thesequen e ( π n ) n of dyadi partitions of [0 , T ] , that is π n = { t n ≤ t n ≤ · · · ≤ t n n = T } , with t ni = i T n , and set, for all s, t ∈ [0 , T ] , π nst = π n ∩ ( s, t ) . Then q st = lim n →∞ P t i ∈ π nst h ( t, t ni ) δg t ni ,t ni +1 for all ≤ s < t ≤ T , and the same kind of arguments as in [6, Theorem 2.2℄ yield our laim q ∈ C λ ([0 , T ]; R ) . We have thus proved that (17) holds in C λ ([0 , T ]; R ) .Step 3. From Proposition 2.3, Step 2 and (13) we have Z ts Z rs h ( r, u ) dg u df r = lim | Π st |→ Z ts n − X i =0 h ( r, t i )( g t i +1 ∧ r − g t i ∧ r ) ! df r = lim | Π st |→ n − X i =0 Z tt i h ( r, t i ) (cid:0) g t i +1 ∧ r − g t i (cid:1) df r = lim | Π st |→ n − X i =0 (cid:20)(cid:18)Z tt i h ( r, t i ) df r (cid:19) (cid:0) g t i +1 − g t i (cid:1) + Z t i +1 t i h ( r, t i ) (cid:0) g t i +1 ∧ r − g t i +1 (cid:1) df r (cid:21) Moreover, thanks to the Hölder properties of f and g , we have n − X i =0 (cid:12)(cid:12)(cid:12)(cid:12)Z t i +1 t i h ( r, t i )( g r − g t i ) df r (cid:12)(cid:12)(cid:12)(cid:12) ≤ C n − X i =0 ( t i +1 − t i ) γ + λ → as | Π st | → , and thus Z ts Z rs h ( r, u ) dg u df r = lim | Π st |→ n − X i =0 (cid:18)Z tt i h ( r, t i ) df r (cid:19) (cid:0) g t i +1 − g t i (cid:1) . Consequently, Step 2 and Theorem 2.5 imply that (15) is satis(cid:28)ed and therefore the proofis omplete. (cid:3)
The following integration by parts and It's formulas will be also needed in the lastpart of this paper.Proposition 2.7. Let f and g be two fun tions in C γ ([0 , T ]; R ) , with γ > / . Then f t g t = f g + Z t f u dg u + Z t g u df u , t ∈ [0 , T ] . Proof. Set q t := f t g t − R t f u dg u − R t g u df u , t ∈ [0 , T ] . It is easy to see that this fun ionbelongs to C γ ([0 , T ]; R ) be ause of the equalities f t g t − f s g s = f s ( δg ) st + g s ( δf ) st + ( δg ) st ( δf ) st and Z ts f u dg u + Z ts g u df u = f s ( δg ) st + g s ( δf ) st + Λ st ( δf δg ) + Λ st ( δgδf ) , ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 9whi h follows from (13). Now, sin e q ∈ C γ ([0 , T ]; R ) , with γ > , q is a onstantfun tion. Otherwise stated, q t = q = f g . Therefore the announ ed result is true. (cid:3) Proposition 2.8. Let g and h be in C γ ([0 , T ] , R ) and f ∈ C b ( R ) . Also let x t = x + R t g s dh s , t ∈ [0 , T ] . Then f ( x t ) = f ( x ) + Z t f ′ ( x u ) g u dh u , t ∈ [0 , T ] . Proof. Pro eeding as in the proof of Proposition 2.7 and using the mean value theorem,we an show that q t = f ( x t ) − Z t f ′ ( x s ) g s dh s , t ∈ [0 , T ] , is a γ -Hölder ontinuous fun tion. Therefore the result holds. (cid:3) Remark 2.9. Proposition 2.8 has been proven in [33℄ using Riemann sums.3. Young delay equationRe all (cid:28)rst that we wish to onsider a di(cid:27)erential equation of the form: y t = ξ + Z t f ( Z yu ) dx u , t ∈ [0 , T ] , (18) Z y = ξ. In the previous equation, the integral has to be interpreted in the Young sense of (13), theinitial ondition ξ is an element of C γ ([ − h, R n ) , the driving noise x is in C γ ([0 , T ]; R d ) ,with γ > / . We seek a solution y in the spa e C λξ, ,T ( R n ) for / < λ < γ , and f is agiven fun tion f : C λ ([ − h, R n ) → R n × d . In this se tion, we shall solve equation (18)thanks to a ontra tion argument, and then study its di(cid:27)erentiability with respe t to thedriving noise x . Of ourse, the main appli ation we have in mind is the ase where x is a d -dimensional fra tional Brownian motion, and this parti ular ase will be onsidered atSe tion 4.3.1. Existen e and uniqueness of the solution. In order to solve equation (18), somesmoothness and boundedness assumptions have to be made on our oe(cid:30) ient f . In fa t,we shall rely on the following hypothesis:Hypothesis 1. There exist a positive onstant M and λ ∈ (1 / , γ ) su h that | f ( ζ ) | ≤ M, and | f ( ζ ) − f ( ζ ) | ≤ M sup θ ∈ [ − h, | ζ ( θ ) − ζ ( θ ) | uniformly in ζ , ζ , ζ ∈ C λ ([ − h, R n ) .A tually we will assume that f satis(cid:28)es a stronger Lips hitz type hypothesis on the spa e C λ ( R n ) . Let us state (cid:28)rst a preliminary result before we ome to this se ond assumption:Lemma 3.1. Let a = ( a , a ) , with ≤ a < a ≤ T , let also Z ∈ C λ ([ a − h, a ]; R n ) andset (cid:2) U ( a ) Z (cid:3) s = f ( Z Zs ) , s ∈ [ a , a ] . U ( a ) is a map from C λ ([ a − h, a ]; R n ) into C λ ([ a , a ]; R n × d ) , satisfying: (cid:13)(cid:13) U ( a ) Z (cid:13)(cid:13) λ, [ a ,a ] ≤ M k Z k λ, [ a − h,a ] . Proof. The proof of this result is an immediate onsequen e of the de(cid:28)nition (6) of Hölder'snorms on C and Hypothesis 1. (cid:3) With this preliminary result in hand, we an now introdu e our se ond hypothesis onthe oe(cid:30) ient f .Hypothesis 2. Taking up the notations of Hypothesis 1, onsider an initial ondition ρ ∈ C λ ([ a − h, a ]) . We assume that, for any N ≥ , there is a positive onstant c N su hthat: kU ( a ) ( Z ) − U ( a ) ( Z ) k λ, [ a ,a ] ≤ c N k Z − Z k λ, [ a − h,a ] , for all ≤ a ≤ a ≤ T and Z , Z ∈ C λρ,a ,a ( R n ) , satisfying max (cid:8) k Z k λ, [ a − h,a ] ; k Z k λ, [ a − h,a ] (cid:9) ≤ N, where λ is given in Hypothesis 1.Observe that Hypothesis 2 holds in parti ular if, for λ > , the map U ( a ) admits aderivative whi h is lo ally bounded, uniformly in a ∈ [0 , T ] .Now that we have stated our main assumptions, the following theorem is the mainresult of this se tion.Theorem 3.2. Under Hypotheses 1 and 2, the delay equation (18) has a unique solutionin C λξ, ,T ( R n ) .Before giving the proof of this theorem, we establish and auxiliary result. This will behelpful in order to get the existen e of an invariant ball under the ontra ting map whi hgives raise to the solution of our equation.Lemma 3.3. Let x ∈ C γ ([ a , a ]; R d ) with γ > / and ≤ a < a , λ ∈ (1 / , γ ) and v ∈ R n . Set a = ( a , a ) , re all notation (7) and de(cid:28)ne V ( a ) : C λ ([ a , a ]; R n × d ) → C λv,a ,a ( R n ) by: (cid:2) V ( a ) Z (cid:3) s = v + J a s ( Z dx ) , s ∈ [ a , a ] , where J a s ( Z dx ) stands for the Young integral de(cid:28)ned by (13). Then kV ( a ) Z k λ, [ a ,a ] ≤ k x k γ (cid:0) k Z k ∞ , [ a ,a ] ( a − a ) γ − λ + c λ + γ k Z k λ, [ a ,a ] ( a − a ) γ (cid:1) , with c λ + γ = (2 λ + γ − − .Proof. Let a ≤ s ≤ t ≤ T . Then Theorem 2.5 point (3) implies that (cid:2) V ( a ) Z (cid:3) t − (cid:2) V ( a ) Z (cid:3) s = J st ( Z dx ) . Our laim is then a dire t onsequen e of Theorem 2.5 point (2) and of the de(cid:28)nition (6). (cid:3)
ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 11Proof of Theorem 3.2: This proof is divided in several steps.Step 1: Existen e of invariant balls. Let us (cid:28)rst onsider an interval of the form [0 , ε ] ,whi h means that, when we in lude the delay of the equation, we shall onsider pro essesde(cid:28)ned on [ − h, ε ] . More spe i(cid:28) ally, let us re all that the spa es C λξ, ,ε ( R n ) have beende(cid:28)ned by relation (8). Then we onsider a map Γ : C λξ, ,ε → C λξ, ,ε , where we have set C λξ, ,ε = C λξ, ,ε ( R n ) for notational sake, de(cid:28)ned in the following way: if z ∈ C λξ, ,ε , then Γ( z ) = ˆ z , where ˆ z t = ξ t for t ∈ [ − h, , and: ( δ ˆ z ) st = J st ( Z dx ) , with Z u = f ( Z zu ) , for s, t ∈ [0 , ε ] . (19)We shall now look for an invariant ball in the spa e C λξ, ,ε for the map Γ .So let us pi k an element z , su h that k z k λ, [ − h,ε ] ≤ N and set Γ( z ) = ˆ z . On [ − h, , wehave ˆ z = ξ , and hen e k δ ˆ z k λ, [ − h, = k δξ k λ, [ − h, ≡ N ξ . We shall thus hoose N ≥ N ξ .On [0 , ε ] , we have now, invoking Lemma 3.3: k δ ˆ z k λ, [0 ,ε ] ≤ k Z k ∞ k x k γ ε γ − λ + c γ,λ k Z k λ, [0 ,ε ] k x k γ ε γ . (20)Furthermore, a ording to Hypothesis 1, we have k Z k ∞ ≤ M and thanks to Lemma 3.1,we also have k Z k λ, [0 ,ε ] ≤ M k z k λ, [ − h,ε ] ≤ M N , by assumption. Then we an re ast theprevious inequality into: k δ ˆ z k λ, [0 ,ε ] ≤ M k x k γ ε γ − λ (cid:2) c γ,λ N ε λ (cid:3) . (21)Let us hoose now ε and N in the following manner (noti e that ε does not depend onthe initial ondition ξ ): ε = [4 M c γ,λ k x k γ ] − /γ ∧ , and N ≥ M k x k γ . (22)With this hoi e of ε, N , inequality (21) be omes k δ ˆ z k λ, [0 ,ε ] ≤ N / . Summarizing the onsiderations above, we have thus found that: ε = [4 M c γ,λ k x k γ ] − /γ ∧ , N ≥ sup { N ξ ; 4 M k x k γ } = ⇒ sup (cid:8) k δ ˆ z k λ, [ − h, ; k δ ˆ z k λ, [0 ,ε ] (cid:9) ≤ N . (23)Consider now s < t , with s ∈ [ − h, and t ∈ [0 , ε ] . Then, owing to the previousrelation, we have: | ( δ ˆ z ) st | ≤ | ( δ ˆ z ) s | + | ( δ ˆ z ) t | ≤ N (cid:0) s λ + t λ (cid:1) ≤ N | t − s | λ , whi h, together with the last inequality, proves that B (0 , N ) in C λξ, ,ε is left invariant by Γ , under the assumptions of (23).Assume now that we have been able to produ e a solution y (1) to equation (18) on theinterval [ − h, ε ] . We try now to iterate the invariant ball argument on [ ε − h ; 2 ε ] . Thearguments above go through with very little hanges: we are now working on delayedHölder spa es of the form C λy (1) ,ε, ε , and the map Γ is de(cid:28)ned by Γ( z ) = ˆ z , with ˆ z = y (1) on [ ε − h ; ε ] , and δ ˆ z having the same expression as in (19) on [ ε, ε ] . We wish to (cid:28)nd a2 JORGE A. LEÓN AND SAMY TINDELball B (0 , N ) in C λy (1) ,ε, ε , left invariant by the map Γ . With the same omputations as forthe interval [ − h, ε ] , the assumptions of inequality (23) be ome: ε = [4 M c γ,λ k x k γ ] − /γ ∧ , N ≥ sup (cid:8) N y (1) ; 4 M k x k γ (cid:9) . Noti e again that we are able to hoose here the same ε as before, by hanging N into N a ording to the value of k y (1) k λ, [ ε − h,ε ] . It is now readily he ked that B (0 , N ) is invariantunder Γ , and this al ulation is also easily repeated on any interval [ kε − h, ( k + 1) ε ] forany k ≥ , until the whole interval [0 , T ] is overed.Step 2: Fixed point argument. We shall suppose here that we have been able to onstru tthe unique solution y to (18) on [ − h ; lε ] , and we shall build the (cid:28)xed point argument on [ lε − h ; ( l + 1) ε ] . On the latter interval, the initial ondition of the paths we shall onsideris ξ l, ≡ y on [ lε − h ; lε ] . If Γ is the map de(cid:28)ned on C λξ l, ,lε, ( l +1) ε by (19), then we knowthat B (0 , N l +1 ) is invariant by Γ .In order to settle our (cid:28)xed point argument, we shall (cid:28)rst onsider an interval of theform [ lε − h ; lε + η ] , for a parameter < η ≤ ε to be determined. On C λξ l, ,lε,lε + η , we de(cid:28)nea map, alled again Γ , a ording to (19). Pi k then two fun tions z , z ∈ C λξ l, ,lε,lε + η , set ˆ z i = Γ( z i ) for i = 1 , and ζ = ˆ z − ˆ z . Then ζ ∈ C λ ,lε,lε + η , and if lε ≤ s < t ≤ lε + η , wehave ( δζ ) st = J st (cid:0) ( Z − Z ) dx (cid:1) , where Z i = f ( Z z i ) . Thus, just like in (20), we have: k δζ k λ, [ lε − h,lε + η ] ≤ k Z − Z k ∞ , [ lε,lε + η ] k x k γ η γ − λ + c γ,λ k Z − Z k λ, [ lε,lε + η ] k x k γ η γ . Furthermore, k Z − Z k ∞ , [ lε,lε + η ] ≤ k Z − Z k λ, [ lε,lε + η ] η λ . Hen e, k δζ k λ, [ lε − h,lε + η ] ≤ (1 + c γ,λ ) k Z − Z k λ, [ lε,lε + η ] k x k γ η γ . We also have Z − Z = f ( Z z ) − f ( Z z ) , and thanks to Hypothesis 2, we obtain: k δζ k λ, [ lε − h,lε + η ] ≤ (1 + c γ,λ ) k x k γ c N l +1 η γ k z − z k λ, [ lε − h,lε + η ] . Therefore, we are able to apply the (cid:28)xed point argument in the usual way as soon as (1 + c γ,λ ) c N l +1 k x k γ η γ ≤ , or η = (cid:2) c γ,λ ) c N l +1 k x k γ (cid:3) − /γ ∧ ε. With this value of η , we are thus able to get a unique solution to (18) on [ lε − h ; lε + η ] .Let us pro eed now to the ase of [ lε + η − h, lε + 2 η ] . The arguments are roughly thesame as in the previous ase, but one has to be areful about the hange in the initial ondition. In fa t, the initial ondition here should be ξ l, ≡ y on [ lε + η − h, lε + η ] .However, we an also hoose to extend this initial ondition ba kward, and set it as ξ l, ≡ y on [ lε − h, lε + η ] . We then de(cid:28)ne the usual map Γ as in (19), and we have to prove that B (0 , N l +1 ) is left invariant by Γ . To this purpose, take z ∈ C λξ l, ,lε + η,lε +2 η in B (0 , N l +1 ) ,and set ˆ z = Γ( z ) . Observe then that, for any t ∈ [ lε + η, lε + 2 η ] , we have ˆ z t = ξ lε + η + Z tlε + η f ( Z zu ) dx u = ξ lε + Z lε + ηlε f ( Z yu ) dx u + Z tlε + η f ( Z zu ) dx u = ξ lε + Z tlε f ( Z zu ) dx u , where we have used the fa t that ξ l, ≡ y on [ lε − h, lε + η ] solves (18). It is now easilyseen that ˆ z is in B (0 , N l +1 ) , and this allows to settle our (cid:28)xed point argument as in theALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 13previous ase, with the same interval length η . This step an now be iterated until thewhole interval [ lε ; ( l + 1) ε ] is overed. (cid:3) y be the solutionof equation (18) on the interval [0 , T ] , with an initial ondition ξ ∈ C λ ([ − h, R n ) . Thenthere exists a stri tly positive onstant c = c ( γ, λ, M, T ) su h that k y k λ, [ − h,T ] ≤ c max (cid:2) k ξ k λ , k x k λ/ ( γ + λ − γ , k x k γ (cid:3) . Proof. From the proof of Theorem 3.2, we know that k y k λ, [ − h,T ] is (cid:28)nite. Let us assumethat this quantity is equal to K , and let us (cid:28)nd an estimate on K . One an begin witha small interval, whi h will be alled again [0 , ε ] , though it won't be the same interval asin the proof of Theorem 3.2. In any ase, taking into a ount that y solves equation (18),we obtain similarly to (20): k δy k λ, [0 ,ε ] ≤ M k x k γ ε γ − λ + c γ,λ M k δy k λ, [ − h,ε ] k x k γ ε γ ≤ M k x k γ ε γ − λ + c γ,λ M K k x k γ ε γ ≡ g ( ε, K ) . (24)Along the same line, for any k ≤ [ T /ε ] , we have k δy k λ, [ kε, ( k +1) ε ] ≤ g ( ε, K ) . Take now s, t ∈ [0 , T ] su h that iε ≤ s < ( i + 1) ε ≤ jε ≤ t < ( j + 1) ε . Set also t i = s , t k = kε for i + 1 ≤ k ≤ j , and t j +1 = t . Then | ( δy ) st | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j X k = i ( δy ) t k t k +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ g ( ε, K ) j X k = i ( t k +1 − t k ) λ ≤ g ( ε, K )( j − i + 1) − λ ( t − s ) λ , where we have used the fa t that r r λ is a on ave fun tion. Note that the indi es i, j above satisfy ( j − i + 1) ≤ T /ε . Plugging this into the last series of inequalities, we endup with k δy k λ, [0 ,T ] ≤ g ( ε, K )(2 T ) − λ ε − λ = (cid:20) M k x k γ ε − γ + c γ,λ M K k x k γ ε γ + λ − (cid:21) (2 T ) − λ . Thus the parameters K and ε satisfy the relation: K ≤ (cid:20) M k x k γ ε − γ + c γ,λ M K k x k γ ε γ + λ − (cid:21) (2 T ) − λ + k ξ k λ , (25)In order to solve (25), hoose ε su h that c γ,λ M k x k γ ε γ + λ − (2 T ) − λ = 12 , that is ε = (cid:2) c γ,λ M k x k γ (2 T ) − λ (cid:3) − / ( γ + λ − . Plugging this relation into (25), we obtain the result when ε < T .Finally,
T < ε if and only if T γ < [2 − λ c γ + λ M || x || γ ] − . Thus, by inequality (24), theproof is omplete. (cid:3) f given by equation (2).Proposition 3.5. Let ν be a (cid:28)nite measure on [ − h, and σ : R n → R n × d a four timesdi(cid:27)erentiable bounded fun tion with bounded derivatives. Then Hypotheses 1 and 2 areful(cid:28)lled for f : C λ ([ − h, R n ) → R n × d de(cid:28)ned by: f ( Z ) = σ (cid:18)Z − h Z ( θ ) ν ( dθ ) (cid:19) , with Z ∈ C λ ([ − h, R n ) .Proof. We (cid:28)rst show that Hypothesis 1 holds. More spe i(cid:28) ally, the ondition | f ( ζ ) | ≤ M being obvious in our ase, we fo us on the se ond ondition of Hypothesis 1. Let Z , Z ∈C λ ([ − h, R n ) . Then there is a onstant C > su h that | f ( Z ) − f ( Z ) |≤ C (cid:12)(cid:12)(cid:12)(cid:12)Z − h ( Z ( θ ) − Z ( θ )) ν ( dθ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cν ([ − h, sup θ ∈ [ − h, | Z ( θ ) − Z ( θ ) | ! . Therefore Hypothesis 1 is satis(cid:28)ed in this ase.Now we prove that U ( a ) is Fré het di(cid:27)erentiable in order to analyze Hypothesis 2. Sin ethe map Z R − h Z ( · + θ ) ν ( dθ ) is easily shown to be a bounded linear operator from C λ ([ a − h, a ]; R n ) into C λ ([ a , a ]; R n ) , we only need to show that σ : C λρ,a ,a ( R n ) → C λ ˆ ρ,a ,a ( R n × d ) , with ˆ ρ , σ ( ρ ) , is Fré het di(cid:27)erentiable in the dire tions of C λ ,a ,a ( R n ) , with derivative [ Dσ ( Z ) ℓ ]( t ) = σ ′ ( Z ( t )) ℓ ( t ) . Towards this end, we have to show that, taking Z ∈ C λρ,a ,a ( R n ) and ℓ ∈C λ ,a ,a ( R n ) , and setting q t = σ ( Z ( t ) + ℓ ( t )) − σ ( Z ( t )) − σ ′ ( Z ( t )) ℓ ( t ) , then lim k ℓ k λ, [ a − h,a → k q k λ, [ a − h,a ] k ℓ k λ, [ a − h,a ] = 0 . (26)In order to prove relation (26), de(cid:28)ne a fun tion b : [0 , → R by: b ( λ, µ ) = Z ( s ) + λℓ ( s ) + µ [ Z ( t ) − Z ( s )] + λµ [ ℓ ( t ) − ℓ ( s )] . Observe then that b (1 ,
1) = Z ( t ) + ℓ ( t ) , b (1 ,
0) = Z ( s ) + ℓ ( s ) , b (0 ,
1) = Z ( t ) and b (0 ,
0) = Z ( s ) . We will also set H ( λ, µ ) = σ ( b ( λ, µ )) . Then σ ( Z ( t ) + ℓ ( t )) − σ ( Z ( t )) − σ ′ ( Z ( t )) ℓ ( t )= σ ( b (1 , − σ ( b (0 , − σ ′ ( b (0 , b (1 , − b (0 , Z ∂ λλ H ( λ, − λ ] dλ, and similarly, we have: σ ( Z ( s ) + ℓ ( s )) − σ ( Z ( s )) − σ ′ ( Z ( s )) ℓ ( s ) = Z ∂ λλ H ( λ, − λ ] dλ. ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 15Hen e, plugging these two relations in the de(cid:28)nition of q , we end up with: ( δq ) st = Z (cid:0) ∂ λλ H ( λ, − ∂ λλ H ( λ, (cid:1) [1 − λ ] dλ = Z ∂ λλµ H ( λ, − λ ] dλ + Z [0 , ∂ λλµµ H ( λ, µ )[1 − λ ][1 − µ ] dλdµ. The al ulation of ∂ λλµ H ( λ, and ∂ λλµµ H ( λ, µ ) is a matter of long and tedious ompu-tations, whi h are left to the reader. Let us just mention that both expressions an bewritten as a sum of terms from whi h a typi al example is: σ ′′′ ( b ( λ, µ )) [( δZ ) st + µ ( δZ ) st ] [ ℓ ( s ) + λ ( δℓ ) st ] ( δℓ ) st . (27)These terms are obviously quadrati in ℓ , and an be bounded uniformly in λ, µ, s, t under the hypothesis σ ∈ C b . Noti e that, in order to bound the term | ℓ ( s ) | in (27), weuse the fa t that ℓ has a null initial ondition, whi h means in parti ular that | ℓ ( s ) | ≤ ( a − a + h ) λ k ℓ k λ, [ a − h,a ] . This (cid:28)nishes the proof of (26). The ontinuity of Dσ ( Z ) andthe existen e of the onstant c N introdu ed in Hypothesis 2 are now a question of trivial onsiderations, and this ends the proof of our proposition. (cid:3) Remark 3.6. The proof of Fre het di(cid:27)erentiability of f was not ne essary for the existen e-uniqueness result, whi h relied on some Lips hitz type ondition. However, this strongerresult turns out to be useful for the Malliavin al ulus part, and this is why we prove ithere. Nevertheless, noti e that Theorem 3.2 holds true for a C b oe(cid:30) ient σ .3.4. Di(cid:27)erentiability of the solution. In this se tion we study the di(cid:27)erentiability ofthe solution of (18) as a fun tion of the integrator x , following losely the methodology of[26℄. In parti ular, our di(cid:27)erentiability result will be a hieved with the help of the map F : C γ , ,T ( R d ) × C λ , ,T ( R n ) → C λ , ,T ( R n ) given by [ F ( k, Z )] t = Z t − J t (cid:16) f ( Z Z +˜ ξ ) d ( x + k ) (cid:17) , t ∈ [0 , T ] (28)where ˜ ξ t = ξ for t ∈ [0 , T ] , and ˜ ξ t = ξ t for t ∈ [ − h, . Here we re all that ξ stands for aninitial ondition in C λ ([ − h, . In this se tion the oe(cid:30) ient f will satis(cid:28)es the following:Hypothesis 3. Set t = (0 , t ) , and re all that the map U ( t ) has been de(cid:28)ned at Lemma3.1. We assume that U ( t ) : C λξ, ,t ( R n ) → C λ ([0 , t ]; R n × d ) is ontinuously Fré het di(cid:27)eren-tiable in the dire tions of C λ , ,t ( R n ) , for some λ ∈ (1 / , γ ) . We all ∇U ( t ) : C λξ, ,t ( R n ) →L ( C λ , ,t ( R n ); C λ , ,t ( R n × d )) its di(cid:27)erential, where L ( C λ , ,t ( R n ); C λ , ,t ( R n × d )) denotes the lin-ear operators from C λ , ,t ( R n ) into C λ , ,t ( R n × d ) . Moreover, for s < t and Z ∈ C λ , ,T ( R n ) , [ ∇U ( t ) ( y )]( Z ) = [ ∇U ( s ) ( y )]( Z ) on [0 , s ] , where y is the solution of equation (18).Remarks 3.7. (1) Noti e that we have shown, during the proof of Proposition 3.5, thatthe weighted delay given by (2) also satis(cid:28)es this last assumption.(2) If Z ∈ C λ , ,t ( R n ) , then k∇U ( t ) ( y )( Z ) k λ, [0 ,t ] ≤ |∇U ( T ) ( y ) |k Z k λ, [0 ,t ] . ˜ Z s = Z s for s ∈ [0 , t ] , and ˜ Z s = Z t for s > t . Therefore Hypothesis 3 implies k∇U ( t ) ( y )( Z ) k λ, [0 ,t ] ≤ k∇U ( T ) ( y )( ˜ Z ) k λ, [0 ,T ] ≤ |∇U ( t ) ( y ) |k Z k λ, [0 ,T ] = ≤ |∇U ( t ) ( y ) |k Z k λ, [0 ,t ] , and our laim is satis(cid:28)ed.We are now ready to prove the di(cid:27)erentiability properties for equation (18):Lemma 3.8. Under the Hypothesis 3, the map F given by (28) is ontinuously Fré hetdi(cid:27)erentiable.Proof. Let us all respe tively D and D the two dire tional derivatives. We (cid:28)rst observethat, for k, g ∈ C γ , ,T ( R d ) and Z ∈ C λ , ,T ( R n ) , we have: F ( k + g, Z ) − F ( k, Z ) + Z · h U ( T ) ( Z + ˜ ξ ) i s dg s = 0 . In other words, the partial derivative D F is de(cid:28)ned by D F ( k, Z )( g ) = − Z · h U ( T ) ( Z + ˜ ξ ) i s dg s = −J · (cid:16) [ U ( T ) ( Z + ˜ ξ )] dg (cid:17) . We shall prove now that D F is ontinuous: onsider k, ˜ k ∈ C γ , ,T ( R d ) and Z, ˜ Z ∈C λ , ,T ( R n ) . For notational sake, set also k · k λ for k · k λ, [0 ,T ] . Then, a ording to Lemma 3.3,we obtain: (cid:13)(cid:13)(cid:13) D F ( k, Z )( η ) − D F (˜ k, ˜ Z )( η ) (cid:13)(cid:13)(cid:13) λ = (cid:13)(cid:13)(cid:13) J (cid:16) [ U ( T ) ( Z + ˜ ξ ) − U ( T ) ( ˜ Z + ˜ ξ )] dη s (cid:17)(cid:13)(cid:13)(cid:13) λ ≤ k η k γ (cid:16)(cid:13)(cid:13)(cid:13) U ( T ) ( Z + ˜ ξ ) − U ( T ) ( ˜ Z + ˜ ξ ) (cid:13)(cid:13)(cid:13) ∞ T γ − λ + C λ + γ T γ (cid:13)(cid:13)(cid:13) U ( T ) ( Z + ˜ ξ ) − U ( T ) ( ˜ Z + ˜ ξ ) (cid:13)(cid:13)(cid:13) λ (cid:17) , whi h, owing to Hypothesis 3, implies that D F is ontinuous.Con erning D F we have, for k ∈ C γ , ,T ( R d ) , Z ∈ C λ , ,T ( R n ) and ˜ Z ∈ C λ , ,T ( R n ) , andthanks to Theorem 2.5: (cid:13)(cid:13)(cid:13) F ( k, Z + ˜ Z ) − F ( k, Z ) − ˜ Z + J (cid:16) [ ∇U ( T ) ( Z + ˜ ξ )]( ˜ Z ) d ( x + k ) (cid:17)(cid:13)(cid:13)(cid:13) λ ≤ k x + k k γ (cid:16)(cid:13)(cid:13)(cid:13) U ( T ) ( Z + ˜ Z + ˜ ξ ) − U ( T ) ( Z + ˜ ξ ) − [ ∇U ( T ) ( Z + ˜ ξ )]( ˜ Z ) (cid:13)(cid:13)(cid:13) ∞ T γ − λ + C λ + γ T γ (cid:13)(cid:13)(cid:13) U ( T ) ( Z + ˜ Z + ˜ ξ ) − U ( T ) ( Z + ˜ ξ ) − [ ∇U ( T ) ( Z + ˜ ξ )]( ˜ Z ) (cid:13)(cid:13)(cid:13) λ (cid:17) . Therefore, making use of Hypothesis 3, we have that: D F ( k, Z )( ˜ Z ) = ˜ Z − Z · ∇U ( T ) ( Z + ˜ ξ )( ˜ Z ) s d ( x s + k s ) . The ontinuity of D F an now be proven along the same lines as for D F , and the omputational details are left to the reader for sake of on iseness. The proof is now(cid:28)nished. (cid:3) The following will be used to show that D F ( k, Z ) is a linear homeomorphism.ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 17Lemma 3.9. Let w ∈ C λ , ,T ( R n ) , y the solution of (18) and assume Hypotheses 1, 2 and3 hold. Then the equation Z t = w t + Z t (cid:0) [ ∇U ( T ) ( y )]( Z ) (cid:1) s dx s , ≤ t ≤ T, (29)has a unique solution Z in C λ , ,T ( R n ) .Proof. Similarly to the proof of Theorem 3.2, we hoose ε ∈ (0 , T ) and set ˜ T : C λ , ,ε ( R n ) →C λ , ,ε ( R n ) given by ˜ T ( Z ) = w + J · ([ ∇U ( ε ) ( y )]( Z ) dx ) . Then, Lemma 3.3 and Remark3.7.(2) yield (cid:13)(cid:13)(cid:13) ˜ T ( Z ) − ˜ T ( ˜ Z ) (cid:13)(cid:13)(cid:13) λ, [0 ,ε ] = (cid:13)(cid:13)(cid:13) J (cid:16) [ ∇U ( ε ) ( y )]( Z − ˜ Z ) dx (cid:17)(cid:13)(cid:13)(cid:13) λ, [0 ,ε ] ≤ k x k λ ε γ − λ (cid:18)(cid:13)(cid:13)(cid:13) ∇U ( ε ) ( y )( Z − ˜ Z ) (cid:13)(cid:13)(cid:13) ∞ , [0 ,ε ] + c λ + γ T λ (cid:13)(cid:13)(cid:13) ∇U ( ε ) ( y )( Z − ˜ Z ) (cid:13)(cid:13)(cid:13) λ, [0 ,ε ] (cid:19) ≤ |∇U ( T ) ( y ) | ε γ − λ k x k λ k Z − ˜ Z k| λ, [0 ,ε ] ( T λ + c λ + γ T λ ) . That is, for ε small enough there exists < C < su h that (cid:13)(cid:13)(cid:13) ˜ T ( Z ) − ˜ T ( ˜ Z ) (cid:13)(cid:13)(cid:13) λ, [0 ,ε ] ≤ C k Z − ˜ Z k| λ, [0 ,ε ] . Hen e, by standard ontra tion arguments, one an (cid:28)nd a unique Z ε ∈ C λ , ,ε ( R n ) su hthat Z εt = w t + Z t (cid:0) [ ∇U ( ε ) ( y )]( Z ε ) (cid:1) s dx s , ≤ t ≤ ε. Now we introdu e ˜ T ε : C λZ ε ,ε, ε ( R n ) → C λZ ε ,ε, ε ( R n ) de(cid:28)ned by ˜ T ε ( Z )( t ) = w t − w ε + Z εε + Z tε ([ ∇U ( ε ) ( y )]( Z )) s dx s , t ∈ [ ε, ε ] . Then, as in the beginning of this proof, we have (cid:13)(cid:13)(cid:13) ˜ T ε ( Z ) − ˜ T ε ( ˜ Z ) (cid:13)(cid:13)(cid:13) λ, [0 , ε ] ≤ C k Z − ˜ Z k| λ, [0 , ε ] . Therefore, there is a unique Z ε ∈ C λZ ε ,ε, ε ( R n ) su h that Z εt = w t + Z t (cid:0) [ ∇U ( ε ) ( y )]( Z ε ) (cid:1) s dx s , ≤ t ≤ ε, due to Hypothesis 3.Finally by indu tion, we an (cid:28)gure out a fun tion Z kε ∈ C λZ ( k − ε , ( k − ε,kε ( R n ) su h that Z kεt = w t + Z t (cid:0) [ ∇U ( k ε ) ( y )]( Z kε ) (cid:1) s dx s , ≤ t ≤ kε. Consequently, by Remark 3.7.(2), it is not di(cid:30) ult to see that Z t = Z kεt for t ∈ [( k − ε, kε ] is the unique solution to equation (29). (cid:3) y be the solution ofequation (18). Then the map h y ( x + h ) is Fré het di(cid:27)erentiable in the dire tions of C γ , ,T ( R d ) , as a C λξ, ,T ( R n ) -valued fun tion. Moreover, for h, k ∈ C γ , ,T ( R d ) , we have [ Dy ( x )( k )] t = Z t U ( T ) ( y ( x )) s dk s + Z t (cid:2) ∇U ( T ) ( y ( x ))( Dy ( x )( k )) (cid:3) s dx s . (30)In parti ular, [ Dy ( x )]( k ) is an element of C λ , ,T ( R n ) .Remark 3.11. Let us re all that equation (30) has a unique solution, thanks to Lemma 3.9.Proof of Proposition 3.10: Like in [26℄, the proof of this result is a onsequen e of theimpli it fun tion theorem, and we only need to show that D F (0 , y ( x ) − ˜ ξ ) is a linearhomeomorphism from C λ , ,T ( R n ) onto C λ , ,T ( R n ) . Indeed, in this ase we dedu e that h y ( x ) is Fré het di(cid:27)erentiable with Dy ( x )( k ) = − (cid:16) D F ( h, y ( x ) − ˜ ξ ) (cid:17) − ◦ D F ( h, y ( x ) − ˜ ξ )( k ) , (31)whi h yields that (30) holds.Finally, noti e that D F (0 , y ( x ) − ˜ ξ ) is bije tive and ontinuous a ording to Lem-mas 3.8 and 3.9. Consequently the open mapping theorem implies that the appli ation D F (0 , y ( x ) − ˜ ξ ) is also a homeomorphism. (cid:3) Interestingly enough, in the parti ular ase of the weighted delay of Se tion 3.3, one an also derive a linear equation for the derivative [ Dy ( x )] t , seen as a Hölder- ontinuousfun tion.Proposition 3.12. Let σ and ν be as in Proposition 3.5. Let also f and y be de(cid:28)nedby (2) and (18), respe tively. Assume that ν is absolutely ontinuous with respe t to theLebesgue measure with Radon-Nykodim derivative in L p ([ − h, for p > / (1 − γ ) . Then,for i ∈ { , . . . , n } and k ∈ C λ , ,T ( R n ) , we have Dy it ( x )( k ) = d X j =1 Z t Φ ijt ( r ) dk jr , where, for j ∈ { i, . . . , d } and i ∈ { , . . . , n } , Φ ij is de(cid:28)ned by the equation Φ ijt ( r ) = ( U ( T ) ( y )) ijt + n X m =1 d X l =1 Z tr (cid:0) ([ ∇U ( T ) ( y )] m ) il (Φ mj ( s )) (cid:1) s dx ls , ≤ r ≤ t ≤ T, (32)and Φ t ( r ) = 0 for all ≤ t < r ≤ T. Remark 3.13. Note that, for ea h s ∈ [0 , T ] equation (32) has a unique solution in C λ ([ s, T ]; R n ) due to Lemma 3.9.ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 19Proof of Proposition 3.12. In order to avoid umbersome matrix notations, we shall provethis result for n = d = 1 : noti e that an easy onsequen e of the proof of Proposition 3.5is that in our parti ular ase, (cid:2) ∇U ( T ) ( Z )( k ) (cid:3) t = σ ′ (cid:18)Z − h Z t + θ ν ( dθ ) (cid:19) (cid:18)Z − h k t + θ ν ( dθ ) (cid:19) . (33)Set now q t = σ ( R − h y t + θ ν ( dθ )) and q ′ t = σ ′ ( R − h y t + θ ν ( dθ )) , and write y = y ( x ) . Thenequation (30) an be read as: [ Dy ( k )] t = Z t q s dk s + U t , with U t = Z t q ′ s (cid:18)Z − h [ Dy ( k )] s + θ ν ( dθ ) (cid:19) dx s . (34)The Fubini type relation given at Lemma 2.6 allows then to show, as in [26, Proposition4℄, that [ Dy ( k )] t = Z t Φ t ( r ) dk r , (35)for a ertain fun tion Φ , λ -Hölder ontinuous in all its variables. In order to identify thepro ess Φ , plug relation (35) into equation (34) and apply Fubini's theorem, whi h yields U t = Z − h ν ( dθ ) Z t q ′ s Z ( s + θ ) + Φ s + θ ( r ) dk r ! dx s . It should be noti ed that this point is where we use the fa t that ν ( dθ ) = µ ( θ ) dθ with ∈ L λ ([ − r, . Indeed, in order to apply Lemma 2.6 to x , k and η F ( η ) = R η − h µ ( θ ) dθ ,we will assume (though this is not ompletely optimal) that F is γ -Hölder ontinuous.However, a simple appli ation of Hölder's inequality yields | F ( η ) − F ( η ) | ≤ c | t − s | ( p − /p k µ k L p ([ − h, . It is now easily seen that the ondition ( p − /p > γ imposes p > / (1 − γ ) .Owing now to a (slight extension of) Lemma 2.6, we an write U t = Z − h ν ( dθ ) Z ( t + θ ) + m t ( r, θ ) dk r , with m t ( r, θ ) = Z tr − θ q ′ s Φ s + θ ( r ) dx s . Apply Fubini's theorem again in order to integrate with respe t to k in the last pla e: weobtain U t = Z t (cid:18)Z − [( t − r ) ∧ h ] m t ( r, θ ) ν ( dθ ) (cid:19) dk r = Z t (cid:18)Z − [( t − r ) ∧ h ] ν ( dθ ) Z tr − θ q ′ s Φ s + θ ( r ) dx s (cid:19) dk r , and going ba k to (34), whi h is valid for any λ -Hölder ontinuous fun tion k , we get that Φ t is de(cid:28)ned on [0 , t ] by the equation Φ t ( r ) = q t + Z − [( t − r ) ∧ h ] (cid:18)Z tr − θ q ′ s Φ s + θ ( r ) dx s (cid:19) ν ( dθ ) , and Φ t ( r ) = 0 if r > t . A last appli ation of Fubini's theorem allows then us to re ast theabove equation as Φ t ( r ) = q t + Z tr q ′ s (cid:18)Z − [ h ∧ ( s − r )] Φ s + θ ( r ) ν ( dθ ) (cid:19) dx s . θ ≤ − ( s − r ) in the above equation, then s + θ ≤ r , whi h means that Φ s + θ ( r ) = 0 . Hen e, we end up with an equation of the form Φ t ( r ) = q t + Z tr q ′ s (cid:18)Z − h Φ s + θ ( r ) ν ( dθ ) (cid:19) dx s , whi h is easily seen to be of the form (32). (cid:3) ˜ f be a mapping from C λξ, ,T ( R n ) into the linear operators from C λ , ,T ( R n ) into C λ ([0 , T ]; R n × d ) su h that, for ≤ a < b ≤ T , ˜ y ∈ C λξ, ,T ( R n ) and ˜ z ∈C λ , ,T ( R n ) ,(1) k ˜ f (˜ y )˜ z k ∞ , [ a,b ] ≤ M k ˜ z k ∞ , [ a − h,b ] .(2) k ˜ f (˜ y )˜ z k λ, [ a,b ] ≤ M k ˜ z k λ, [ a − h,b ] + M k ˜ y k λ, [ a − h,b ] k ˜ z k ∞ , [ a − h,b ] . Also let y be the solution of the equation (18), w ∈ C λ , ,T ( R n ) and z ∈ C λ , ,T ( R n ) thesolution of the equation z t = w t + Z t ( ˜ f ( y ) z )( t ) dx t , t ∈ [0 , T ] . Then k z k λ, [0 ,T ] ≤ c k w k λ, [0 ,T ] D γ,λ e c D γ,λ , for two stri tly positive onstants c i = c i ( T, γ, λ, M ) , i = 1 , and D γ,λ = ( k ξ k λ k x k γ ) / ( γ + λ ) + k x k /γγ + k x k (2 λ + γ − / (( γ + λ )( γ + λ − γ . Remarks 3.15. (1) Observe that if f is as in Proposition 3.5 and ˜ f = ∇U ( T ) , thenstraightforward al ulations show that Conditions (1) and (2) in the Proposition aresatis(cid:28)ed.(2) The fa t that z = 0 implies that k z k ∞ , [0 ,T ] ≤ c T λ k w k λ, [0 ,T ] D γ,λ e c D γ,λ . (3) Let λ = γ . Then ( γ + 2 λ − / (( γ + λ )( γ + λ − in Proposition 3.14 is smaller than2 for γ > H , where H = (7 + √ / ≈ . . This is the threshold above whi h ourgeneral delay equation will admit a smooth density.(4) The unusual threshold H above stems from the ontinuous dependen e of the solu-tion on its past, represented by the measure ν . In ase of a dis rete delay of the form σ ( y t , y t − r , . . . , y t − r q ) , we shall see that all our onsiderations are valid for any H > / .Proof of Proposition 3.14. We (cid:28)rst onsider two generi positive numbers k ∈ N and ε ,su h that ( k + 1) ε ≤ T . Then Theorem 2.5, point (2), and Conditions (1) and (2) imply k z − w k λ, [ kε, ( k +1) ε ] ≤ k ˜ f ( y ) z k ∞ , [ kε, ( k +1) ε ] k x k γ ε γ − λ + c γ,λ k ˜ f ( y ) z k λ, [ kε, ( k +1) ε ] k x k γ ε γ ≤ M k z k ∞ , [0 , ( k +1) ε ] k x k γ ε γ − λ + c γ,λ M k x k γ (cid:0) k z k λ, [0 , ( k +1) ε ] + k z k ∞ , [0 , ( k +1) ε ] k y k λ, [0 ,T ] (cid:1) ε γ . ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 21The following (arguably non optimal) bound on k z k ∞ , [0 , ( k +1) ε ] an now be easily veri(cid:28)edby indu tion: k z k ∞ , [0 , ( k +1) ε ] ≤ k +1 X i =1 k +1 − i k z − z ( i − ε k ∞ , [( i − ε,iε ] ≤ k +1 X i =1 k +1 − i k z k λ, [( i − ε,iε ] . This yields k z − w k λ, [ kε, ( k +1) ε ] ≤ M k x k γ ε γ k +1 X i =1 k +1 − i k z − z ( i − ε k λ, [( i − ε,iε ] ! + c γ,λ M k x k γ ε γ (cid:0) k z k λ, [0 ,kε ] + k z k λ, [ kε, ( k +1) ε ] (cid:1) + c γ,λ M k x k γ k y k λ, [0 ,T ] ε γ + λ k +1 X i =1 k +1 − i k z − z ( i − ε k λ, [( i − ε,iε ] ! . (36)Now the proof an be split in three steps.Step 1. Bounds depending on ε . Let ε = ( T + [6 M k x k γ (1 + c γ,λ )] /γ + [6 M k x k γ c γ,λ k y k λ, [0 ,T ] ] / ( γ + λ ) − ∧ T. (37)Note that in this ase, inequality (36) yields k z k λ, [ kε, ( k +1) ε ] ≤ k w k λ, [ kε, ( k +1) ε ] + M k x k γ ε γ k X i =1 k +2 − i k z k λ, [( i − ε,iε ] ! + c γ,λ M k x k γ ε γ k z k λ, [0 ,kε ] + ε λ k y k λ, [0 ,T ] k X i =1 k +2 − i k z k λ, [( i − ε,iε ] ! ≤ k w k λ, [ kε, ( k +1) ε ] + k X i =1 k +2 − i k z k λ, [( i − ε,iε ] (cid:0) M k x k γ ε γ + c γ,λ M k x k γ ε γ + c γ,λ M k x k γ ε γ + λ k y k λ, [0 ,T ] (cid:1) ≤ k w k λ, [ kε, ( k +1) ε ] + k X i =1 k +1 − i k z k λ, [( i − ε,iε ] , (38)where we have used (37) in the last step.Step 2. Bounds for k z k λ, [ kε, ( k +1) ε ] . Here we will use indu tion on k to show that k z k λ, [( i − ε,iε ] ≤ i X j =1 i +1 − j k w k λ, [( j − ε,jε ] . (39)By (38) we have that this inequality holds for i = 1 . Therefore we an assume that(39) holds for any positive integer i less o equal than k to show that it is also true for i = k + 1 .2 JORGE A. LEÓN AND SAMY TINDELThe inequalities (38) and (39) lead us to write k z k λ, [ kε, ( k +1) ε ] ≤ k w k λ, [ kε, ( k +1) ε ] + k X i =1 k +1 − i i X j =1 i +1 − j k w k λ, [( j − ε,jε ] ≤ k w k λ, [ kε, ( k +1) ε ] + k X j =1 k w k λ, [( j − ε,jε ] k +2 − j k X i =1 i ≤ k w k λ, [ kε, ( k +1) ε ] + k X j =1 k w k λ, [( j − ε,jε ] k +3 − j . Now it is easy to see that (39) also holds for i = k + 1 .Step 3. Final bound. Let k su h that k ε < T < ( k + 1) ε . Then, by Step 2 we have k z k λ, [0 ,T ] ≤ k w k λ, [0 ,T ] k X k =1 k X j =1 k +1 − j ≤ k w k λ, [0 ,T ] ( k ) k +1 ≤ k w k λ, [0 ,T ] (2 T /ε ) T ε − +3 . Thus the proof is (cid:28)nished by plugging relation (37) into the last expression, and invokingProposition 3.4. (cid:3)
The following result is a slight extension of Proposition 3.14, allowing to take intoa ount the ase of onstant but non vanishing fun tions.Corollary 3.16. Let ˜ f , D γ,λ , w and y be as in Proposition 3.14. Furthermore, assumethat ˜ f is a mapping from C λξ, ,T ( R n ) into the linear operators from the onstant fun tionson [ − h, T ] into C λ ([0 , T ]; R n × d ) satisfying the Conditions (1) and (2) of Proposition 3.14when ˜ z is a onstant fun tion. Then the solution of the equation z t = c + w t + Z t ( ˜ f ( y ) z )( t ) dx t , t ∈ [0 , T ] , satis(cid:28)es the inequality k z k λ, [0 ,T ] ≤ c (cid:13)(cid:13)(cid:13)(cid:13) w + Z · ( ˜ f ( y )˜ c )( t ) dx t (cid:13)(cid:13)(cid:13)(cid:13) λ, [0 ,T ] D γ,λ e c D γ,λ , where ˜ c stands for the onstant fun tion ˜ c t ≡ c .Proof. The proof is an immediate onsequen e of Proposition 3.14. Indeed, we only needto observe that z t − ˜ c t = w t + Z t ( ˜ f ( y )˜ c )( t ) dx t + Z t ( ˜ f ( y )( z − ˜ c ))( t ) dx t , t ∈ [0 , T ] , where ˜ c ( t ) = c , t ∈ [0 , T ] . (cid:3) ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 234. Delay equations driven by a fra tional Brownian motionHere we onsider the Young sto hasti delay equation y t = ξ + Z t f ( Z yt ) dB t , ≤ t ≤ T, Z y = ξ, (40)where B = { B t ; 0 ≤ t ≤ T } is a d -dimensional fra tional Brownian motion (fBm) withparameter H ∈ (1 / , . The oe(cid:30) ient f satis(cid:28)es Hypotheses 1-3 and ξ is a givendeterministi fun tion in C γ ([ − h, R n ) , for some λ < γ < H . Remember that λ ∈ (1 / , H ) is introdu ed at the beginning of Se tion 3.The fBm B is a entered Gaussian pro ess with the ovarian e R H ( t, s ) δ i,j = E ( B is B jt ) = 12 δ i,j ( s H + t H − | t − s | H ) . In parti ular, B has ν -Hölder ontinuous paths for any exponent ν < H . Consequently,from Theorem 3.2 and Hypothesis 1-3, equation (40) has a unique C λξ, ,T ( R n ) -pathwisesolution.Here, our main goal is to analyze the existen e of a smooth density of the solutionof equation (40). This will be done via the Malliavin al ulus or sto hasti al ulus ofvariations.4.1. Preliminaries on Malliavin al ulus. In this subse tion we introdu e the frame-work and the results that we use in the remaining of this paper. Namely, we give sometools of the Malliavin al ulus for fra tional Brownian motion. Towards this end, wesuppose that the reader is familiar with the basi fa ts of sto hasti analysis for Gaussianpro esses as presented, for example, in Nualart [23℄.Hen eforth, we will onsider the abstra t Wiener spa e introdu ed in Nualart andSaussereau [26℄, in order to take advantage of the relation between the Fré het derivativesof the solution to equation (40) (see Proposition 3.10) and its derivatives in the Malliavin al ulus sense (see [23℄, Proposition 4.1.3). This abstra t Wiener spa e is onstru ted asfollows (for a more detailed exposition of it, the reader an onsult [26℄).We assume that the underlying probability spa e (Ω , F , P ) is su h that Ω is the Bana hspa e of all the ontinuous funtions C ([0 , T ]; R d ) , whi h are zero at time , endowed withthe supremum norm. P is the only probability measure su h that the anoni al pro ess { B t ; 0 ≤ t ≤ T } is a d -dimensional fBm with parameter H ∈ (1 / , and the σ -algebra F is the ompletion of the Borel σ -algebra of Ω with respe t to P .Two important tools related to the fBm B are the ompletion H of the R d -valued stepfun ions E with respe t to the inner produ t h ( [0 ,t ] , . . . , [0 ,t d ] ) , ( [0 ,s ] , . . . , [0 ,s d ] ) i = P di =1 R H ( s i , t i ) and the isometry K ∗ H : H → L ([0 , T ] d ) , whi h satis(cid:28)es K ∗ H (( [0 ,t ] , . . . , [0 ,t d ] ) = ( [0 ,t ] ( · ) K H ( t , · ) , . . . , [0 ,t d ] K H ( t d , · )) , where K H ( t, s ) = c H s / − H R ts ( u − s ) H − / u H − / du is a kernel verifying R H ( t, s ) = Z t ∧ s K H ( t, r ) K H ( s, r ) dr. K ∗ H an be represented in the two following ways: [ K ∗ H ϕ ] t = Z Tt ϕ r ∂ r K ( r, t ) dr = c H s / − H [ I H − / T − ( u H − / ϕ u )] t , (41)where I αT − stands for the fra tional integration of order α on [0 , T ] (see [24℄ for furtherdetails).The isometry K ∗ H allows us to introdu e the version of the Reprodu ing Kernel Hilbertspa e H H asso iated with the pro ess B . Namely, Let K H be given by K H : L ([0 , T ]; R d ) → H H := K H ( L ([0 , T ]; R d )) , ( K H h )( t ) = Z t K H ( t, s ) h ( s ) ds. The spa e H is ontinuously and densely embedded in Ω . Indeed, it is not di(cid:30) ult to seethat the operator R H : H → H H de(cid:28)ned by R H φ = Z · K H ( · , s )( K ∗ H φ )( s ) ds embeds H ontinuously and densely into Ω , be ause, as it was pointed out in [26℄, R H ( φ ) is H -Hölder ontinuous. Thus, we have that (Ω , H , P ) is an abstra t Wiener spa e.Now we introdu e the derivative in the Malliavin al ulus sense of a random variable.We say that a random variable F is a smooth fun tional in S if it has the form F = f ( B ( h ) , . . . , B ( h n )) , where h , . . . , h n ∈ H and f and all its partial derivatives have polynomial growth. Thederivative of this smooth fu tional is the H -valued random variable given by D F = n X i =1 ∂f∂x i ( B ( h ) , . . . , B ( h n )) h i . For p > , the operator D is losable from L p (Ω) into L p (Ω; H ) (see [23℄). The losure ofthis operator is also denoted by D and its domain by D ,p , whi h is the ompletion of S with respe t to the norm k F k p ,p = E ( | F | p ) + E ( kD F k p H ) . The operator D has the lo al property (i.e., D F = 0 on A ⊂ Ω if A F = 0 ). This allowsus to extend the domain of the operator D as follows. We say that F ∈ D ,ploc if there isa sequen e { (Ω n , F n ) , n ≥ } ⊂ F × D ,p su h that Ω n ↑ Ω w.p.1 and F = F n on Ω n . Inthis ase, we de(cid:28)ne D F = D F n on Ω n .It is known that, in the abstra t Wiener spa e (Ω , H , P ) , we an onsider the dif-ferentiability of random variable F in the dire tions of H . That is, we say that F is H -di(cid:27)erentiable if for almost all ω ∈ Ω and h ∈ H , the map ε F ( ω + ε R H h ) is di(cid:27)eren-tiable. The following result due to Kusuoka [14℄ (see also [23℄, Proposition 4.1.3) will befundamental in the study of the existen e of smooth densities of the solution of equation(40).Proposition 4.1. Let F be an H -di(cid:27)erentiable random variable. Then F belongs to thespa e D ,ploc , for any p > .ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 25We will apply this result to the solution of equation (40) as follows. Note that for ϕ ∈ H , we have the inequality | ( R H ϕ ) i ( t ) − ( R H ϕ ) i ( s ) | = (cid:0) E [ | B it − B is | ] (cid:1) / k ϕ k H ≤ k ϕ k H | t − s | H . Consequently, Proposition 3.10 (see also Lemma 4.2 below) implies that the randomvariable y t de(cid:28)ned in equation (40) is also H -di(cid:27)erentiable, whi h, together with Propo-sition 4.1, yields that y it belongs to D ,ploc for every t ∈ [0 , T ] , p > and i ∈ { , . . . , n } .Moreover, the relation between the H -derivative and D is given by (see also Lemma 4.3), hD y it , h i H = ddε y it ( ω + ε R H h ) | ε =0 , h ∈ H . (42). More generally, if ω X ( ω ) is in(cid:28)netely Fré het diferentiable in the dire tions of C λ , ,T ( R ) , then for a smooth random variable X , then hD n X, h ⊗ · · · ⊗ h n i H n = D R H h ,..., R H h n X = ∂∂ε . . . ∂∂ε n X ( ω + ε R h + . . . + ε n R h n ) | ε = ... = ε n =0 . t ∈ [0 , T ] , the random variable y t introdu ed in equation (40) has a density.Let us start with two important te hni al tools. The (cid:28)rst one relates the derivative ofthe ve tor-valued quantity y t with the derivative of y as a fun tion.Lemma 4.2. Let y be the solution of (40) and t ∈ [0 , T ] . Then almost surely, h y t ( B + h ) is Fré het di(cid:27)erentiable from C λ , ,T ( R d ) into R n . Furthermore Dy t ( B )( h ) = [ Dy ( B )( h )] t . Proof. The proof is an immediate onsequen e of | y t ( x + h ) − y t ( x ) − ( Dy ( x )( h )) ( t ) | = | y t ( x + h ) − y t ( x ) − ( Dy ( x )( h )) ( t ) − y ( x + h ) − y ( x ) − ( Dy ( x )( h )) (0) |≤ k y ( x + h ) − y ( x ) − Dy ( x )( h ) k λ t λ , with x, h ∈ C λ , ,T ( R d ) . (cid:3) Lemma 4.3. Let y be the solution of (40). Then y it belongs to D , loc for every t ∈ [0 , T ] and i ∈ { , . . . , n } . Moreover, for h ∈ H , we have hD y it , h i H = (cid:2) Dy i ( B )( R H h ) (cid:3) t . (43)Proof. By Proposition 4.1 and Lemma 4.2, we have already shown that y it is in D , loc forevery t ∈ [0 , T ] and i ∈ { , . . . , n } .Furthermore, by (42) and Lemma 4.2, we have hD y it , h i H = D R H h y it = Dy it ( B )( R H h ) = (cid:0) Dy i ( B )( R H h ) (cid:1) ( t ) . Thus, the proof is omplete. (cid:3)
We now use the ideas of Nualart and Saussereau [26℄ to state one of the main resultsof this se tion:6 JORGE A. LEÓN AND SAMY TINDELTheorem 4.4. Let us assume that Hypotheses 1-3 hold, re all that ξ is the (fun tional) ini-tial ondition of equation (40), and assume that the spa e spanned by { ( f ( ξ ) j , . . . , f ( ξ ) nj );1 ≤ j ≤ d } is R n . Then for t ∈ (0 , T ] , the random variable y t given by (40) is absolutely ontinuous with respe t to the Lebesgue measure on R n .Proof. As in [26℄ (proof of Theorem 8), we have that y it belongs to D , loc . Therefore weonly need to see that the Malliavin ovarian e matrix Q ijt := hD y it , D y jt i H (44)is invertible almost surely.For v ∈ R n , following [26℄ (proof of Theorem 8), we have v T Q t v = ∞ X m =1 |h Dy ( B )( R H h m )( t ) , v i R n | , where { h n , m ≥ } is a omplete orthonormal system of H .Now assume that the Malliavin matrix Q t is not almost surely invertible. Then, on theset of stri tly positive probability where Q t is not invertible, there exists v ∈ R n , v = 0 su h that v T Q t v = 0 . Moreover, re alling our notation (28), it is lear from equation (31)that D F ( k, Z ) is a linear homomorphism. Hen e, we obtain that h D F (0 , y ( B − ˜ ξ ))( R H h m )( t ) , v i R n = − (cid:28)Z t U ( T ) ( y ( B )) s d R H h m ( s ) , v (cid:29) R n = − n X i =1 d X j =1 v i Z t (cid:0) U ( T ) ( y ( B )) (cid:1) ijs d R H h jm ( s )= − n X i =1 h v i (cid:0) U ( T ) ( y ( B )) (cid:1) i [0 ,t ] , h m i H , for all m ≥ , where the last equality follows from [26℄. For t > , taking into a ount the de(cid:28)nition of U ( T ) given at Lemma 3.1, we obtain that P ni =1 v i f ij ( ξ ) = 0 , whi h ontradi ts the fa tthat R n oin ides with the spa e spanned by { ( f ( ξ ) j , . . . , f ( ξ ) nj ); 1 ≤ j ≤ d } . So we have that the Malliavin matrix Q t is invertible for any t ∈ (0 , T ] , as we wished toprove. (cid:3) L − p moments of its Malliavin matrix. Towards this aim, it will be useful to produ e anequation solved by the Malliavin derivative of the solution y t of equation (40). This is ontained in the following Lemma:Lemma 4.5. Under the onditions of Proposition 3.12, let y be the solution to equa-tion (40). Assume furthermore that B is a fBm with Hurst parameter H > H , where H ALLIAVIN CALCULUS FOR FRACTIONAL DELAY EQUATIONS 27is de(cid:28)ned at Remark 3.15. Then y t ∈ D ,p for any p ≥ , and Φ t ( r ) := D r y t is the uniquesolution to the following equation: Φ t ( r ) = [ U ( T ) ( y )] t + V t ( r ) , where V ijt ( r ) = n X m =1 d X l =1 Z tr (cid:0) ([ ∇U ( T ) ( y )] m ) il (Φ mj ( s )) (cid:1) s dB ls , (45)with the additional onstraint Φ t ( r ) = 0 for all ≤ t < r ≤ T. Proof. The equation followed by D y is a dire t onsequen e of relation (43) and Proposi-tion 3.12. The fa t that y t ∈ D ,p when H > H stems now from Proposition 3.14. (cid:3) Now we are able to state the se ond main result of this se tion, for whi h we need anadditional notation: for two a non-negative matri es
M, N ∈ R n × n , we write M ≥ N when the matrix M − N is non-negative.Theorem 4.6. Let f, σ , ν and B as in Lemma 4.5. Assume that σ has bounded deriva-tives of any order and that σ ( η ) σ ( η ) ∗ ≥ ε Id R n , for all η , η ∈ R n . (46)Then, for t ∈ (0 , T ] , y t has a C ∞ -density.Proof. The proof follows losely the lines of [15, Theorem 3.5℄, whi h is lassi al in theMalliavin al ulus setting, and we shall thus pro eed without giving too many details.Nevertheless, we shall divide our proof in two steps.Step 1: Let Q t be the Malliavin matrix of y t , de(cid:28)ned by (44). The standard onditionsto verify in order to get a C ∞ density are: (i) y t ∈ D ∞ , and (ii) [det( Q t )] − ∈ L p for all p ≥ . Condition (i) is obtained by iterating the derivatives of y , similarly to what isdone in [26℄, so that we will fo us on point (ii).In order to he k that [det( Q t )] − ∈ L p , we bound P ( | [det( Q t )] | − ≥ µ ) for µ largeenough, and invoke the fa t that P (cid:0) | [det( Q t )] | − ≥ µ (cid:1) ≤ P (cid:18) Q t (cid:3) µ Id R n (cid:19) . In the sequel of the proof, we will evaluate the right hand side of the above inequality.Step 2: In order to bound Q t from below, the basi idea is to use de omposition (45)for the Malliavin derivative of y . In this de omposition, the term [ U ( T ) ( y )] t is boundeddeterministi ally from below under the non-degenera y ondition (46), while V is a highly(cid:29)u tuating quantity, sin e it is given by a sto hasti integral with respe t to B .One an formalize the previous heuristi onsiderations in the following way: L t = (cid:13)(cid:13) U ( T ) ( y ) [0 ,t ] (cid:13)(cid:13) H = (cid:13)(cid:13) K ∗ H (cid:0) U ( T ) ( y ) [0 ,t ] (cid:1)(cid:13)(cid:13) L ([0 ,t ]; R n ) . Thanks to relation (41), one an show that L t = c H n X l =1 Z t s − H Z ts Z ts ( r − s ) H − / ( u − s ) H − / r H − / u H − / h q ∗ r q u , e l i dudrds, { e l ; l = 1 , . . . , n } stands for the anoni al basis of R n , and where we have set q s = σ ( R − h y s + θ ν ( dθ )) as in the proof of Proposition 3.12. Therefore, ondition (46)yields, for a onstant c whi h may hange from line to line, L t ≥ c ε (cid:18)Z t s − H Z ts Z ts ( r − s ) H − / ( u − s ) H − / r H − / u H − / dudrds (cid:19) Id R n ≥ c εt H Id R n . A ording to relation (45), it is now readily he ked that Q t ≥ L t − k V t k H Id R n . Thus, for any stri tly positive number α , there exists a universal onstant c su h that P (cid:18) Q t (cid:3) cαεt H Id R n (cid:19) ≤ P (cid:18) k V t k H Id R n ≥ cαεt H (cid:19) ≤ (cid:18) cαεt H (cid:19) p E [ k V t k p H ] α p . It is now enough to observe that E [ k V t k p H ] is a (cid:28)nite quantity for any p ≥ , owing toProposition 3.14, to on lude the proof. (cid:3) Remark 4.7. As mentioned before, the restri tion
H > H for the smoothness of thedensity of the random variable y t is due to the ontinuous dependen e of our oe(cid:30) ient f on the past of the solution. Indeed, in ase of a dis rete delayed oe(cid:30) ient of the form σ ( y t , y t − r , . . . , y t − r q ) , with q ≥ and r < · · · < r q ≤ h , it an be seen that equation (40) an be redu ed to an ordinary di(cid:27)erential equation driven by B . This allows to apply the riterions given in [13℄, whi h are valid up to H = 1 / .In order to get onvin ed of this fa t, onsider the simplest dis rete delay ase, that isan equation of the form ξ + Z t σ ( y t , y t − r ) dB t , ≤ t ≤ T, (47)with r > . The initial ondition of this pro ess is given by ξ ∈ C γ on [ − r, , and wealso assume that σ and B are real valued. Without loss of generality, one an assumethat T = m r for m ∈ N ∗ . In this ase, set y ( k ) = { y s + kr ; s ∈ [0 , r ) } , and adopt the samenotation for B . Then one an re ast (47) as y t ( k ) = y r ( k −
1) + Z t σ ( y u ( k ) , y u ( k − dB u ( k ) , t ∈ [0 , r ] , k ≤ m − . (48)Setting now y = ( y (1) , . . . , y ( m )) t , B = ( B (1) , . . . , B ( k )) t and de(cid:28)ning ˆ σ : R m → R m,m by ˆ σ ( η (1) , . . . , η ( m )) = Diag( σ ( η (1)) , . . . , σ ( η ( m ))) , we an express (48) in a matrix form as y t = y + Z t ˆ σ ( y u (1) , . . . , y u ( m )) d B u , , t ∈ [0 , r ] . (49)This is now an ordinary equation driven by a m -dimensional fBm B . Whenever | σ ( η ) | ≥ ε > and H > / , one an apply the non-degenera y riterion of [13℄ in order tosee that y t posesses a smooth density for any t ∈ (0 , T ] . The ase of a ve tor valuedoriginal equation (47) an also be handled through umbersome matrix notations. 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