Rough Volterra equations 2: Convolutional generalized integrals

Transcription

1 Stochatic Procee and their Application 121 (2011) ough Volterra equation 2: Convolutional generalized integral Aurélien Deya, Samy Tindel Intitut Élie Cartan Nancy, Nancy-Univerité, B.P. 239, Vandœuvre-lè-Nancy Cedex, France eceived 26 June 2010; received in revied form 20 April 2011; accepted 2 May 2011 Available online 7 May 2011 Abtract We define and olve Volterra equation driven by a non-differentiable ignal, by mean of a variant of the rough path theory which allow u to handle generalized integral weighted by an exponential coefficient. The reult are applied to a tandard rough path x = (x 1, x 2 ) C γ 2 (m ) C 2γ 2 (m,m ), with γ > 1/3, which include the cae of fractional Brownian motion with Hurt index H > 1/3. c 2011 Elevier B.V. All right reerved. MSC: 60H05; 60H07; 60G15 Keyword: ough path theory; Stochatic Volterra equation; Fractional Brownian motion 1. Introduction Thi paper i part of an ambitiou ongoing project which aim at offering a new point of view on multidimenional tochatic calculu, via the emi-determinitic rough path approach initiated by Lyon [24]. We tackle the iue of the non-linear Volterra ytem yt i = a i + σ i0 (t, u, y u ) du + 0 m j=1 0 σ i j (t, u, y u ) dx j u, i = 1,..., d, t [0, T ],(1) where T tand for an arbitrary horizon, x : [0, T ] m a multidimenional γ -Hölder path, a d an initial condition and σ i j : [0, T ] 2 d mooth enough function. Correponding author. addree: deya@iecn.u-nancy.fr (A. Deya), tindel@iecn.u-nancy.fr (S. Tindel) /$ - ee front matter c 2011 Elevier B.V. All right reerved. doi: /j.pa

2 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) The (ordinary) Volterra equation providing a relevant model in many biological or phyical ituation, it i not urpriing that it noiy verion ha already given birth to a great number of paper. A firt analyi when x i a Brownian motion i contained in the pioneering work [6,7], and thi wa then generalized to the cae of a emimartingale in [31]. If the coefficient σ i j are alo een a random function, which often happen to be more appropriate, ome anticipative tochatic calculu technique are required in order to olve the ytem, and we refer the reader to [1,28,30] for the main reult in thi direction. It hould be mentioned at thi point that the lat of thoe reference [30] i motivated by financial model of capital growth rate, which goe beyond the claical phyical or biological application of Volterra equation. Several author alo enviaged the poibility of a ingularity for the application u < t σ (t, u,.) a t tend to u [10,11,37], while example of a o-called backward tochatic Volterra equation have recently appeared in the literature [38,40], timulated (here again) by new financial application [39]. Beide, one can find in [34,21,43] tudie of infinite-dimenional verion of (1), often linked to the context of tochatic partial differential equation. It i finally worth noticing that the behavior of the olution to the Itô Volterra equation i now deeply undertood, through the conideration of numerical cheme [35,42] or the exitence of large deviation [17,33,27,42] and Straen law [29] reult. In thi background, it eem quite natural to wonder whether the interpretation and reolution of (1) can be extended to a non-emimartingale driving proce x. The exitence of a theoretical olution would for intance allow u to tudy the influence of a more general gauian noie in the aymptotic equilibria oberved in [4,2,3,5]. The interet in a generalization of the ytem ha alo been recently reinforced by the emergence, in the field of nanophyic, of a model involving a Volterra ytem perturbed by a fractional Brownian motion (fbm in the equel) with Hurt index H different from 1/2 [22,23]. In the latter reference, the fractional proce only intervene through an additive noie: the reolution of the ytem (1) in it general form would here open the way to a ophitication of the model. The particular cae where x tand for a fbm with Hurt index H > 1/2 ha been thoroughly treated in [16]: the integral i therein undertood in the Young ene. Notice that in thi ituation, [8] provide a lightly different approach to the equation, baed on fractional calculu technique. If one wihe to go one tep further in the procedure and conider a γ -Hölder path with γ 1/2, the rough path method mut come into the picture. However, the claical rough path theory introduced by Lyon and Qian [25] (ee alo the recent formulation in [18]) i motly deigned to handle the cae of diffuion type equation, and there ha been an intenive activity during the lat couple of year in order to extend thee emi-pathwie technique to other ytem, uch a delay equation [26] or PDE [9,20]. The current article fit into thi global project, and we hall ee how to modify the original rough path etting in order to handle ytem like (1). The method then lead to what appear to u a the firt reult on the exitence and uniquene of a global olution ever hown for the rough Volterra equation (1), in the cae where γ < 1 2. Our reult more exactly applie to the convolutional Volterra equation y i t = a i + m j=1 0 φ(t u)σ i j (y u ) dx j u, i = 1,..., d, t [0, T ], (2) where φ : and σ i j : d are mooth enough application. Notice that we have included the drift term in the um, by auming that the firt component of x coincide with the identity function. In pite of it pecificity, the formulation (2) cover mot of the aforementioned

3 1866 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) model (it i in particular the model under conideration in [22,23]). The main reult of thi paper can be tated in the following way: Theorem 1.1. Aume that the path x : [0, T ] m allow the contruction of a geometric 2-rough path x = (x 1, x 2 ) C γ 2 (m ) C 2γ 2 (m,m ) for ome coefficient γ > 1/3. If φ C 3 (; ) and σ i j C 3,b ( d ; ) for all i = 1,..., d, j = 1,..., m, then the ytem (2), interpreted thank to Propoition 5.5 and 6.2, admit a unique global olution y in the pace of controlled path introduced in [19] (ee Definition 2.5). Moreover, the Itô map aociated with the ytem i locally Lipchitz continuou: if y (rep. ŷ) tand for the olution of the ytem driven by x (rep. ˆx) with initial condition a (rep. â), then where N [y ŷ; C γ 1 (d )] a c x, x â γ + N [x ˆx; C 1 (m )] + N [x 2 ˆx 2 ; C 2γ 2 (m,m )], (3) c x, x = C N [x; C γ 1 (m )], N [ ˆx; C γ 1 (m )], N [x 2 ; C 2γ 2 (m,m )], N [ˆx 2 ; C 2γ 2 (m,m )], for ome function C : ( + ) + growing with it four argument. Beyond the interpretation and reolution of the fractional Volterra ytem, the continuity reult (3) i likely to offer implified proof of the claical reult (large deviation, upport theorem) obtained in the (tandard) Brownian cae. For the ake of conciene, we hall leave thi procedure in abeyance, though (thi hould follow the line of Chapter 19 in [18]). A firt attempt to olve the determinitic ytem (2) ha been initiated in [16] by reorting to the tandard rough path formalim. A evoked earlier, the method turn out to be ucceful in the Young cae (γ > 1/2) with the exitence of a unique global olution. Unfortunately, it give an incomplete anwer for the problem in the rough cae (γ 1/2), allowing a local reolution only. The difficultie raied by the extenion of the path have been extenively commented on in [16]. They are eentially due to the dependence of the ytem with repect to the pat of the trajectory. To figure out thi phenomenon, remember that the uual reolution framework in rough path theory i a (well-choen) pace of Hölder path (or path with bounded p-variation). Here, the variation of the (potential) olution y between two time < t are given by yt i yi = φ(t u) σ i j (y u ) dxu j + 0 [φ(t u) φ( u)] σ i j (y u ) dx j u, (4) and through the latter integral, the problem in quetion pop out: the variation of y between a time (preent) and a time t (future) are linked to the pat ([0, ]) of the path. In the Young cae, the right-hand ide of (4) can be etimated by an affine function of y, which allow one to overcome the dependence on the pat and ettle a global fixed-point argument. The reaoning no longer hold true when γ 1/2, the etimate thi time giving rie to an (at leat) quadratic term in y. Let u ay a few word about the trategy that we have adopted in thi paper in order to exhibit a global olution when γ (1/3, 1/2]: (i) Firt, we will reformulate (2) (when x i differentiable) by writing φ a the Fourier tranform of a function φ L 1 (), that i to ay uing the repreentation φ(v) = dξ S v (ξ) φ(ξ), S v (ξ) e 2iπξv, v [0, T ]. (5)

4 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Thank to the Fubini theorem, the ytem (2) can now be equivalently preented a: for all i = 1,..., d, yt i = a i + dξ φ(ξ)ỹ t i (ξ), ỹi t (ξ) = S t u (ξ) dxu j σ i j (y u ), t [0, T ]. (6) 0 Owing to the additivity property S t+t (ξ) = S t (ξ)s t (ξ), it i eaily een that for any fixed ξ, ỹ i t (ξ) ỹi (ξ) = S t u (ξ) dxu i σ i j (y u ) + A t (ξ)ỹ i (ξ), (7) with A t (ξ) S t (ξ) 1, and the dependence w.r.t. the pat ([0, ]) i here reduced to a dependence w.r.t. the preent () only, which make it eaier to control on ucceive patching interval I 1, I 2,.... Therefore, the ytem will firt be olved under the form (7), before we go back to the original etting (2). (ii) The tranition from y to ỹ i however not cot free: we leave the Euclidean context of (2) to enter the framework of functional-valued path. For intance, the definition of a Hölder path will then be relative to a norm of function to be made precie (ee (26)). Beide, oberve that the expreion ỹt i (ξ) = S t u (ξ) dxu j σ i j (y u ), ỹ0 i = 0, (8) 0 i quite cloe to the mild formulation of an evolution equation: in order to analyze thi ytem, we have drown our inpiration from the method and formalim developed in [20] for a cla of rough partial differential equation. In particular, the interpretation of the rough integral will involve an adaptation of the notion of 2-rough path to the background under conideration here: the tandard path (x 1, x 2 ) will be replaced (in a firt phae at leat) by a convolutional path ( X x, X ax, X xx ), given, when x i differentiable, by the three formula (i, j = 1,..., m) X x,i t (ξ) X xx,i j t (ξ) S t u (ξ) dx i u, X t Ax,i (ξ) A tu (ξ) dxu i, (9) S t u (ξ) dxu i (x u j x i ). (10) If x i a Hölder path, thoe three definition are a priori only formal, but once we have admitted the exitence of thoe integral (ee for intance Hypothei 5 for a more precie tatement), we can reort to an extenion procedure for the integral t S t u(ξ) dxu j σ i j (y u ) imilar to the one ued in the analyi of ordinary ytem, and baed on the intervention of an invere operator Λ (Propoition 3.8). The extenion of the three expreion in (9) and (10) will be analyzed at the end of the paper (Section 6): for the ake of conciene, the quetion will actually be reduced to a looe integration by part argument. (iii) In the cae 1/3 < γ 1/2, the reaoning that lead u to the exitence of a global olution conit in a technical patching argument (Section 5) baed on the following obervation: in pite of the implification uggeted by (7), the ytem keep ome dependence w.r.t. the pat through the preent. Conequently, if one want to patch together local olution ỹ (k) on ucceive time interval I k = [l k, l k+1 ], one mut control the Hölder norm of ỹ (k), and alo the initial condition ỹ (k) l k. The general principle of the reaoning i contained in the proof of Theorem 5.10, but it actually lean on the control obtained in Propoition

5 1868 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) It i worth noticing that the general cheme of the proof in quetion, a well a the cheme of the proof of Theorem 4.3, are referred to in [14,13] in relation to the tudy of rough PDE model. Here i how our article i organized. We recall ome baic definition of algebraic integration in Section 2, and we adapt thoe notion to the convolutional context in Section 3.2. Section 4 i devoted to the impler cae of Young equation, which allow u to explain our method with le technical apparatu. Then at Section 5 we move to the rough cae of our Volterra equation, and explain all the detail of the method that we have choen in order to olve it. Finally, we apply our theory to (tandard) rough path in Section Algebraic integration Thi ection i devoted to recalling the very baic element of the algebraic integration theory introduced in [19], in order to fix notation for the remainder of the paper Increment A explained in [19], the extenion of the integral temming from the tandard differential ytem dyt i = dx j t σ i j (y t ) i baed on the notion of increment, together with an elementary operator δ acting on them. The notion of increment can be introduced in the following way: for two arbitrary real number l 2 > l 1 0, a vector pace V, and an integer k 1, we denote by C k (V ) the et of continuou function g : [l 1, l 2 ] k V uch that g t1 t k = 0 whenever t i = t i+1 for ome i k 1. Such a function will be called a (k 1)-increment, and we will et C (V ) = k 1 C k (V ). The operator δ alluded to above can be een a an operator acting on k-increment, and i defined a follow on C k (V ): k+1 δ : C k (V ) C k+1 (V ) (δg) t1 t k+1 = ( 1) i+1 g t1 ˆt i t k+1, (11) i=1 where ˆt i mean that thi particular argument i omitted. Then a fundamental property of δ, which i eaily verified, i that δδ = 0, where δδ i conidered a an operator from C k (V ) to C k+2 (V ). We will ue the notation ZC k (V ) = C k (V ) Ker δ and BC k (V ) = C k (V ) Im δ. Some imple example of action of δ, which will be the one that we will really ue throughout the paper, are obtained by letting g C 1 and h C 2. Then, for any t, u, [l 1, l 2 ], we have (δg) t = g t g, and (δh) tu = h t h tu h u. (12) The above-mentioned ordinary ytem i then of coure equivalent to y 0 = a, (δy i ) t = dx j u σ i j (y u ). (13) Furthermore, it i readily checked that the complex (C, δ) i acyclic, i.e. ZC k+1 (V ) = BC k (V ) for any k 1. In particular, the following baic property, which we label for further ue, hold true: Lemma 2.1. Suppoe that k 1 and h ZC k+1 (V ). Then there exit a (non-unique) f C k (V ) uch that h = δ f.

6 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Oberve that Lemma 2.1 implie that all the element h C 2 (V ) uch that δh = 0 can be written a h = δ f for ome (non-unique) f C 1 (V ). Thu we get a heuritic interpretation of δ C2 (V ): it meaure how much a given 1-increment i far from being an exact increment of a function (i.e. a finite difference). Let u now introduce a convenient notation for the product of increment: Definition 2.2. Let V and W be two normed pace and I a ubinterval of [0, T ]. If g C k (I ; L(V, W )) and h C l (I ; W ), for ome k, l N, we define the product gh a the (k+l 2)- increment (with value in W ) given by the following formula: for all t 1 t 2... t k+l 1, (gh) t1...t k+l 1 g t1...t k h tk t k+1...t k+l 1. (14) Notice again that our future dicuion will mainly rely on k-increment with k 2, for which we will ue ome analytical aumption. Namely, we meaure the ize of thee increment uing Hölder norm defined in the following way: for f C 2 (V ) et f µ f t V up,t [l 1,l 2 ] t µ, and Cµ 1 (V ) f C 2 (V ); f µ <. In the ame way, for h C 3 (V ), et h tu V h γ,ρ up,u,t [l 1,l 2 ] u γ t u ρ (15) h µ inf h i ρi,µ ρ i ; h = h i, 0 < ρ i < µ, i i where the lat infimum i taken over all equence {h i C 3 (V )} uch that h = i h i and for all choice of the number ρ i (0, z). Then µ i eaily een to be a norm on C 3 (V ), and we define C µ 3 (V ) h C 3 (V ); h µ <. Finally, let u et C 1+ 3 (V ) µ>1 C µ 3 (V ), and remark that the ame kind of norm can be conidered on the pace ZC 3 (V ), leading to the definition of ome pace ZC µ 3 (V ) and ZC 1+ 3 (V ). In order to avoid ambiguitie, we hall denote by N [ f ; Cκ j ] the κ-hölder norm on the pace C j, for j = 1, 2, 3. For ζ C j (V ), we alo et N [ζ ; C 0 j (V )] = up [l 1 ;l 2 ] j ζ V. With thi notation in mind, the following propoition i a baic reult which i at the core of our approach to pathwie integration (ee [19] for the original proof of the reult, baed on the Stoke theorem, and [20] for a implified verion): Theorem 2.3 (The Sewing Map). Suppoe that µ > 1. For any h ZC µ 3 ([0, 1]; V ), there exit a unique Λh C µ 2 ([0, 1]; V ) uch that δ(λh) = h. Furthermore, Λh µ c µ N [h; C µ 3 (V )], (16) with c µ = µ k=1 k µ. Thi give rie to a linear continuou map Λ : ZC µ 3 ([0, 1]; V ) C µ 2 ([0, 1]; V ) uch that δλ = Id ZC µ 3 ([0,1];V ). The following corollary give a firt relation between the tructure that we have jut introduced and generalized integral, in the ene that it connect the operator δ and Λ with iemann um.

7 1870 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Corollary 2.4 (Integration of Small Increment). For any 1-increment g C 2 (V ), uch that δg C 1+ 3, et δ f = (Id Λδ)g. Then n (δ f ) t = lim g ti+1 t i, Π t 0 i=0 where the limit i over any partition Π t = {t 0 = t,..., t n = } of [t, ] whoe meh tend to zero. The 1-increment δ f i the indefinite integral of the 1-increment g. Proof. For any partition Π t = { = t 0 < t 1 < < t n = t} of [, t], write (δ f ) t = n (δ f ) ti+1 t i = i=0 n g ti+1 t i i=0 n Λ ti+1 t i (δg). i=0 Oberve now that for ome µ > 1 uch that δg C µ 3, n Λ ti+1 t i (δg) V i=0 n Λ ti+1 t i (δg) V N [Λ(δg); C µ 2 (V )] Π t µ 1 t, i=0 and a a conequence, lim Πt 0 ni=0 Λ ti+1 t i (δg) = Diection of a tandard rough integral Let u ay a few word about the way in which the tool introduced in the previou ubection interact with each other to lead to an interpretation of the rough integral t dxi u zi u. In a firt phae, thoe tool enable a real diection of the ordinary verion of the integral (when x and poibly z are differentiable). For intance, by combining the elementary decompoition t dxi u zi u = (δxi ) t z i + t dxi u (δzi ) u with the relation δ dx i (δz i ) = (δx i )(δz i ), one deduce from Theorem 2.3 the expreion dx i u zi u = (δxi ) t z i + Λ t (δx i )(δz i ). It i now readily checked that if x, z C γ 1, with γ > 1/2 (the Young cae), the right-hand ide of the latter equality till make ene: the development i then legitimately choen a a definition for the rough integral. When γ 1/2, a deeper analyi of the ordinary integral i required. In order to bring the procedure to a ucceful reult, the cla of potential integrand z ha to be retricted to a particular et of pre-integrated path, that will be met again in Section 5: Definition 2.5. Suppoe that I i a ubinterval of [0, T ] and x C γ 1 (I ; m ) with γ > 1/3. For any l N, a path y C 1 (I ; l ) i aid to be γ -controlled on I, with value in l, if it increment δy can be decompoed in the following way: for all < t I, (δy i ) t = (δx j x, ji ) t y + yt,i, avec y x C γ 1 (I ; l,m ) et y C 2γ 2 (I ; l ). (17) The et of γ -controlled path will be denoted by Q γ x (I ; l ) and provided with the eminorm N [y; Q γ x (I ; l )] N [y; C γ 1 (I ; l )] + N [y x ; C 1 (I ; l,m )] + N [y ; C 2γ 2 (I ; l )], (18)

8 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Then we define Q γ x (I ; k,l ) (k N ) a the et of path y C 1 (I ; k,l ) uch that y i = y i. Q γ x (I ; l ) for all i = 1,..., k, and we aociate with the element of thi et the quantity N [y; Q γ x (I ; k,l )] k i=1 N [y i ; Q γ x (I ; l )]. If x i differentiable and z Q γ x, a quick algebraic computation how that, by etting x 2,i j t t dxi u (δx j ) u, we get t dxi u zi u = (δxi ) t z i +x2,i j x, ji t z +r t, with δr = (δx i ) z,i +x 2,i j δz x, ji, and o dx i u zi u = (δxi ) t z i + x2,i j t x, ji z + Λ t (δx i ) z,i + x 2,i j δz x, ji. (19) The right-hand ide of the latter equality can now be extended to any 2-rough path x = (δx, x 2 ) C γ 2 C2γ 2 with γ > 1/3, that i to ay to any γ -Hölder path x allowing the contruction of a Lévy area x 2,i j t t dxi u (δx j ) u (ee [25] for a thorough definition), a hypothei which i for intance known to be atified by a fractional Brownian motion with Hurt index H > 1/3 (ee [12] or [36]). In fact, if one i permitted to retrict the cla of integrand to Q γ x, it i becaue the latter pace i large and table enough to make poible the interpretation and reolution of the ordinary rough ytem (δy i ) t = t dx j u σ i j (y u ) therein, for a ufficiently mooth vector field σ. It i indeed not difficult to ee that if y Q γ x and σ C 2,b, then z σ (y) Q γ x, while (19) immediately how that dx z Q γ x. All of thoe conideration will be kept in mind when analyzing the ytem (2). 3. Algebraic convolutional integration We already announced thi in the introduction: in order to reduce the dependence of equation (2) with repect to the pat, we will appeal to a preliminary rewriting of the ytem, baed on the repreentation of φ a the Fourier tranform of a function φ. The reulting formulation will be cloe to the model tudied in [20]: jut a in the latter reference, it ugget a natural adaptation of the tandard algebraic formalim preented in the previou ection Tranformation of the ordinary ytem Aume in thi ubection that x i differentiable. Let u go back for a hort while to the tranformation ketched out in the introduction, and which tarted from the aumption that φ could be written a in (5). Note here and now that thi hypothei i actually not very retrictive. Indeed, inofar a we are working with a finite fixed horizon T, only the behavior of φ on [0, T ] matter, and it i poible to replace, in (2), φ with a compactly upported function φ T uch that φ [0,T ] = φ T [0,T ]. If φ i aumed to be continuou on, then φ T can be picked in L 2 (), and in thi cae φ T = F φ, with φ = φ T = F 1 φ T L 2 (), where F tand for the Fourier tranform. In fact, under the hypothee of Theorem 1.1 (φ C 3 ()), it i eay to how that φ i integrable (ee Propoition 6.6). Neverthele, for the time being, we record thi condition in the following hypothei: Hypothei 1. We aume, in thi ection and the two following, that the function φ admit the repreentation (5), for ome function φ L 1 ().

9 1872 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) We are then allowed to apply the Fubini theorem and aert that the ytem (2) i equivalent to yt i = a i + ỹt i (ξ) φ(ξ) dξ t (20) ỹt i (ξ) = S t v (ξ)dxv j σ i j (y v ). 0 Beide, a we alo evoked in the introduction, the increment (δỹ i ) t (ξ) ỹt i(ξ) ỹi (ξ) are governed by the equation (δỹ i (ξ)) t = = where we have et A t (ξ) S t (ξ) 1. S t v (ξ)dx j v σ i j (y v ) + A t (ξ) S t v (ξ)dx j v σ i j (y v ) + A t (ξ)ỹ j (ξ), 0 S t v (ξ)dx j v σ i j (y v ) Notice now that the firt term t S t v(ξ)dxv j σ i j (y v ) above i really imilar to what one obtain in the diffuion cae, namely an integral of the form t (ee (13))). However, the econd term A t (ξ)ỹ (ξ) i a little clumy for further expanion. Hence, a traightforward idea i to make it diappear by jut etting ( δỹ i ) t (ξ) (δỹ i ) t (ξ) A t (ξ)ỹ i (ξ). (22) Then the lat equation can be read a ( δỹ i ) t (ξ) = t S t v(ξ)dxv j σ i j (y v ), and the ytem (20) become yt i = a i + ỹt i (ξ) φ(ξ) dξ (23) ( δỹ i ) t (ξ) = S t v (ξ)dxv j σ i j (y v ), with the initial condition ỹ 0 0. In the equel, we hall eentially focu on the path ỹ, by merging the two equation of the lat ytem into a ingle one: ỹ 0 = 0, ( δỹ i ) t (ξ) = S t v (ξ) dx j v (21) σ i j T a, φ (ỹ v ), (24) where the operator T a,φ i defined by T a, φ (ϕ) a + dη φ(η)ϕ(η). (25) The original olution path y can then be recovered in an obviou way, o it will be ufficient to olve the Volterra equation under the more uitable form (24), with a right-hand ide written a an integral from to t with repect to x (compare with (13)). Actually, if we take the liberty of focuing on δ rather than on the tandard increment δ, it i becaue the former operator alo make poible the building of an integration theory, by mean

10 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) of an inverion mapping imilar to Λ, and that will be denoted by Λ (ee Propoition 3.8). Thi i what we mean to elaborate on in the following ubection Convolutional increment Notice that, due to the fact that S t1 t 2 (ξ) i tudied only for t 1 > t 2, our integration domain will be of the form S n = S n ([l 1, l 2 ]), where S n tand for the n-implex S n = {(t 1,..., t n ) : l 2 t 1 t 2 t n l 1 }. For any Banach pace E, the notation C n ([l 1, l 2 ]; E) will henceforth refer to the et of path h which are continuou on S n, with value in E, and uch that h t1...t k = 0 if there exit i j for which t i = t j. According to the (firt) definition (22), δ i uppoed to act on functional-valued path. Let u anticipate here the next ection by introducing the pace of function that will pontaneouly arie during the tudy of (24) (ee for intance Propoition 4.2). Thoe are the L 1 -type pace induced by the norm N [ g; L β (V )] = N [ g; L β, φ (V )] dξ φ(ξ) (1 + ξ β ) g(ξ) V, (26) where β > 0 i a fixed parameter and V a Euclidean pace. Then we define C k,β (I ; V ) C k (I ; L β (V )). (27) The tandard incremental operator δ act on thoe pace through the obviou formula: If h C k,β (I ; V ), (δ h) t1...t k+1 (ξ) δ( h(ξ)) t1...t k+1, ξ. (28) A for δ, it can be naturally extended to any C k,β (I ; V ) (k N ): Definition 3.1. Let I be an interval of + and V a Euclidean pace. For any β > 0, we define the equence of operator δ k : C k,β (I ; V ) C k+1,β (I ; V ) by the formula: if h C k,β (I ; V ), then for all ξ, ( δ k h) t1...t k+1 (ξ) (δ k h) t1...t k+1 (ξ) A t1 t 2 (ξ) h t2...t k+1 (ξ), (t 1,... t k+1 ) S k+1 (I ). (29) In particular, if < u < t I, ( δ 1 h) t (ξ) = h t (ξ) S t (ξ) h (ξ), ( δ 2 h) tu (ξ) = h t (ξ) h tu (ξ) S t u (ξ) h u (ξ). For the ake of clarity, we hall ue the ame notation δ for the operator δ k, k N. emark 3.2. In the ret of the paper, we will explicitly write down the pace variable ξ only when there might be confuion. Thu, we will for intance imply write δ h = δ h a h. The convention given by (14) for product of increment can be tranlated in thi context a follow: Lemma 3.3. If relation M C n,β (I ; k,l ) and L C m (I ; l ), then the product ( M L) t1...t m+n 1 (ξ) M t1...t n (ξ) L tn...t m+n 1, M L, defined by the

11 1874 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) belong to C m+n 1,β (I ; k ). Moreover, when n = 2, the following algebraic relation are atified: δ( M L) = δ M L M δl, et δ( M L) = δ M L M δl. (30) Proof. The firt part of the aertion i obviou. A for the algebraic relation when n = 2, the firt one i immediate, while for the econd one, it uffice to notice that δ( M L) t1...t m+2 = δ( M L) t1...t m+2 A t1 t 2 M t2 t 3 L t3...t m+2 = (δ M) t1 t 2 t 3 L t3...t m+2 M t1 t 2 (δl) t2...t m+2 A t1 t 2 M t2 t 3 L t3...t m+2 = [(δ M) t1 t 2 t 3 A t1 t 2 M t2 t 3 ]L t3...t m+2 ( M δl) t1...t m+2. With thi notation and thee preliminary reult in hand, we are in a poition to prove that the tarting property of tandard algebraic integration (ummed up in Section 2), namely the cohomological relation δδ = 0, remain true for δ: Propoition 3.4. δ δ = 0. More preciely, for any β > 0 and any k N, Im δ C k,β (I ;V ) = Ker δ C k+1,β (I ;V ). Proof. If F C k,β (I ; V ), then uing the relation δδ = 0 and the reult of Lemma 3.3, we deduce that δ δ F = (δ A) [(δ A) F] = δδ F δ(a F) A δ F + A A F = δ A F + A δ F A δ F + A A F = A A F δ A F. It i then readily checked, owing to the additivity S t S t = S t+t, that (δ A) tu = A tu A u, (t, u, ) S 3 (I ), which give δ δ F = 0. Now, if C C k+1,β (I ; V ) i uch that δ C = 0, we et B t1...t n C t1...t n, for ome arbitrary time I. Then [ δ B] t1...t n+1 = [δ C] t1...t n+1 + ( 1) n+1 C t1...t n+1 A t1 t 2 C t2...t n = [ δ C] t1...t n+1 + ( 1) n+1 C t1...t n+1 = ( 1) n+1 C t1...t n+1. Therefore, by etting D ( 1) n+1 B, we get δ D = C. emark 3.5. A traightforward iteration of the relation δ δ = 0 lead to the following formula: for any partition { = t 0 < t 1 <... < t n = t} of [, t], for any f C 1,β ([, t]; V ), n 1 ( δ f ) t = S t ti+1 ( δ f ) ti+1 t i. (31) i=0 Thi kind of decompoition will be appealed to everal time in the equel, epecially in the proof of Lemma 3.7 and Corollary 3.9. In ome way, thi i the convolutional analog of the uual telecopic um (δ f ) t = n 1 i=0 (δ f ) t i+1 t i. The cochain complex ( C k,β (I ; V ), δ) will tand for the tructure at the core of all the contruction in thi paper. Let u try to give an idea of the relevance of thi tructure in the

12 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) context of equation (24). To thi end, we et, for two mooth path f : [0, T ] W, g : [0, T ] L(W, V ), J t ( dg f )(ξ) and for any mooth path h : [0, T ] 2 W, J t ( dg h)(ξ) S t u (ξ) dg u f u, ξ A, (32) S t u (ξ) dg u h u. (33) The uual Chale relation δ dg f = 0 become here: Propoition 3.6. With the notation of (32) and (33), one ha, if f : [0, T ] W and g : [0, T ] L(W, V ) tand for two differentiable path, δ J ( dg f ) = 0, δ J ( dg δ f ) = J ( dg) δ f. (34) Proof. Thi i a matter of traightforward computation: if < u < t, δ J ( dg f ) = J t( dg f ) J tu ( dg f ) S t u J u ( dg f ), tu and S t u J u ( dg f ) = u S t v dg v f v, which eaily yield δ J ( dg f ) = 0. In the ame way, δ J ( dg δ f ) = S t v dg v (δ f ) v S t v dg v (δ f ) vu = S t v dg v (δ f ) u. u u u 3.3. Convolutional Hölder pace and the Λ map In order to cope with (24), the definition of a (generalized) Hölder path preented in the previou ection ha to be adapted to the convolutional formalim that we have jut introduced. We firt define, for all (fixed) parameter µ, β, γ > 0, any interval I of + and any Euclidean pace V, C µ 2,β ỹ (I ; V ) C 2,β (I ; V ) : N [ỹ; C µ N [ỹ t ; L β (V )] 2,β (I ; V )] up <t I t µ < C µ 1,β (I ; V ) {ỹ C 1,β (I ; V ) : δỹ C µ 2,β (I ; V )}. (35) A for path with three variable, we define, a in the tandard cae, the intermediate pace C (γ,β) (I ; V ) induced by the norm 3,β N [ h; C (γ,ρ) 3,β (I ; V )] up <u<t I N [ h tu ; L β (V )] t u γ u ρ, and then et C µ 3,β (I ; V ) 0 α µ 3,β (I ; V ). We alo provide the latter pace with the norm N [ h; C µ 3,β (I ; V )] inf i C α,µ α N [h i ; C (ρ i,µ ρ i ) 3,β (I ; V )]; h = i h i, 0 < ρ i < µ,.

13 1876 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) It i worth noticing that the elementary reult aerting that Im δ 1 C µ 2 (V ) = {0} if µ > 1 admit a direct analog: Lemma 3.7. Fix β > 0. If µ > 1, then Im δ C 1,β (I ;V ) C µ 2,β (V ) = {0}. Proof. Suppoe that M = δ f Im δ C 1,β (I ;V ) C µ 2,β (V ). According to (31), we can write, for all < t, M t = n 1 i=0 S t t i+1 M ti+1 t i, for any partition Π t = { = t 0 < t 1 < < t n = t} of [, t]. Since S t (ξ) = 1, thi entail n 1 N [ M t ; L β (V )] N [ M ti+1 t i ; L β (V )] N [ M; C µ 2,β (V )] t Π t µ 1, i=0 and the latter etimate tend to 0 a meh Π t tend to 0. With all of thoe reult in hand, it i now eay to follow the ame line a in the proof of Theorem 2.3 in order to etablih the exitence of an invere operator for δ (ee [20] for a imilar adaptation): Propoition 3.8. Suppoe that µ > 1, β > 0, I i an interval of + and V i a Euclidean pace. For all h Ker δ C 3,β (I ;V ) C µ 3,β (I ; V ), there exit a unique path Λ h C µ 2,β (I ; V ) uch that δ( Λ h) = h. Moreover, the following contraction property hold true: N [ Λ h; C µ 2,β (I ; V )] c µ N [ h; C µ 3,β (I ; V )], (36) with c µ a contant that only depend on µ. Thi tatement give birth to a continuou linear mapping uch that Λ : Ker δ C 3,β (I ;V ) C µ 3,β (I ; V ) C µ 2,β (I ; V ) δ Λ = Id Ker δ C 3,β (I ;V ) C µ 3,β (I ;V ) and Λ δ = IdC µ 2,β (I ;V ). (37) We alo have the following equivalent of Corollary 2.4 at our dipoal: Corollary 3.9. Let g C 2,β (I ; V ) be uch that δ g δ f (Id Λ δ) g, then ( δ f ) t = lim Π t 0 i=0 n S t ti+1 g ti+1 t i in L β, C µ 3,β (I ; V ), for ome coefficient µ > 1. If where the limit i over any partition Π t = {t 0 = t,..., t n = } of [t, ] whoe meh tend to zero. Proof. Here again, it uffice to ue the ame argument a in the tandard cae (Corollary 2.4), tarting from the decompoition (31).

14 4. The Young cae A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) emember that we firt wih to olve the ytem in the form (24), which can alo be written, with the notation (32), a ỹ 0 0, δỹ i = J dx j σ (y) i j, y u = T a, φ (ỹ u) = a + dξ φ(ξ)ỹ u (ξ). (38) For the time being, the right-hand ide of the latter equality only make ene for a differentiable path x. The aim of thi ection i to extend the definition of the equation to a γ -Hölder path x with γ > 1/2, and then olve it with the reulting interpretation. To thi end, we will follow the ame general trategy a in the tandard cae (Section 2.2), which begin with a diection of the ordinary integral Heuritic conideration and interpretation of the ytem Let u aume for the moment that x and ỹ are differentiable (in time) and let u ucceively et y T a, φ (ỹ), zi j σ i j (y), o the integral under conideration here i given by J ( dx j z i j ). Before we turn to the diection procedure for thi integral, it i important to ponder about the regularity that one can expect for z, or equivalently for y (we will uppoe that σ i mooth enough), when x and ỹ become non-differentiable. To anwer the quetion, oberve the decompoition (δy i ) t = dξ φ(ξ) (δỹ) t (ξ) = dξ φ(ξ) ( δỹ i ) t (ξ) + dξ φ(ξ) A t (ξ) ỹ i (ξ). (39) A ỹ tand for the (potential) olution of (38)) and S t (ξ) = 1, δỹ i expected to inherit the regularity of x, or otherwie tated, ( δỹ) t (ξ) c x t γ (uniformly in ξ), which would lead, a we have aumed that dξ φ(ξ) < (Hypothei 1), to an etimate uch that dξ φ(ξ) ( δỹ i ) t (ξ) c x t γ. To retrieve t -increment from the term dξ φ(ξ) A t (ξ) ỹ i (ξ), we hall lean on the elementary etimate A t (ξ) = S t (ξ) 1 c γ t γ ξ γ. (40) Thi i where the pace L β (V ) defined by (26) occur. Indeed, from (40), one ha dξ φ(ξ) A t (ξ)ỹ (ξ) c γ t γ N [ỹ ; L γ ( d )]. (41) Going back to decompoition (39), we ee that, by tarting with a path ỹ that take value in L γ ( d ), we hould retrieve a path y, and then a path z, both Hölder continuou in the claical ene. Thoe conideration (that will be made precie through Propoition 4.2) will help u in the diection procedure of the integral J ( dx j z i j ). Indeed, we will no longer heitate to let the tandard increment δz come (back) into the picture, and we will thu tart, jut a in the diffuion cae, with the decompoition (x i till aumed to be differentiable) J t ( dx j z i j ) = J ( dx j ) z i j + J t ( dx j δz i j ), (42)

15 1878 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) where J t ( dx j ) J t ( dx j 1) = t S t u dxu j. When x become rough (that i to ay γ -Hölder with 0 < γ < 1), the integral t S t u dxu j can till be undertood a a Young integral ([41]). In the pirit of the rough path methodology and by anticipating the computation of Propoition 4.1 and Theorem 4.3 below, we will make the following more precie hypothei: Hypothei 2. Suppoe that x C γ 1 ([0, T ]; m ), with γ > 1/2. We aume the exitence of a equence x ε of differentiable path that atifie N [x ε x; C γ 1 ([0, T ]; m )] ε 0 0, and uch that the aociated equence of path X xε,i t (ξ) S t u (ξ) dx ε,i u converge to X t x,i (ξ) t S t u(ξ) dxu i (undertood a a Young integral) w.r.t. the topology of the pace C γ 2,γ ([0, T ]; m ). In particular, X x C γ 2,γ ([0, T ]; m ) and δ X x = 0. If x i differentiable, we aume that thi reult hold true for x ε x. Propoition 4.1. Let x : [0, T ] m be a path that atifie Hypothei 2, and I be a ubinterval of [0, T ]. For any z C γ 1 (I ; d,m ) and ξ, et Then: J ( dx j z i j )(ξ) X x, j (ξ) z i j + Λ( X x, j δz i j )(ξ) = (Id Λ δ)( X x, j z i j )(ξ). (43) (1) J ( dx j z i j ) i well-defined a an element of C γ 2,γ (I ; d ), and it coincide with the uual iemann integral t S t v(ξ) dx v z v when x i differentiable. (2) The following etimate hold true (remember that we have et N [z; C1 0(I ; d,m )] up I z ): N [J ( dx z); C γ 2,γ (I ; d )] c x N [z; C 0 1 (I ; d,m )] + I γ N [z; C γ 1 (I ; d,m )] (3) For all < t I, J t ( dx j z i j ) = lim n 1 Π t 0 k=0. (44) S t tk+1 X x, j t k+1,t k z i j t k in L γ, (45) where the limit i taken over any partition Π t = {t 0 = t,..., t n = } of [, t] whoe meh tend to 0. Proof. To how that the increment defined by (43) coincide with the iemann integral t S t u(ξ) dxu j zu i j in the cae where x i differentiable, let u go back to the decompoition (42), that can alo be written a J t ( dx j δz i j ) = J t ( dx j z i j ) X x, j t By applying δ to the two ide of the relation, and then uing (34) and (30), we get δ J ( dx j z i j ) = δ X x, j z i j + X x, j δz i j = X x, j δz i j, z i j.

16 and o, via (37), A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) J ( dx j δz i j ) = Λ X x, j δz i j, which enable u to recover (43). The fact that formula (43) i well-defined in C γ 2,γ i a traightforward conequence of Hypothei 2. Indeed, owing to the latter hypothei, we know that X x δz C 2γ 3,γ (I ; d ) Ker δ C2,γ (I ; d ), and we are thu in a poition to apply Λ. The etimate (44) i then due to the contraction property (36). A for the expreion (45), it tem from Corollary 3.9. In order to give ene to the ytem (38) through the definition (43), we will rely on the following propoition, which actually ummarize the above conideration: Propoition 4.2. Suppoe that I = [l 1, l 2 ] i a ubinterval of [0, T ] and σ C 2,b ( d ; d,m ). For any ỹ C γ 1,γ (I ; d ), we et y T a, φ (ỹ) and define N [ỹ; C 1,γ (I ; d )] N [ỹ; C 1,γ 0 (I ; d )] + N [ỹ; C γ 1,γ (I ; d )], with N [ỹ; C 0 1,γ (I ; d )] up I N [ỹ ; L γ ( d )]. Then σ (y) C γ 1 (I ; d,m ) and N [σ (y); C γ 1 (I ; d,m )] c σ N [ỹ; C 1,γ (I ; d )]. (46) Moreover, if ỹ (1), ỹ (2) C γ 1,γ (I ; d ) are uch that ỹ (1) l 1 = ỹ (2), then N [σ (y (1) ) σ (y (2) ); C1 0 (I ; d,m )] c σ I γ N [ỹ (1) ỹ (2) ; C 1,γ (I ; d )], (47) N [σ (y (1) ) σ (y (2) ); C γ 1 (I ; d,m )] c σ 1 + N [ỹ (2) ; C 1,γ (I ; d )] l 1 N [ỹ (1) ỹ (2) ; C 1,γ (I ; d )]. (48) Proof. By uing (40), we get δ(σ (y)) t Dσ dξ φ(ξ) (δỹ) t (ξ) Dσ dξ φ(ξ) ( δỹ) t (ξ) + dξ φ(ξ) A t (ξ) ỹ (ξ) c γ Dσ t γ N [ỹ; C γ 1,γ ] + N [ỹ; C 1,γ 0 ], which correpond to (46). The inequality (47) can be obtained in the ame way, after noticing that, for any I, σ (y (1) ) σ (y (2) ) Dσ dξ φ(ξ) δ(ỹ (1) ỹ (2) ) l1 (ξ). A for (48), thi i a conequence of the claical etimate δ(σ (y (1) ) σ (y (2) )) t Dσ δ(y (1) y (2) ) t + D 2 σ δ(y (2) ) t y t (1) y t (2) + y (1) y (2).

17 1880 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Solving the equation Propoition 4.1, together with Propoition 4.2, provide a reaonable interpretation of (38). We can now tate the main reult of thi ection: Theorem 4.3. Let x be a path that atifie Hypothei 2. If σ C 2,b ( d ; d,m ), then the Eq. (38), interpreted with Propoition 4.1 and 4.2, admit a unique olution in the pace C γ 1,γ ([0, T ]; d ) defined by (35). Proof. Conider a contant ε > 0, l N, and aume that we have already contructed a olution ỹ (l) C γ 1,γ ([0, lε]). If l = 0, then ỹ(0) = ỹ (0) 0 = 0. The proof will conit in howing that one can extend ỹ (l) into a olution ỹ (l+1) C γ 1,γ ([0, (l + 1)ε]), by mean of a fixed-point argument. Step 1: The exitence of invariant ball. Let ỹ C γ 1,γ ([0, (l + 1)ε]) be uch that ỹ [0,lε] = ỹ (l), and denote by z = Γ (ỹ) the element of C 1,γ ([0, (l + 1)ε]) characterized by z [0,lε] = ỹ (l) and for all, t [0, (l + 1)ε], ( δ z) t = J t dx σ (y), where, a in Propoition 4.2, y T a, φ (ỹ) (remember the notation of (25)). Firt, the etimate (44) provide N [ z; C γ 1,γ ([lε, (l + 1)ε])] c x N [σ (y); C1 0 ([0, (l + 1)ε])] + εγ N [σ (y); C γ 1 ([0, (l + 1)ε])], which, together with (46), give N [ z; C γ 1,γ ([lε, (l + 1)ε])] c1 x,σ If 0 lε t (l + 1)ε, we ue (31) to deduce 1 + ε γ N [ỹ; C 1,γ ([0, (l + 1)ε])]. N [( δ z) t ; L γ ] N [( δ z) t,lε ; L γ ] + N [( δ z) lε, ; L γ ] 2 max N [ z; C γ 1,γ ([lε, (l + 1)ε])], N [ỹ(l) ; C γ 1,γ ([0, lε])] t γ. (49) Beide, for any [0, (l + 1)ε], z = ( δ z) 0, and o N [ z; C 1,γ 0 ([0, (l + 1)ε])] N [ z; 1,γ ([0, (l + 1)ε])]T γ. (50) C We are thu led to et 1/γ ε 4c 1 x,σ (1 + T γ ) N l+1 max 2(1 + T γ )N [ỹ (l) ; C γ 1,γ ([0, lε])], 4c1 x,σ (1 + T γ ). Indeed, for uch value, it i readily checked from (49) and (50) that if N [ỹ; N l+1, then N [ z; C γ 1,γ ([0, (l +1)ε])] N l+1 1+T γ and N [ z; N [ z; C 1,γ ([0, (l + 1)ε])] N l+1. In other word, the ball B N l+1 C C 1,γ ([0, (l+1)ε])] C1,γ 0 ([0, (l +1)ε])] N l+1 1+T γ T γ, and hence ỹ (l),(l+1)ε = {ỹ 1,γ ([0, (l + 1)ε]) : ỹ [0,lε] = ỹ (l), N [ỹ; 1,γ ([0, (l + 1)ε])] N l+1} i invariant under Γ. C

18 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) The independence of ε with repect to ỹ (l) will allow u to repeat the procedure (with the ame ε) and thu to get a equence of radii (N k ) k 1 uch that the et B N k are invariant under ỹ (k),kε Γ. Of coure, the definition of the latter application ha to be adapted to each of thoe et. Step 2: The contraction property. We are now going to look for a plitting of [lε, (l + 1)ε] into ubinterval [lε, lε + η], [lε + η, lε + 2η],... of the ame length η (that could depend on ε and l), on which Γ i a contraction mapping. Let ỹ a, ỹ b C γ 1,γ ([0, lε +η]) be uch that ỹa [0,lε] = ỹb [0,lε] = ỹ(l), N [ỹ a ; C 1,γ ([0, lε +η])] N l+1, N [ỹ b ; C 1,γ ([0, lε + η])] N l+1, and et z a Γ (ỹ a ), z b Γ (ỹ b ), where Γ i defined jut a in Step 1, but retricted to ([0, lε + η]). By uing (44) again, we deduce C γ 1,γ N [ z a z b ; C γ 1,γ ([lε, lε + η])] c γ,x N [σ (y a ) σ (y b ); C1 0 ([lε, lε + η])] and then, according to (47) and (48), + η γ N [σ (y a ) σ (y b ); C γ 1 ([lε, lε + η])], N [ z a z b ; C γ 1,γ ([lε, lε + η])] c2 x,σ {1 + N l+1} η γ N [ỹ a ỹ b ; C 1,γ Since the path ỹ a ỹ b, z a z b vanih on [0, lε], the latter etimate implie ([lε, lε + η])]. N [ z a z b ; C γ 1,γ ([0, lε + η])] c2 x,σ {1 + N l+1} η γ N [ỹ a ỹ b ; C γ 1,γ ([0, lε + η])]. Beide, ( z a z b ) = δ( z a z b ),lε, o N [ z a z b ; C 1,γ 0 ([0, lε +η])] N [ za z b ; η])]η γ. Therefore, N [ z a z b ; C 1,γ ([0, lε + η])] c 2 x,σ {1 + N l+1} (1 + T γ )η γ N [ỹ a ỹ b ; C 1,γ C 1,γ ([0, lε + ([0, lε + η])]. (51) Fix η inf ε, (2c 2 x,σ {1 + N l+1} (1 + T γ )) 1/γ o a to make Γ a trict contraction of the et {ỹ C 1,γ ([0, lε + η]) : ỹ [0,lε] = ỹ (l), N [ỹ; C 1,γ ([0, lε + η])] N l+1}. Uing the invariance of B N l+1, it i eaily een that the latter et i invariant under Γ too (ee ỹ (l),(l+1)ε Lemma 4.4 below). Conequently, there exit a unique fixed point in thi et, that we denote by ỹ (l),η. Inofar a η doe not depend on ỹ (l), the reaoning remain true on the (invariant) et {ỹ C 1,γ ([0, lε + 2η]) : ỹ [0,lε+η] = ỹ (l),η, N [ỹ; C 1,γ ([0, lε + 2η])] N l+1}. Thu, ỹ (l),η can be extended into a olution ỹ (l),2η defined on [0, lε + 2η] and by iterating the procedure until the interval [lε, (l + 1)ε] i covered, we get the expected extenion ỹ (l+1). The uniquene of the olution can be eaily hown with the argument of Step 2 (replace z a, z b with ỹ a, ỹ b in (51)). The detail are left to the reader. Lemma 4.4. With the notation of the previou proof, the et {ỹ C 1,γ ([0, lε + η]) : ỹ [0,lε] = ỹ (l), N [ỹ; C 1,γ ([0, lε + η])] N l+1} i invariant under Γ.

19 1882 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Proof. Conider an element ỹ in the et in quetion and define z Γ (ỹ). Then define ỹt if t lε + η ŷ t = S t (lε+η) ỹ lε+η if t [lε + η, (l + 1)ε]. The path ŷ i clearly continuou and accordingly belong to C 1,γ ([0, (l + 1)ε]). Moreover, if, t [lε + η, (l + 1)ε], ( δỹ) t = 0, while if lε + η t, ( δŷ) t = S t (lε+η) ( δỹ) lε+η,, o N [ŷ; C γ 1,γ ([0, (l + 1)ε])] N [ỹ; C γ 1,γ ([0, lε + η])]. Since N [ŷ; C 1,γ 0 ([0, (l + 1)ε])] N [ỹ; C 1,γ 0 ([0, lε + η])], we deduce N [ŷ; C 1,γ ([0, (l + 1)ε])] N [ỹ; C 1,γ ([0, lε + η])] N l+1, which mean that ŷ B N l+1 ỹ l,(l+1)ε. According to Step 1 of the previou proof, B N l+1 i invariant ỹ l,(l+1)ε under Γ, and o, if ẑ Γ (ŷ), then N [ẑ; C 1,γ ([0, (l + 1)ε])] N l+1. It i now obviou that z = ẑ [0,lε+η], which finally lead to N [ z; C 1,γ ([0, lε + η])] N [ẑ; C 1,γ ([0, (l + 1)ε])] N l+1. To conclude with thi ection, let u go back to the original etting of the equation: Corollary 4.5. Under Hypothei 2, and auming that σ C 2,b ( d ; d,m ), the ytem (2), interpreted with Propoition 4.1, admit a unique olution y in C γ 1 ([0, T ]; d ). Proof. If ỹ tand for the olution of (38) given by Theorem 4.3, it uffice to et, for any t [0, T ], y t T a, φ (ỹ t). The detail are left to the reader. 5. The rough cae Our aim till conit in tudying the ytem (38), but we will uppoe in thi ection that the Hölder coefficient γ of x belong to (1/3, 1/2]. Definition (43) doe not make ene any longer, and ome development at order 2 are required. To thi end, we will reort to the ame trategy a in the diffuion cae (ee Section 2.2), divided into two tep: (1) Identifying the algebraic tructure of the potential olution ỹ, which will lead to the introduction of a pace Q of controlled path. (2) Extending the integral of the ytem above x C γ 1 when ỹ Q Convolutional controlled path Let u tart with ome heuritic conideration. A in the Young cae, the ytem will be analyzed in the form (remember the notation (32)) ỹ 0 0, δỹ i = J dx j σ (y) i j, y u = T a, φ (ỹ u) := a + dξ φ(ξ)ỹ u (ξ). (52) Aume for the moment that x i a differentiable path. Eq. (52) admit in thi cae a unique olution ỹ, whoe (convolutional) increment can be expanded into with ( δỹ i ) t (ξ) = X x, j t (ξ) = S t u (ξ) dxu j, S t u (ξ) dxu j σ i j (y u ) = X x, j t (ξ)σ i j (y ) + r t i (ξ), (53) r t i (ξ) = S t u (ξ) dxu j (δσ i j (y)) u. (54)

20 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) Thi elementary decompoition already let emerge the tructure likely to replace Q γ x (Definition 2.5) emerge in the convolutional etting. Let u go a little bit deeper into the analyi of (53): if x and y are γ -Hölder (γ (1/3, 1/2]), it i natural to expect that, on the one hand, X x belong to a pace uch that C γ 2,β ([0, T ]; m ), for ome coefficient β > 0, and on the other hand, r C 2γ 2,β ([0, T ]; d ). For ome technical reaon that will pop out in the proof of Theorem 5.10, we hall actually be prompted to take β = 1 in order to exhibit a global olution for (52). Notation: For ake of clarity, we henceforth ue the hort form C γ k (I ; V ) C γ k,1 (I ; V ), k {1, 2, 3}. (55) A in the previou ection, let u label the appropriate regularity aumption relative to the path X x : Hypothei 3. Suppoe that x C γ 1 ([0, T ]; m ), with γ (1/3, 1/2]. We aume that there exit a equence x ε of differentiable path that atifie N [x ε x; C γ 1 ([0, T ]; m )] ε 0 0, and uch that the equence of path defined by X xε,i t (ξ) S t u (ξ) dx ε,i u converge to X t x,i (ξ) t S t u(ξ) dxv i (undertood a a Young integral) w.r.t. the topology of C γ 2 ([0, T ]; m ). In particular, X x C γ 2 ([0, T ]; m ) and δ X x = 0. If x i a differentiable path, we aume that thi reult hold true for x ε x. With the decompoition (53) in mind, the mot natural and conitent framework in which to tudy the ytem (52) i the following: Definition 5.1. Aume that Hypothei 3 i atified. For any interval I of [0, T ], we decribe a a convolutional controlled path on I, with value in d, any element ỹ in C γ 1 (I ; d ) whoe convolutional increment can be written a ( δỹ i ) t = X x, j t ỹ x,i j + ỹ,i t, with ỹ x C γ 1 (I ; d,m ) and ỹ C 2γ 2 (I ; d ). (56) The et of convolutional controlled path on I will be denoted by Q γ x (I ; d ) and we provide the latter pace with the eminorm N [ỹ; Q γ x (I ; d )] N [ỹ; C γ 1 (I ; d )] + N [ỹ x ; C 0 1 (I ; d,m )] + N [ỹ x ; C γ 1 (I ; d,m )] + N [ỹ ; C 2 2κ (I ; d )]. (57) emark 5.2. It may be worth noticing that in pite of it notation, the path ỹ x defined through (56) take value in a Euclidean pace, and not in a functional pace. In order to give ene to the ytem (52) when ỹ Q γ x (I ; k ), it i now important to identify the algebraic tructure of the integrand σ (y u ), where y u T a, φ (ỹ u). To begin with, oberve that

21 1884 A. Deya, S. Tindel / Stochatic Procee and their Application 121 (2011) if δỹ admit the decompoition (56), then the increment of y can be written a (δy i ) t = dξ φ(ξ)(δỹ i ) t (ξ) = dξ φ(ξ)( δỹ i ) t (ξ) + dξ φ(ξ)a t (ξ)ỹ i (ξ) = dξ φ(ξ) X x, j t (ξ)ỹ x,i j + dξ φ(ξ)ỹ t,i (ξ) + dξ φ(ξ)a t (ξ)ỹ i (ξ) = X x, j t ỹ x,i j + dξ φ(ξ)ỹ t,i (ξ) + dξ φ(ξ)a t (ξ)ỹ i (ξ), (58) where X x, j t dξ φ(ξ) X x, j t (ξ) i well-defined a an element of C γ 2 ([0, T ]; m ), thank to Hypothei 3. Let u analyze (58) a far a Hölder continuity i concerned. For the lat term of the compoition, remember the obviou etimate A t (ξ) c ξ t, which entail here dξ φ(ξ)a t (ξ)ỹ i (ξ) t N [ỹ ; L 1 ], and conequently ugget that the path under conideration i quite mooth. Beide, the regularity aumption on ỹ,i immediately give dξ φ(ξ)ỹ t,i (ξ) t 2γ N [ỹ; Q γ x ]. With thoe two control in hand, it would be tempting to enviage an algebraic tructure uch that {y : (δy i ) t = X x, j t y x,i j + yt,i, with y x C γ 1 (m,l ) and y C 2γ 2 (k )}. It i indeed poible to how that the latter et i invariant when compoing the path with a mooth enough mapping, which would enure the tranition between y and σ (y). Neverthele, a little bit more ubtle analyi of (58) lead to more convenient algebraic handling. It actually uffice to oberve that the path X x can be decompoed a X t x (ξ) = S t u (ξ) dx u = (δx) t + A tu (ξ) dx u. When x C γ 1 (m ), the latter tranformation i at thi point purely formal. Let u record thi through the following theoretical hypothei, that will be examined in detail in Section 6: Hypothei 4. Under Hypothei 3, we aume that the equence of path defined by X Axε,i t (ξ) A tu (ξ) dx ε,i u, converge w.r.t. to the topology of the pace 2,0 (m ) (we recall that thi pace ha been defined in Section 3.3). In particular, C 1+γ X Ax C 1+γ 2,0 (m ) and X t x (ξ) = x1 t + X t Ax (ξ), (59) where we have defined, according to [25], x 1 δx. If x i a differentiable path, we aume that thi reult hold true for x ε x.