Volterra equations driven by rough signals 2: higher order expansions
aa r X i v : . [ m a t h . P R ] F e b VOLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 2:HIGHER ORDER EXPANSIONS
FABIAN A. HARANG, SAMY TINDEL, AND XIAOHUA WANG
Abstract.
We extend the recently developed rough path theory for Volterra equations from [1]to the case of more rough noise and/or more singular Volterra kernels. It was already observedin [1] that the Volterra rough path introduced there did not satisfy any geometric relation, similarto that observed in classical rough path theory. Thus, an extension of the theory to more irregulardriving signals requires a deeper understanding of the specific algebraic structure arising in theVolterra rough path. Inspired by the elements of ”non-geometric rough paths” developed in [12]and [2] we provide a simple description of the Volterra rough path and the controlled Volterraprocess in terms of rooted trees, and with this description we are able to solve rough volterraequations in driven by more irregular signals. Introduction
Background and description of the results.
Volterra equations of the second kind aretypically given on the form y t = y + Z t k ( t, s ) b ( y s ) ds + Z t k ( t, s ) σ ( y s ) dx s , y ∈ R d (1.1)where b and σ are sufficiently smooth functions, x : [0 , T ] → R d is a α -H¨older continuous pathwith α ∈ (0 , k and k are two possibly singular kernels, behaving like | t − s | − γ for some γ ∈ [0 ,
1) whenever s → t . Such equations frequently appear in mathematical models for naturalor social phenomena which exhibits some form of memory of its own past as it evolves in time (seee.g. [3] and the references therein). Most recently, Volterra equations of this form have becomevery popular in the modelling of stochastic volatility for financial asset prices. In this case thekernels k ( t, s ) and k ( t, s ) are typically assumed to be very singular when s → t , and the path x is assumed to be a sample path of a Gaussian process (see e.g. [5, 10, 4]).Whenever the driving noise x is sampled from a Brownian motion (or some other continuoussemi-martingale), one may use traditional probabilistic techniques from stochastic analysis (seee.g. [13, 15]) in order to make sense of equations like (1.1). However, for more general driving noise x with rougher regularity than a Brownian motion, very little is known about solutions to Volterraequations. Inspired by the theory of rough paths [11], it is desirable to solve equation (1.1) in apurely pathwise sense relying only on the analytic behaviour of the sample paths of x . This wouldallow to remove the probabilistic restrictions imposed by classical stochastic analysis. However,due to the non-local nature of the equations induced by the kernels k and k , the theory of rough Date : February 23, 2021.
Key words and phrases.
Volterra equations, Rough path theory, Singular integral equations, Regularity struc-tures.
MSC2020: 45D05, 45G05, 60L20, 60L30, 60L70Acknowledgments:
S. Tindel is supported by the NSF grant DMS-1952966. F. Harang gratefully acknowledgesfinancial support from the STORM project 274410, funded by the Research Council of Norway . paths can not directly be applied in order to solve singular Volterra equations of the form of(1.1). Indeed, the fundamental algebraic relations satisfied by a a classical rough path do not holdwhen the signal is influenced by a possibly singular kernel. Let us mention at this point a fewcontributions in the rough paths realm trying to overcome this obstacle:(i) The articles [7, 8] handle some cases of rough Volterra equations thanks to an elaborationof traditional rough paths elements. However, the analysis was only valid for kernels with nosingularities.(ii) The paper [14] focuses on Volterra equations from a para-controlled calculus perspective. Thiselegant method is unfortunately restricted to first order rough paths type expansions, with inherentlimits on both the irregularity of the driving process x and the singularity of the kernel k .(iii) The contribution [4] investigates Volterra equations through the lens of regularity structures.Although only the strategy of the construction is outlined therein, we believe that a mere appli-cation of regularity structures techniques would only yield local existence and uniqueness results.It should also be mentioned that renormalization techniques are invoked in [4].As the reader might see, the rough paths analysis of Volterra equations is thus far from beingcomplete.With those preliminary notions in mind, in the recent article [1] we initiated a rough pathinspired study of singular Volterra equations, in a reduced form of (1.1) given by u t = u + Z t k ( t, r ) f ( u r ) dx r , (1.2)where f is a sufficiently regular function, x is a H¨older continuous path, and k is a singular kernel.To this end, we define∆ n := ∆ n ([ a, b ]) = { ( x , . . . , x n ) ∈ [ a, b ] n | a ≤ x < · · · < x n ≤ b } . (1.3)Next we introduce a class of two parameter paths z : ∆ → R d , needed to capture the possiblesingularity and regularity imposed by the kernels k and k and the driving noise x in (1.1). Thesepaths will then constitute the fundamental building blocks of the framework. The canonicalexample of such path is given by z τt := Z t k ( τ, s ) dx s , where t ≤ τ ∈ [0 , T ] . (1.4)For the moment, we may assume that x is a sufficiently regular path x : [0 , T ] → R d , and k ( t, s ) isan integrable (but possibly singular) kernel when s → t , so that the above integral makes pathwisesense. We observe in particular that t z tt is just a standard Volterra integral (commonly referredto as a Volterra process in stochastic analysis). Heuristically one may think that the regularityarising from the mapping τ z τt is induced by the behaviour of the kernel k while the regularityof the mapping t z τt is inherited by the regularity of x . By construction of a Volterra sewinglemma, we observed that this was indeed the case, even when x is only α -H¨older continuous forsome α ∈ (0 , OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 3 product ∗ , playing the role as the tensor product ⊗ in the classical rough path signature. Thesignature is then given as a family three-variable functions { ( s, t, τ ) z n,τts } n ∈ N , where, in the caseof smooth x , each term is given by z n,τts = Z ∆ n ([ s,t ]) k ( τ, r n ) . . . k ( r , r ) dx r ⊗ · · · ⊗ dx r n , (1.5)where we recall that ∆ n ([ s, t ]) is defined by (1.3). The algebraic structure associated with suchiterated integrals resembles that of the tensor algebra of rough path theory, but where the tensorproduct is replaced by the convolution product. Together with Volterra signatures, we defineda class of controlled Volterra paths. Combining those two notions, it allowed to give a pathwiseconstruction of solutions to Volterra equations of the form (1.1). Similarly to the theory of roughpaths, the number of iterated integrals needed in order to give a pathwise definition of a roughVolterra integral is strongly dependent on the regularity of the path x ∈ C α ([0 , T ]; R d ) and thesingularity of the kernel k . Under the assumption that | k ( t, s ) | behaves like | t − s | − γ when s → t ,the investigation in [1] was limited to the case when α − γ > , and thus only considers the firsttwo components of the Volterra signature.Our article [1] therefore left two important open questions, related to both the algebraic andprobabilistic perspectives on rough paths theory: (i) Algebraic aspects: Are there suitable algebraic relations describing the Volterra signature whichare adaptable to prove existence and uniqueness of (1.1) in the case when α − γ < ? (ii) Probabilistic aspects: For what type of stochastic processes { x t ; t ∈ [0 , T ] } and singular kernels k does there exist a collection of iterated integrals of the form of (1.5) almost surely satisfying therequired algebraic and analytic relations?The current article has to be seen as a step towards the answer of the algebraic problem mentionedabove. Namely we investigate the case when α − γ < , and leave the probabilistic problem for afuture work.The rough Volterra picture gets significantly more involved when introducing a rougher signal x or a more singular kernel k . Indeed, the main challenge lies in the fact that the Volterra signaturedoes not satisfy any geometric type property, in contrast with the classical rough paths situation.That is, classical integration by parts does not hold for Volterra iterated integrals, and thereforewe do not have a relation of the form z ,τts + ( z ,τts ) T = z ,τts ∗ z , · ts , where ( · ) T denotes the transpose. Thus in order to consider α − γ lower than , one needs toresort to different techniques than what is standard in the theory of rough paths.Inspired by Martin Hairer’s theory of regularity structures, we will in this article show thatthe Volterra signature is given with a Hopf algebraic type structure. Hence with the help of adescription by rooted trees for the Volterra rough path, we are able to describe the necessaryalgebraic relations desired for the Volterra rough stochastic calculus. We will limit the scope ofthe current article to the case when α − γ > , and show that in order to prove existence anduniqueness of (1.1) in a ”Volterra rough path” sense, one needs to introduce two more iteratedintegrals, as well as two more controlled Volterra derivatives than what is needed in the case α − γ > . We believe that the techniques developed here are an important stepping stonetowards the goal of providing a rough paths framework for Volterra equations of the form of (1.1)in the general regime α − γ > F. HARANG, S. TINDEL, AND X. WANG
Organization of the paper.
In section 2 we provide the necessary assumptions and pre-liminary results from [1]. In particular, we give the definition of Volterra paths, recall the Volterrasewing lemma and the convolution product between Volterra paths. Those results will play acentral role for our subsequent analysis. Section 3 is devoted to the extension of the sewing lemmafrom the previous section to the case of two singularities, and we will apply this to create a thirdorder convolution product between Volterra rough paths. In Section 4 we motivate the use ofrooted trees to describe the Volterra rough path, and give a definition of controlled Volterra pro-cesses analogously. With this definition we prove both the convergence of a rough Volterra integralwith respect to controlled Volterra paths, and that compositions of (sufficiently) smooth functionswith a controlled Volterra path are again controlled Volterra paths. We conclude Section 4 witha proof of existence and uniqueness of Volterra equations driven by rough signals in the rougherregime.1.3.
Frequently used notation.
We reserve the letter E to denote a Banach space, and we letthe norm on E be denoted by | · | E . In subsequent sections, E will typically be given as R d or L ( R m , R d ) (The space of linear operators from R m to R d ). We will write a . b , whenever thereexists a constant C > a ≤ Cb . Thespace of continuous functions f : X → Y is denoted by C ( X, Y ). Whenever the codomain is notimportant, we use the shorter notation C ( X ). To denote that there exists a constant C whichdepends on a parameter p , we write a . p b . For a one parameter path f : [0 , T ] → E , we write f ts := f t − f s , with a slight abuse of notation, we will later also use this notation for two variablefunctions of the form f : [0 , T ] → R d , where f ts means evaluation in the point ( s, t ) ∈ [0 , T ] .We believe that it will always be clear from context what is meant. For α ∈ (0 , C α ([0 , T ]; E ) the standard space of α -H¨older continuous functions from [0 , T ] into E , equippedwith the norm k f k C α := | f | E + k f k α , where k f k α denotes the classical H¨older seminorm given by k f k α := sup ( s,t ) ∈ ∆ | f ts || t − s | γ . (1.6)Whenever the domain and codomain is otherwise clear from the context, we will use the shorthand notation C α . We recall here that the n -simplex was already defined in (1.3). Throughoutthe article, we will frequently use the following simple bounds: for ( s, u, t ) ∈ ∆ and γ >
0, then | t − u | γ . | t − s | γ and | t − s | − γ . | t − u | − γ . Assumptions and fundamentals of Volterra Rough Paths
We will start by presenting the necessary assumptions on the Volterra kernel k , as well as thedriving noise x in (1.2). A full description (together with proofs) for the results recalled in thissection can be found in [1].Let us begin to present a working hypothesis for the type of kernels k , seen in (1.2), that wewill consider in this article. OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 5
Hypothesis 2.1.
Let k be a kernel k : ∆ → R , we assume that there exists γ ∈ (0 , such thatfor all ( s, r, q, τ ) ∈ ∆ ([0 , T ]) and η, β ∈ [0 , we have | k ( τ, r ) | . | τ − r | − γ | k ( τ, r ) − k ( q, r ) | . | q − r | − γ − η | τ − q | η | k ( τ, r ) − k ( τ, s ) | . | τ − r | − γ − η | r − s | η | k ( τ, r ) − k ( q, r ) − k ( τ, s ) + k ( q, s ) | . | q − r | − γ − β | r − s | β | k ( τ, r ) − k ( q, r ) − k ( τ, s ) + k ( q, s ) | . | q − r | − γ − η | τ − q | η . In the sequel a kernel fulfilling condition the Hypothesis 2.1 will be called Volterra kernel of order γ .Remark . We limit our investigations in this article to the case of real valued Volterra kernels k for conciseness. The Volterra sewing lemma, and most results relating to Volterra rough pathsare however easily extended to general Volterra kernels k : ∆ → L ( E ) for some Banach space E ,by appropriate change of the bounds in 2.1, see e.g. [9, 6] where the Volterra sewing lemma from[1] is readily applied in an infinite dimensional setting.As mentioned in the introduction, one of the key ingredients in [1] is to consider processes( t, τ ) z τt indexed by ∆ (where we recall that the simplex ∆ n was defined in (1.3)). We beginthis section with a recollection of the H¨older space containing such processes and introduce theVolterra sewing Lemma 2.10, we will then move on to introduce the convolution product anddiscuss its relation with the Volterra signature.2.1. The space of Volterra paths.
We begin this section by recalling the topology used tomeasure the regularity of processes like (1.4), and give a simple motivation for the introduction ofthis type of space.
Definition 2.3.
Let E be a Banach space, and consider ( α, γ ) ∈ (0 , with α − γ > . Wedefine the space of Volterra paths of index ( α, γ ) , denoted by V ( α,γ ) (∆ ; R d ) , as the set of functions z : ∆ → E, given by ( t, τ ) z τt , with the condition z τ = z ∈ E for all τ ∈ (0 , T ] , and satisfying k z k ( α,γ ) = k z k ( α,γ ) , + k z k ( α,γ ) , , < ∞ . (2.1) In (2.1) , the 1-norm and (1,2)-norm are respectively defined as follows: k z k ( α,γ ) , := sup ( s,t,τ ) ∈ ∆ | z τts | E [ | τ − t | − γ | t − s | α ] ∧ | τ − s | α − γ , (2.2) k z k ( α,γ ) , , := sup ( s,t,τ ′ ,τ ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,α − γ ) | z ττ ′ ts | E | τ − τ ′ | η | τ ′ − t | − η + ζ ([ | τ ′ − t | − γ − ζ | t − s | α ] ∧ | τ ′ − s | α − γ − ζ ) , (2.3) with the convention z τts = z τt − z τs and z ττ ′ s = z τs − z τ ′ s . In addition, under the mapping z
7→ | z | + k z k ( α,γ ) , the space V ( α,γ ) (∆ ; E ) is a Banach space. Whenever the domain and codomain is otherwise clear from the context, we will simply write V ( α,γ ) := V ( α,γ ) (∆ n ; R d ). Throughout the article, we will typically let the Banach space E be givenby R d or L ( R d ). F. HARANG, S. TINDEL, AND X. WANG
Remark . As will be proved in Theorem 2.11 below, the typical example of path in V ( α,γ ) isgiven by z τt defined as in (1.4), with suitable assumption on k and x . Note also that C α ([0 , R d ) ⊂V ( α,γ ) (∆ ([0 , R d ) for any γ ∈ [0 , x ∈ C α , define z τt = x t . Using that | t − s | α ≤ | τ − s | α , it is readily checked that | z τts | . | τ − t | − γ | t − s | α ∧ | τ − s | α . Furthermore, z ττ ′ ts = 0, and thus k z k ( α,γ ) < ∞ for any γ ∈ (0 , Remark . We will also consider functions u : ∆ → R d , which, with a slight abuse of notation,will be denoted by the mapping ( s, t, τ ) u τts . We then define the space V ( α,γ ) (∆ ; R d ) analogouslyas in Definition 2.3, but where the increments of the path ( t, τ ) z τt in the lower variable,appearing in (2.2) and (2.3), is simply replaced by the evaluation u τts and u τts − u τ ′ ts respectively. Remark . Similarly as for the classical H¨older spaces, we have the following elementary embed-ding: for β < α ∈ (0 , β − γ >
0, we have V ( α,γ ) ֒ → V ( β,γ ) . Indeed, suppose y ∈ V ( α,γ ) , it isreadily checked that | y τts | . | τ − t | − γ | t − s | α | ∧ | τ − s | α − γ ≤ T α − β ( | τ − t | − γ | t − s | β ∧ | τ − s | β − γ ) , and thus k y k ( β,γ ) , ≤ T α − β k y k ( α,γ ) , . Similarly, one can also show that k y k ( β,γ ) , , ≤ T α − β k y k ( α,γ ) , , ,and thus k y k ( β,γ ) ≤ T α − β k y k ( α,γ ) .The following lemma gives useful embedding results for V ( α,γ ) related to variations in the sin-gularity parameter γ . Lemma 2.7.
Let α, γ ∈ (0 , with α > γ , and recall that ρ = α − γ . Then for the spaces V ( α,γ ) given in Definition 2.3, the following inclusion holds true: V (3 ρ + γ,γ ) ⊂ V (3 ρ +2 γ, γ ) ⊂ V (3 ρ +3 γ, γ ) . (2.4) Proof.
We will prove the second relation: V (3 ρ +2 γ, γ ) ⊂ V (3 ρ +3 γ, γ ) , the first relation being provedin a similar way. Moreover, in order to prove that V (3 ρ +2 γ, γ ) ⊂ V (3 ρ +3 γ, γ ) , we will show that k z k (3 ρ +3 γ, γ ) ≤ k z k (3 ρ +2 γ, γ ) , for the ( α, γ ) − norms introduced in Definition 2.3. Also recall thatthe ( α, γ ) − norms are defined by (2.2) and (2.3). For sake of conciseness we will just prove that k z k (3 ρ +3 γ, γ ) , ≤ k z k (3 ρ +2 γ, γ ) , , (2.5)and leave the similar bound for the (1 , − norm to the reader.In order to prove (2.5), we refer again to (2.2). From this definition, it is readily checked that(2.5) can be reduced to prove the following relation: | τ − t | − γ | t − s | ρ +3 γ ∧ | τ − s | ρ . | τ − t | − γ | t − s | ρ +2 γ ∧ | τ − s | ρ . (2.6)The proof of (2.6) will be split in 2 cases, according to the respective values of | τ − t | and | t − s | .In the sequel C designates a strictly positive constant. Case 1: | τ − t | ≤ C | t − s | . Let us write | τ − s | ρ = | τ − s | ρ +2 γ | τ − s | − γ . Then if | τ − t | ≤ C | t − s | , one has | τ − s | ρ +2 γ = | τ − t + t − s | ρ +2 γ . | t − s | ρ +2 γ . Hence we get | τ − s | ρ . | t − s | ρ +2 γ | τ − s | − γ . | t − s | ρ +2 γ | τ − t | − γ . (2.7)Relation (2.6) is then immediately seem from (2.7). OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 7
Case 2: | τ − t | > C | t − s | . In this case write | t − s | ρ +2 γ | τ − t | − γ = | t − s | ρ +3 γ | τ − t | − γ (cid:18) | τ − t || t − s | (cid:19) γ . Then resort to the fact that | τ − t | ≥ C | t − s | in order to get | τ − t | γ | t − s | − γ ≥ C γ . This yields | t − s | ρ +2 γ | t − s | − γ & | t − s | ρ +3 γ | τ − t | − γ , from which (2.6) is readily checked.Combining Case 1 and Case 2, we have thus finished the proof of (2.6). As mentioned above,this implies that (2.5) is true and achieves our claim (2.4). (cid:3) Volterra Sewing lemma.
We begin with a recollection of the space of abstract Volterraintegrands, to which the Volterra sewing Lemma 2.10 will apply. The typical path in this spaceexhibits different types of regularities/singularities in its arguments, similarly to Definition 2.3.As a necessary ingredient in the subsequent definition we introduce a particular notation, whichwill frequently be used throughout the article.
Notation 2.8.
Recall that the simplex ∆ n is defined by (1.3) . For a path g : ∆ → R d and ( s, u, t ) ∈ ∆ , we set δ u g ts = g ts − g tu − g us (2.8) We will consider δ as an operator from C (∆ ) to C (∆ ) , where C (∆ n ) denotes the spaces of con-tinuous functions on ∆ n . Definition 2.9.
Let α ∈ (0 , , γ ∈ (0 , with α − γ > . We also consider two coefficients κ ∈ (0 , ∞ ) and β ∈ (1 , ∞ ) . Denote by V ( α,γ )( β,κ ) (cid:0) ∆ ; R d (cid:1) , the space of all functions Ξ : ∆ → R d such that k Ξ k V ( α,γ )( β,κ ) = k Ξ k ( α,γ ) + k δΞ k ( β,κ ) < ∞ , (2.9) where δ is introduced in (2.8) , where the quantity k Ξ k ( α,γ ) is given by (2.1) (see also Remark 2.5)and where k δΞ k ( β,κ ) = k δΞ k ( β,κ ) , + k δΞ k ( β,κ ) , , (2.10) with k δΞ k ( β,κ ) , := sup ( s,m,t,τ ) ∈ ∆ | δ m Ξ τts | [ | τ − t | − κ | t − s | β ] ∧ | τ − s | β − κ (2.11) k δΞ k ( β,κ ) , , := sup ( s,m,t,τ ′ ,τ ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,β − κ ) | δ m Ξ ττ ′ ts || τ − τ ′ | η | τ ′ − t | − η + ζ ([ | τ ′ − t | − κ − ζ | t − s | β ] ∧ | τ ′ − s | β − κ − ζ ) . (2.12) In the sequel the space V ( α,γ )( β,κ ) will be our space of abstract Volterra integrands. With these two Volterra spaces in hand, we are ready to recall the Volterra sewing Lemmawhich can be found, together with a full proof, in [1, Lemma 21].
Lemma 2.10.
Consider four exponents β ∈ (1 , ∞ ) , κ ∈ (0 , , α ∈ (0 , and γ ∈ (0 , such that β − κ ≥ α − γ > . Let V ( α,γ )( β,κ ) and V ( α,γ ) be the spaces given in Definition 2.9 and Definition 2.3respectively. Then there exists a linear continuous map I : V ( α,γ )( β,κ ) (cid:0) ∆ ; R d (cid:1) → V ( α,γ ) (cid:0) ∆ ; R d (cid:1) such that the following holds true. F. HARANG, S. TINDEL, AND X. WANG (i) The quantity I ( Ξ τ ) ts := lim |P|→ P [ u,v ] ∈P Ξ τvu exists for all ( s, t, τ ) ∈ ∆ , where P is a genericpartition of [ s, t ] and |P| denotes the mesh size of the partition. Furthermore, we define I ( Ξ τ ) t := I ( Ξ τ ) t , and we have that I ( Ξ τ ) ts = I ( Ξ τ ) t − I ( Ξ τ ) s .(ii) For all ( s, t, τ ) ∈ ∆ we have |I ( Ξ τ ) ts − Ξ τts | . k δΞ k ( β,κ ) , (cid:16)h | τ − t | − κ | t − s | β i ∧ | τ − s | β − κ (cid:17) , (2.13) while for ( s, t, τ ′ , τ ) ∈ ∆ we get (cid:12)(cid:12)(cid:12) I ( Ξ ττ ′ ) ts − Ξ ττ ′ ts (cid:12)(cid:12)(cid:12) . k δΞ k ( β,κ ) , , | τ − τ ′ | η | τ ′ − t | − η + ζ (cid:16)h | τ ′ − t | − κ − ζ | t − s | β i ∧ | τ ′ − s | β − κ − ζ (cid:17) . (2.14)Lemma 2.10 is applied in [1] in order to get the construction of the path ( t, τ ) z τt introducedin (1.4). We recall this result here, since z is at the heart of our future considerations. Theorem 2.11.
Let x ∈ C α and k be a Volterra kernel of order − γ satisfying Hypothesis 2.1, suchthat ρ = α − γ > . We define an element Ξ τts = k ( τ, s ) x ts . Then the following holds true:(i) There exists some coefficients β > and κ > with β − κ = α − γ such that Ξ ∈ V ( α,γ )( β,κ ) ,where V ( α,γ )( β,κ ) is given in Definition 2.9. It follows that the element I ( Ξ τ ) obtained in Lemma2.10 is well defined as an element of V ( α,γ ) and we set z τts ≡ I ( Ξ τ ) ts = R ts k ( τ, r ) dx r .(ii) For ( s, t, τ ) ∈ ∆ z satisfies the bound | z τts − k ( τ, s ) x ts | . (cid:2) | τ − t | − γ | t − s | α (cid:3) ∧ | τ − s | ρ , and in particular it holds that k z k ( α,γ ) , < ∞ .(iii) For any η ∈ [0 , and any ( s, t, q, p ) ∈ ∆ we have | z pqts | . | p − q | η | q − t | − η + ζ (cid:16)h | q − t | − γ − ζ | t − s | α i ∧ | q − s | ρ − ζ (cid:17) , where z pqts = z pt − z qt − z ps + z qs . In particular it holds that k z k ( α,γ ) , , < ∞ .Remark . Thanks to Theorem 2.11, we know that a typical example of a Volterra path in V ( α,γ ) is given by the integral R ts k ( τ, r ) dx r , as mentioned in Remark 2.4.2.3. Convolution product in the rough case α − γ > . A second crucial ingredient in theVolterra formalism put forward in [1] is the notion of convolution product. In this section we showhow this mechanism is introduced for first and second order convolutions, where we recall thatsecond order convolutions were enough to handle the case ρ = α − γ > in [1].Let us first introduce a piece of notation which will prevail throughout the paper. Notation 2.13.
In the sequel we will often consider products of the form y s z τts , where y and z τ are increments lying respectively in C ([0 , T ]) and C (∆ ) . For algebraic reasons due to our roughVolterra formalism, we will write this product as [( z τts ) ⊺ y ⊺ s ] ⊺ (2.15) For obvious notational reason, we will simply abbreviate (2.15) into z τts y s In the same way, products of 3 (or more) elements of the form f ′ ( y s ) y s z τts will be denoted as z τts y s f ′ ( y s ) without further notice. OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 9
We now recall how the convolution with respect to z τ is obtained, borrowing the followingproposition from [1, Theorem 25]. Proposition 2.14.
We consider two Volterra paths z ∈ V ( α,γ ) ( R d ) and y ∈ V ( α,γ ) ( L ( R d )) as givenin Definition 2.3, where we recall that α, γ ∈ (0 , . Define ρ = α − γ , and assume ρ > . Then theconvolution product of the two Volterra paths y and z is a bilinear operation on V ( α,γ ) ( R d ) givenby z τtu ∗ y · us = Z t>r>u dz τr y rus := lim |P|→ X [ u ′ ,v ′ ] ∈P z τv ′ u ′ y u ′ us . (2.16) The integral in (2.16) is understood as a Volterra-Young integral for all ( s, u, t, τ ) ∈ ∆ . Moreover,the following two inequalities holds for any η ∈ [0 , , ζ ∈ [0 , ρ ) and ( s, u, t, τ, τ ′ ) ∈ ∆ : | z τtu ∗ y · us | . k z k ( α,γ ) , k y k ( α,γ ) , , (cid:0)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ (cid:1) (2.17) (cid:12)(cid:12)(cid:12) z τ ′ τtu ∗ y · us (cid:12)(cid:12)(cid:12) . k z k ( α,γ ) , , k y k ( α,γ ) , , | τ ′ − τ | η | τ − t | − η + ζ (cid:16)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ − ζ (cid:17) (2.18)In addition to Proposition 2.14, the rough Volterra formalism relies on a stack of iteratedintegrals verifying convolutional type algebraic identities. Thanks to Proposition 2.14 we can nowstate the main assumption about this stack of integrals, which should be seen as the equivalent ofChen’s relation in our Volterra context. Hypothesis 2.15.
Let z ∈ V ( α,γ ) be a Volterra path as given in Definition 2.3. For n such that ( n + 1) ρ + γ > , we assume that there exists a family { z j,τ ; j ≤ n } such that z j,τts ∈ ( R m ) ⊗ j , z = z and verifying δ u z j,τts = j − X i =1 z j − i,τtu ∗ z i, · us = Z ts d z j − i,τtr ⊗ z i,rus , (2.19) where the right hand side of (2.19) is defined in Proposition 2.14. In addition, we suppose thatfor j = 1 , . . . , n we have z j ∈ V ( jρ + γ,γ ) . The last notation we need to recall from [1] is the concept of second order convolution product.To this aim, we first introduce some basic notation about increments.
Notation 2.16.
We will denote by u , a function u : ∆ → L (( R d ) ⊗ , R d ) with two upper indices,namely, ∆ ∋ ( s, τ , τ ) u τ ,τ s ∈ R d . The notation u , highlights the order of integration in future computations. We now specify the kind of topology we will consider for functions of the form u , . Definition 2.17.
Let W ( α,γ )2 denote the space of functions u : ∆ → L (( R d ) ⊗ , R d ) with a fixedinitial condition u p,q = u , endowed with the norm (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) := (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , + (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , . (2.20) The right hand side of (2.20) is defined as follows, recalling the convention ρ = α − γ : (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , := sup ( s,t,τ ) ∈ ∆ | u τ,τts | (cid:2) | τ − t | − γ | t − s | α (cid:3) ∧ | τ − s | ρ , (2.21) and (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , := (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , ,> + (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , ,< , (2.22) where the norms k u , k ( α,γ ) , , ,> and k u , k ( α,γ ) , , ,< are respectively defined by (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , ,> = sup ( s,t,r ,r ,r ′ ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,α − γ ) | u r ′ ,r ts − u r ′ ,r ts | h η,ζ ( s, t, r , r , r ′ ) , (2.23) (cid:13)(cid:13) u , (cid:13)(cid:13) ( α,γ ) , , ,< = sup ( s,t,r ′ ,r ,r ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,α − γ ) | u r ,r ′ ts − u r ,r ′ ts | h η,ζ ( s, t, r , r , r ′ ) , (2.24) where the function h is defined by h η,ζ ( s, t, r , r , r ′ ) = | r − r | η | min( r , r , r ′ ) − t | − η + ζ × (cid:16)h | min( r , r , r ′ ) − t | − γ − ζ | t − s | α i ∧ | min( r , r , r ′ ) − s | α − γ − ζ (cid:17) . (2.25) Remark . In the sequel we will need to estimate differences of functions u · , · : ∆ → L (( R m ) ⊗ , R m )of the form | u τ,qt − u τ,pt | . Those differences can be handled thanks to Definition 2.17 as follows: | u τ,qt − u τ,pt | ≤ | u τ,q − u τ,p | + | u τ,qt − u τ,pt |≤ k u k ( α,γ ) , , | q − p | η | p − t | − η + ζ (cid:16)h | p − t | − γ − ζ | t | α i ∧ | p | ρ − ζ (cid:17) . (2.26)Since ζ ∈ [0 , ρ ) and η ∈ [0 , η = ζ , that is | u τ,qt − u τ,pt | . k u k ( α,γ ) , , | q − p | ζ . k u k ( α,γ ) , , . we also have, for any τ ∈ [0 , T ], | u τ,τt − u τ,τ | ≤ k u k ( α,γ ) , (cid:2) | τ − t | − γ | t | α ∧ | τ | ρ (cid:3) . k u k ( α,γ ) , . (2.27)With the above definition at hand, we are now ready to recall the construction of second orderconvolution products in the rough case α − γ > . Theorem 2.19.
Let z ∈ V ( α,γ ) be as given in Definition 2.3 with α, γ ∈ (0 , satisfying ρ = α − γ > . We assume that z fulfills Hypothesis 2.15 with n = 2 . Consider a function y : ∆ →L (( R d ) ⊗ , R d ) with k y , k ( α,γ ) , , < ∞ and y , = y , for a fixed initial condition y ∈ L (( R d ) ⊗ , R d ) .For all fixed ( s, t, τ ) ∈ ∆ we have that z ,τts ∗ y , s := lim |P|→ X [ u,v ] ∈P z ,τvu y u,us + ( δ u z ,τvs ) ∗ y , s (2.28) is a well defined Volterra-Young integral. It follows that ∗ is a well defined bi-linear operation be-tween the three parameters Volterra function z and a -parameter path y . Moreover, the followinginequality holds (cid:12)(cid:12) z ,τts ∗ y , s − z ,τts y s,ss (cid:12)(cid:12) . k y , k ( α,γ ) , , (cid:0) k z k (2 ρ + γ,γ ) , + k z k ( α,γ ) , , k z k ( α,γ ) , (cid:1) × (cid:0)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ (cid:1) . (2.29) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 11
Remark . By Hypothesis 2.15, the term ( δ u z ,τvs ) ∗ y , s in the right hand side of (2.28) can berewritten as z ,τvu ∗ z , · us ∗ y , s , where the convolution with z ,τ is defined through (2.16) and the inside integral concerns thesecond variable in y , . As an example, if k , x are smooth functions and z ,τvs = R vs k ( τ, r ) dx r , thenthis convolution is understood in the following way z ,τvu ∗ z , · us ∗ y , s = Z vu k ( τ, r ) dx r ⊗ Z us k ( r , r ) dx r y r ,r s . Remark . Recalling that ρ = α − γ , notice that Proposition 2.14 and Theorem 2.19 tell ushow to define the n ’th order convolution products under the condition ρ > . We will follow asimilar strategy to define third order convolution products and construct our solution to equation(1.2) with ρ > in the subsequent section.3. Volterra rough paths for α − γ > This section is devoted to the generalization of the concepts introduced in Section 2 to accom-modate the case of Volterra rough paths with regularity ρ = α − γ > . One of the main issuesencountered in this direction is to define third order convolution structures. To this end, we willstate a version of our Volterra sewing Lemma 2.10 extended to the case of two types of Volterrasingularities.3.1. Volterra sewing lemma with two singularities.
With the aim of extending the Volterrasewing Lemma 2.10 with one singularity to an increment exhibiting two singularities, we firstintroduce a new space of abstract integrands.
Definition 3.1.
Let α ∈ (0 , and γ ∈ (0 , with α − γ > . We also consider three coefficients ( β, κ, θ ) , with ( κ + θ ) ∈ (0 , and β ∈ (1 , ∞ ) . Denote by V ( α,γ )( β,κ,θ ) (∆ ; R d ) , the space of allfunctions of the form ∆ ∋ ( v, s, t, τ ) ( Ξ τv ) ts ∈ R d such that the following norm is finite: k Ξ k V ( α,γ )( β,κ,θ ) = k Ξ k ( α,γ ) + k δΞ k ( β,κ,θ ) . (3.1) In equation (3.1) , the operator δ is introduced in (2.8) , the quantity k Ξ k ( α,γ ) is given by (2.1) andthe term k δΞ k ( β,κ,θ ) takes the double singularity into account. Namely we have k δΞ k ( β,κ,θ ) = k δΞ k ( β,κ,θ ) , + k δΞ k ( β,κ,θ ) , , , where k δΞ k ( β,κ,θ ) , := sup ( v,s,m,t,τ ) ∈ ∆ | δ m ( Ξ τv ) ts | h | τ − t | − κ | t − s | β | s − v | − θ i ∧ | τ − v | β − κ − θ , (3.2) and the term k δΞ k ( β,κ,θ ) , , is defined by k δΞ k ( β,κ,θ ) , , := sup ( v,s,m,t,τ ′ ,τ ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,β − κ − θ ) (cid:12)(cid:12) δ m ( Ξ ττ ′ v ) ts (cid:12)(cid:12) f ( v, s, t, τ ′ , τ ) , (3.3) where the function f is given by f ( v, s, t, τ ′ , τ ) = | τ − τ ′ | η | τ ′ − t | − η + ζ (cid:16)h | τ ′ − t | − κ − ζ | t − s | β | s − v | − θ i ∧ | τ ′ − v | β − κ − θ − ζ (cid:17) . (3.4) Notice that we will use V ( α,γ )( β,κ,θ ) as a space of abstract Volterra integrands with a double singu-larity. With this new space V ( α,γ )( β,κ,θ ) at hand, we are ready to state the Volterra sewing Lemma withtwo singularities alluded to above. Lemma 3.2.
Consider five exponents ( α, γ ) , and ( β, κ, θ ) , with β ∈ (1 , ∞ ) , ( κ + θ ) ∈ (0 , , α ∈ (0 , and γ ∈ (0 , such that α − γ > . Let V ( α,γ )( β,κ,θ ) and V ( α,γ ) be the spaces givenin Definition 3.1 and Definition 2.3 respectively. Then there exists a linear continuous map I : V ( α,γ )( β,κ,θ ) (cid:0) ∆ ; R d (cid:1) → V ( α,γ ) (cid:0) ∆ ; R d (cid:1) such that the following holds true. (i) The quantity I ( Ξ τv ) ts := lim |P|→ P [ u,w ] ∈P ( Ξ τv ) wu exists for all ( v, s, t, τ ) ∈ ∆ , where P is ageneric partition of [ s, t ] and |P| denotes the mesh size of the partition. Furthermore, we define I ( Ξ τv ) t := I ( Ξ τv ) t , and have I ( Ξ τv ) ts = I ( Ξ τv ) t − I ( Ξ τv ) s . (ii) For all ( v, s, t, τ ) ∈ ∆ we have |I ( Ξ τv ) ts − ( Ξ τv ) ts | . k δΞ k ( β,κ,θ ) , (cid:16)h | τ − t | − κ | t − s | β | s − v | − θ i ∧ | τ − v | β − κ − θ (cid:17) , (3.5) while for ( v, s, t, τ ′ , τ ) ∈ ∆ we get (cid:12)(cid:12)(cid:12) I ( Ξ ττ ′ v ) ts − ( Ξ ττ ′ v ) ts (cid:12)(cid:12)(cid:12) . k δΞ k ( β,κ,θ ) , , f ( v, s, t, τ ′ , τ ) , (3.6) where f is the function given by (3.4) .Proof. This is an extension of [1, Lemma 21]. Let us consider the n-th order dyadic partition P n of [ s, t ] where each set [ u, w ] ∈ P n has length 2 − n | t − s | . We define the n -th order Riemann sumof Ξ τv , denoted I n ( Ξ τv ) ts , as follows I n ( Ξ τv ) ts = X [ u,w ] ∈P n ( Ξ τv ) wu . Our aim is to show that the sequence {I n ( Ξ τv ); n ≥ } converges to an element I ( Ξ τv ) which fulfillsrelation (3.5). To this aim we begin to consider the difference I n +1 ( Ξ τv ) − I n ( Ξ τv ). A series ofelementary computations reveals that I n +1 ( Ξ τv ) ts − I n ( Ξ τv ) ts = − X [ u,w ] ∈P n δ m ( Ξ τv ) wu , (3.7)where m = w + u and where we recall that δ is given by relation (2.8). Plugging relation (3.2) into(3.7), it is easy to check that X [ u,w ] ∈P n | δ m ( Ξ τv ) wu | . k δΞ k ( β,κ,θ ) , X [ u,w ] ∈P n | τ − w | − κ | u − v | − θ | w − u | β . (3.8)We will upper bound the right hand side above. Invoking the fact that β > | w − u | =2 − n | t − s | , for u, w ∈ P n we write X [ u,w ] ∈P n | τ − w | − κ | u − v | − θ | w − u | β ≤ − n ( β − | t − s | β − X [ u,w ] ∈P n | τ − w | − κ | u − v | − θ | w − u | . (3.9)With the definition of Riemann sums in mind, the term X [ u,w ] ∈P n | τ − w | − κ | u − v | − θ | w − u | OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 13 in the right hand side of (3.9) can be dominated by the following finite integral (recall that κ + θ < Z ts | τ − x | − κ | x − v | − θ dx. In addition, some elementary calculations show that the above integral can be upper bounded asfollows, Z ts | τ − x | − κ | x − v | − θ dx . | τ − t | − κ | s − v | − θ | t − s | ∧ | t − v | − κ − θ . (3.10)Plugging the inequality (3.10) into (3.9), we thus get X [ u,w ] ∈P n | τ − w | − κ | u − v | − θ | w − u | β . − n ( β − (cid:16)h | τ − t | − κ | s − v | − θ | t − s | β i ∧ | τ − v | β − κ − θ (cid:17) . Then taking (3.8) into account, relation (3.7) can be recast as |I n +1 ( Ξ τv ) ts − I n ( Ξ τv ) ts | . − n ( β − k δΞ k ( β,κ,θ ) , (cid:16)h | τ − t | − κ | s − v | − θ | t − s | β i ∧ | τ − v | β − κ − θ (cid:17) . (3.11)Since β >
1, then (3.11) implies that the sequence {I n ( Ξ τv ); n ≥ } is Cauchy. It thus convergesto a quantity I ( Ξ τv ) ts which satisfies (3.5). The rest of this proof is the same as [11, Lemma 4.2],which means that the element I ( Ξ τv ) has finite k · k ( β,κ,θ ) , norm. The proof of relation (3.6) isvery similar to (3.5), and left to the reader for sake of conciseness. We just define an increment Ξ τ,τ ′ v instead of Ξ τv and then proceed as in (3.7)-(3.11). The proof is now complete. (cid:3) Third order convolution products in the rough case α − γ > . In this section weestablish a proper definition of third order convolution products. Let us first introduce the classof integrands we shall consider for those products.
Notation 3.3.
Similarly to Notation 2.16, we denote by u , , a function u : ∆ → L (( R d ) ⊗ , R d ) given by ( s, τ , τ , τ ) u τ ,τ ,τ s . To motivate the upcoming analysis and in order to get a better intuition of what is meantby third order convolution products, let us first give a definition of the third order convolutionproduct for smooth functions, and prove a useful relation for the construction of this convolution.
Definition 3.4.
Let x be a continuously differentiable function and consider a Volterra kernel k which fulfills Hypothesis 2.1 with γ < . Let also f : ∆ → L (( R m ) ⊗ , R m ) be a smooth functiongiven in Notation 3.3. Then recalling our Notation 2.13 for τ ≥ t > s ≥ v the convolution z ,τts ∗ f , , v is defined by z ,τts ∗ f , , v = Z t>r >s k ( τ, r ) dx r ⊗ Z r >r >s k ( r , r ) dx r ⊗ Z r >r >s k ( r , r ) dx r f r ,r ,r v . (3.12) Lemma 3.5.
Under the same conditions as in Definition 3.4, let z ,τts ∗ f , , s be the incrementgiven by (3.12) . Consider ( s, t ) ∈ ∆ and a generic partition P of [ s, t ] . Then we have z ,τts ∗ f , , s = lim |P|→ X [ u,v ] ∈P z ,τvu ∗ f , , s + (cid:0) δ u z ,τvs (cid:1) ∗ f , , s . (3.13) Proof.
Starting from expression (3.12), it is readily seen that z ,τts ∗ f , , s = X [ u,v ] ∈P Z v>r >u k ( τ, r ) dx r ⊗ Z r >r >s k ( r , r ) dx r ⊗ Z r >r >s k ( r , r ) dx r f r ,r ,r s . Then for each [ u, v ] ∈ P , divide the region { v > r > u } ∩ { r > r > r > s } into { v > r > r > r > u } ∪ { v > r > r > u > r > s } ∪ { v > r > u > r > r > s } . This yields a decomposition of z ,τts ∗ f , , s of the form z ,τts ∗ f , , s = X [ u,v ] ∈P A τvu + B τvu + C τvu , where A τvu , B τvu , and C τvu are respectively given by A τvu = Z v>r >u k ( τ, r ) dx r ⊗ Z r >r >u k ( r , r ) dx r ⊗ Z r >r >u k ( r , r ) dx r f r ,r ,r s B τvu = Z v>r >u k ( τ, r ) dx r ⊗ Z r >r >u k ( r , r ) dx r ⊗ Z u>r >s k ( r , r ) dx r f r ,r ,r s C τvu = Z v>r >u k ( τ, r ) dx r ⊗ Z u>r >s k ( r , r ) dx r ⊗ Z r >r >s k ( r , r ) dx r f r ,r ,r s . We recognize the term A τvu as the expression z ,τvu ∗ f , , s given by Definition 3.12. Moreover, wecan check that B τvu = z ,τvu ∗ z , · us ∗ f , , s , and C τvu = z ,τvu ∗ z , · us ∗ f , , s . Then since z ,τ satisfies (2.19),we have B τvu + C τvu = ( δ u z ,τvs ) ∗ f , , s . This finishes the proof of our claim (3.13). (cid:3) In order to generalize the notion of convolution product beyond the scope of Definition 3.4 toaccommodate rough signals x , let us introduce the kind of norm we shall consider for processeswith 3 upper variables of the form u , , , and in that connection introduce another Volterra-H¨olderspace equipped with this new norm. Definition 3.6.
Let W ( α,γ )3 denote the space of functions u : ∆ → L (( R d ) ⊗ , R d ) as given inNotation 3.3 with u τ ,τ ,τ = u ∈ L (( R d ) ⊗ , R d ) and such that k u , , k ( α,γ ) < ∞ , where the norm k u , , k ( α,γ ) is defined by (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) := (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , + (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , , . (3.14) More specifically, the k · k ( α,γ ) , and k · k ( α,γ ) , , , norms in (3.14) are respectively defined by (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , := sup ( s,t,τ ) ∈ ∆ | u τ,τ,τts | [ | τ − t | − γ | t − s | α ] ∧ | τ − s | ρ , (3.15) and (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , , := (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , + (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , + (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , . (3.16) In the right hand side of (3.16) , similarly to (2.23) - (2.24) , we have set k u , , k ( α,γ ) , , as the sum k u , , k ( α,γ ) , , ,> + k u , , k ( α,γ ) , , ,< , with (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , ,> = sup ( s,t,r,r ,r ,r ′ ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,α − γ ) | u r ′ ,r ,rts − u r ′ ,r ,rts | h η,ζ ( s, t, r , r , r, r ′ ) , (3.17) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 15 (cid:13)(cid:13) u , , (cid:13)(cid:13) ( α,γ ) , , ,< = sup ( s,t,r,r ′ ,r ,r ) ∈ ∆ η ∈ [0 , ,ζ ∈ [0 ,α − γ ) | u r ,r ′ ,rts − u r ,r ′ ,rts | h η,ζ ( s, t, r , r , r, r ′ ) . (3.18) Here we define h as follows: h η,ζ ( s, t, r , r , r, r ′ ) = | r − r | η | min( r , r , r, r ′ ) − t | − η + ζ × (cid:16)h | min( r , r , r, r ′ ) − t | − γ − ζ | t − s | α i ∧ | min( r , r , r, r ′ ) − s | α − γ − ζ (cid:17) . (3.19) Moreover, the norms k u , , k ( α,γ ) , , and k u , , k ( α,γ ) , , in (3.16) are defined similarly to rela-tions (3.17) - (3.18) .Remark . Notice that Definition 3.6 has been introduced so that the increments y u,u,u − y r,r,r can be controlled by (3.16). Indeed, we have for any η ∈ [0 ,
1] and ζ ∈ [0 , ρ ) | y u,u,uts − y r,r,rts | = | y u,u,uts − y u,r,rts + y u,r,rts − y r,r,rts | ≤ | y u,u,uts − y u,r,rts | + | y u,r,rts − y r,r,rts |≤ (cid:0) k y k ( α,γ ) , , + k y k ( α,γ ) , , (cid:1) | u − r | η | r − t | − η + ζ (cid:16)h | r − t | − γ − ζ | t − s | α i ∧ | r − s | ρ − ζ (cid:17) . k y k ( α,γ ) , , , | u − r | η | r − t | − η + ζ (cid:16)h | r − t | − γ − ζ | t − s | α i ∧ | r − s | ρ − ζ (cid:17) ≤ k y k ( α,γ ) , , , | u − r | η | r − t | − η + ζ | r − s | ρ − ζ . (3.20)Hence similarly to (2.27), we let η = ζ and we obtain | y u,u,uts − y r,r,rts | . k y k ( α,γ ) , , , . (3.21)Thanks to Hypothesis 2.15 and Definition 3.6, we can now state a general convolution productfor functions defined on ∆ . As mentioned above, it has to be seen as a generalization of Definition3.4 to a rough context. Theorem 3.8.
Let z ∈ V ( α,γ ) with α, γ ∈ (0 , satisfying ρ = α − γ > , as given in Definition 2.3.We assume that z fulfills Hypothesis 2.15 with n=3. Consider a function y : ∆ → L (( R m ) ⊗ , R m ) as given in Notation 3.3 such that k y , , k ( α,γ ) , , , < ∞ and y , , = y , where k y , , k ( α,γ ) , , , isdefined by (3.16) . Then with Notation 2.13 in mind, we have for all fixed ( s, t, τ ) ∈ ∆ that z ,τts ∗ y , , s = lim |P|→ X [ u,v ] ∈P z ,τvu y u,u,us + (cid:0) δ u z ,τvs (cid:1) ∗ y , , s . (3.22) is a well defined Volterra-Young integral. It follows that ∗ is a well defined bi-linear operationbetween the three parameters Volterra function z and a -parameter path y . Moreover, we havethat (cid:12)(cid:12) z ,τts ∗ y , , s − z ,τts y s,s,ss (cid:12)(cid:12) . k y , , k ( α,γ ) , , , (cid:0) k z k (3 ρ + γ,γ ) , + k z k ( α,γ ) , , k z k ( α,γ ) , + k z k ( α,γ ) , , k z k ( α,γ ) , (cid:1) (cid:0)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ (cid:1) . (3.23) Remark . Similarly to Remark 2.20, the term ( δ u z ,τvs ) ∗ y , , s is defined thanks to the fact that(according to relation (2.19)) δ u z ,τvs ∗ y , , s = z ,τvu ∗ z , · us ∗ y , , + z ,τvu ∗ z , · us ∗ y , , , (3.24)and the convolutions with respect to z ,τ , z ,τ in (3.24) are respectively defined by Theorem 2.14and Theorem 2.19. Proof of Theorem 3.8.
We first prove (3.22). To this aim, for a generic partition P of [ s, t ] letus denote by I P the approximation of the right hand side of (3.22). Specifically we set I P := P [ u,v ] ∈P ( Ξ τs ) vu , where ( Ξ τs ) vu = z ,τvu y u,u,us + (cid:0) δ u z ,τvs (cid:1) ∗ y , , s . (3.25)We now compute δ r ( Ξ τs ) vu in order to check that the extended Volterra sewing Lemma 3.2 can beapplied in our context. Recall that δ r ( Ξ τs ) vu = ( Ξ τs ) vu − ( Ξ τs ) vr − ( Ξ τs ) ru , for all τ > v > r > u > s. Moreover, we know from Hypothesis 2.15 that δ r z ,τvu = z ,τvr ∗ z , · ru + z ,τvr ∗ z , · ru . Therefore, a few elementary computations reveal that δ r (cid:0) z ,τvu y u,u,us (cid:1) = − z ,τvr ( y r,r,rs − y u,u,us ) + (cid:0) z ,τvr ∗ z , · ru + z ,τvr ∗ z , · ru (cid:1) y u,u,us (3.26) δ r (cid:0)(cid:0) δ u z ,τvs (cid:1) ∗ y , , s (cid:1) = − (cid:0) z ,τvr ∗ z , · ru + z ,τvr ∗ z , · ru (cid:1) ∗ y , , s , (3.27)Combining (3.26) and (3.27), we thus get δ r ( Ξ τs ) vu = − (cid:0) Q vru + Q vru + Q vru (cid:1) , (3.28)where the quantities Q vru , Q vru , Q vru are defined by Q vru = z ,τvr ( y r,r,rs − y u,u,us ) Q vru = z ,τvr ∗ z , · ru ∗ (cid:0) y , , s − y u,u,us (cid:1) Q vru = z ,τvr ∗ z , · ru ∗ (cid:0) y , , s − y u,u,us (cid:1) We will bound each of the above terms separately.Applying (3.20) with ζ = 0, and invoking the definition of k z k (3 ρ + γ,γ ) , in (2.2), and using that r ∈ [ u, v ] we have for any η ∈ [0 , (cid:12)(cid:12) Q vru (cid:12)(cid:12) . (cid:13)(cid:13) y , , (cid:13)(cid:13) ( α,γ ) , , , (cid:13)(cid:13) z (cid:13)(cid:13) (3 ρ + γ,γ ) , | u − s | − η | τ − v | − γ | v − u | ρ + γ + η , (3.29)We then choose η such that 3 ρ + γ + η > η + γ <
1, which is alwayspossible since ρ >
0, to obtain the desired regularity. For the term Q vru , we invoke the bound in(2.29), and observe that (cid:12)(cid:12) Q vru (cid:12)(cid:12) ≤ (cid:12)(cid:12) z ,τvr (cid:12)(cid:12) (cid:12)(cid:12) z ,rru ∗ ( y r,r, s − y u,u,us ) (cid:12)(cid:12) + k ˆ y k ( α,γ ) , , (cid:0) k z k α,γ ) + k z k (2 ρ + γ,γ ) (cid:1) | τ − v | − γ | v − u | ρ + γ ∧ | τ − u | ρ (3.30)where ˆ y l,wru = z , · ru ∗ ( y l,w, s − y u,u,us ), and we will need to find a bound for k ˆ y k ( α,γ ) , , . Note thatconvolution only happens in the first term of y l,w, s − y u,u,us . By (2.18) it follows that k ˆ y k ( α,γ ) , , . k z k ( α,γ ) , k y , , k ( α,γ ) , , | v − u | η | u − s | − η . Furthermore, from (2.17) it is readily checked that (cid:12)(cid:12) z ,rru ∗ ( y r, , s − y u,u,us ) (cid:12)(cid:12) . k z k ( α,γ ) , k y r, , − y u,u,us k ( α,γ ) , , | r − u | ρ We continue to investigate the first terms in (3.30). From the above regularity estimate it followsthat (cid:12)(cid:12) z ,τvr (cid:12)(cid:12) (cid:12)(cid:12) z , · ru ∗ (cid:0) y , , s − y u,u,us (cid:1)(cid:12)(cid:12) . (cid:13)(cid:13) z (cid:13)(cid:13) (2 ρ + γ,γ ) (cid:13)(cid:13) z (cid:13)(cid:13) ( α,γ ) k y k ( α,γ ) | τ − v | − γ | v − u | ρ + γ + η | u − s | − η . OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 17
Combining our estimates for the different terms on the right hand side of (3.30), we have that (cid:12)(cid:12) Q vru (cid:12)(cid:12) . (cid:13)(cid:13) y , , (cid:13)(cid:13) ( α,γ ) , , , (cid:13)(cid:13) z (cid:13)(cid:13) (2 ρ + γ,γ ) (cid:13)(cid:13) z (cid:13)(cid:13) ( α,γ ) | τ − v | − γ | v − u | ρ + γ + η | u − s | − η , (3.31)By similar computations as for the bound for Q , we obtain a bound for Q given by (cid:12)(cid:12) Q vru (cid:12)(cid:12) . (cid:13)(cid:13) y , , (cid:13)(cid:13) ( α,γ ) , , , (cid:13)(cid:13) z (cid:13)(cid:13) ( α,γ ) (cid:13)(cid:13) z (cid:13)(cid:13) (2 ρ + γ,γ ) | τ − v | − γ | v − u | ρ + γ + η | u − s | − η . (3.32)Plugging (3.29)-(3.32) into (3.28), we have thus obtained | δ r ( Ξ τs ) vu | . C y, z | τ − v | − γ | u − s | − η | v − u | ρ + γ + η , (3.33)where the constant C y, z used above is given explicitly as c y, z = (cid:13)(cid:13) y , , (cid:13)(cid:13) ( α,γ ) , , , (cid:16) k z k (3 ρ + γ,γ ) + 2 (cid:13)(cid:13) z (cid:13)(cid:13) (2 ρ + γ,γ ) (cid:13)(cid:13) z (cid:13)(cid:13) ( α,γ ) (cid:17) . Starting from (3.33), one can now check that k δΞ k (3 ρ + γ + η,γ,η ) , < ∞ , (3.34)where the norm in the left hand side of (3.34) is defined by (3.2). In the same way, we let thepatient reader check that k Ξ k (3 ρ + γ + η,γ,η ) , , < ∞ , where the k · k (3 ρ + γ + η,γ,η ) , , norm is introducedin (3.3). Since we have chosen η such that 3 ρ + γ + η > γ + η <
1, we can apply Lemma3.2 to the increment Ξ, which directly yields our claims (3.5) and (3.6). (cid:3)
Remark . The general convolution z ,τ ∗ y , , s is given in (3.22), for a path y defined on ∆ .If we wish to consider the convolution restricted to a path y , s defined on ∆ , a natural way toproceed is to define z ,τts ∗ y , s := z ,τts ∗ ˆ y , , , with ˆ y r ,r ,r = y r ,r . This means that the path ˆ y has no dependence in r . Therefore resorting to the notations (2.21)-(2.22), and (3.15)-(3.16), it is not difficult to check that (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,> = (cid:13)(cid:13) y , (cid:13)(cid:13) ( α,γ )1 , ,< , (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,< = 0 , (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,> = (cid:13)(cid:13) y , (cid:13)(cid:13) ( α,γ )1 , ,> , (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,< = 0 , (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,> = (cid:13)(cid:13) y , (cid:13)(cid:13) ( α,γ )1 , ,> , (cid:13)(cid:13) ˆ y , , (cid:13)(cid:13) ( α,γ ) , , ,< = (cid:13)(cid:13) y , (cid:13)(cid:13) ( α,γ )1 , ,< . Hence we have k ˆ y , , k ( α,γ ) , , , . k y , k ( α,γ ) , , , where k ˆ y , , k ( α,γ ) , , , is given in (3.16) and thenorm k y , k ( α,γ ) , , is introduced in (2.22). This will be invoked for our rough path constructionsin the upcoming section. Remark . In our applications to rough Volterra equations we will consider the case ρ = α − γ ∈ ( , ). Therefore it is sufficient to show that the convolution product ∗ can be performed on thethird level of a Volterra rough path.4. Stochastic calculus for Volterra rough paths
In this section we carry out some of the main steps leading to a proper differential calculus in aVolterra context. That is, we show how to integrate a Volterra controlled process in Section 4.1,and we solve Volterra type equations in Section 4.2.
Volterra controlled processes and rough Volterra integration.
We begin with aproper definition of rough Volterra integration in rough case α − γ > . As usual in roughintegration theory, one needs to specify a proper class of processes which can be integrated withrespect to the driving noise. As we will see, a non-geometric rough path type theory based ontree type expansions are needed, in order to construct a well defined rough Volterra integral. Wetherefore begin with some motivation for tree type expansions for iterated integrals.4.1.1. Tree expansions setting.
In Hypothesis 2.15, we have introduced the notion of a convo-lutional rough path z above z . While z satisfies the Chen type relation (2.19), it cannot beconsidered as a geometric rough path (see e.g. [11]). The reader might check for instance that fora path z ,τ given by the mapping ( t, τ ) R t k ( τ, r ) dx r , the component z ,τ will not satisfy thecomponent-wise relation (cid:0) z ,τts (cid:1) ii = 12 (cid:0) ( z ,τts ) i (cid:1) , i = 1 , · · · , d. Hence in order to define a rough path type calculus of order 2 related to z τ , we have to invoketechniques related to non geometric settings. The standard language in this kind of context isrelated to the Hopf algebra of trees. We give a brief account on those notions in the currentsection, referring to [2] for further details.Let T be the set of rooted trees with at most 3 vertices, whose vertices are decorated by labelsfrom the alphabet { , . . . , d } . A full description of the undecorated version of T is given by T = n , , , o . (4.1)In the sequel we will use the operation [ · ] on trees. Namely for σ , . . . , σ m ∈ T and a ∈ { , . . . , d } .we define σ = [ σ · · · σ k ] a as the tree for which σ , . . . , σ k are attached to a new root with label a .For instance in the unlabeled case we have[ 1 ] = [ ] = [ ] = [ ] = . It is thus readily checked that any tree in T can be constructed iteratively from smaller trees thanksto the operation [ · ]. Let us also mention that we always assume that the order of the branches ineach tree does not matter, in the sense that [ σ · · · σ m ] i = [ σ π · · · σ π m ] i for all permutations π of { , . . . , m } .The set T can be turned into a Hopf algebra when equipped with a suitable coproduct andantipode. This elegant structure is applied and discussed in detail in [2], but is not necessary in ourcontext. However, we shall use some of the notation contained in [2] for our future computations. Notation 4.1.
For any tree σ ∈ T , the quantity | σ | denotes the numbers of vertices in σ . Wecall the set F a forest consisting of elements with 2 vertices or less. Namely, F is defined by F = { , , } . Remark . Note that the operation [ · ] sends the set { } ∪ F into T .4.1.2. Tree indexed rough path and controlled processes.
We have already introduced the family { z j,τ , j = 1 , , } in Hypothesis 2.15. These objects will be identified with tree indexed objectsbelow. On top of this family, our computations will also hinge on an additional function called OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 19 z ,τ . Similarly to (3.12), whenever x is a continuously differentiable function and k satisfiesHypothesis 2.1, the increment z ,τ is defined by z ,τts = Z ts k ( τ, r ) (cid:18)Z rs k ( r, l ) dx l (cid:19) ⊗ (cid:18)Z rs k ( r, w ) dx w (cid:19) ⊗ dx r . (4.2)However, for a generic rough signal x we need some more abstract assumptions which are sum-marized below. Hypothesis 4.3.
Let z ∈ V ( α,γ ) be a Volterra path as given in Definition 2.3. Recall that α, γ satisfies ρ + γ > where ρ = α − γ . We assume the existence of a family z = { z σ,τ , σ ∈ T } suchthat z σ,τts ∈ ( R m ) ⊗| σ | . This family is defined by z ,τ = z ,τ , z ,τ = z ,τ , z ,τ = z ,τ , where z ,τ , z ,τ , z ,τ are introduced in Hypothesis 2.15. Moreover the increment z ,τ satisfies thealgebraic relation δ u z ,τts = 2 z ,τtu ∗ z , · us + z ,τtu ∗ ( z , · us ) ⊗ , (4.3) where the right hand side of (4.3) is defined in Proposition 2.14. Analytically, we require each z σ,τ to be an element of V ( | σ | ρ + γ,γ ) , and we define ||| z ||| ( α,γ ) := X σ ∈T k z σ k ( | σ | ρ + γ,γ ) . (4.4) Remark . Note that |||·||| does not define a seminorm of any sort, but is rather meant as aconvenient way to collect the seminorm terms concerning z σ for σ ∈ T .Together with the elements in Hypothesis 4.3, we will also make use of the family { z δ ; δ ∈ F } in the sequel. To this aim, let us now introduce the element z . Notation 4.5.
As stated in Hypothesis 4.3, we have set z ,τ = z ,τ . In addition, we also define z ,τ as z ,τts = Z ts k ( τ, r ) dx r Z ts k ( τ, l ) dx l = ( z ,τts ) ⊗ . (4.5) Therefore we can recast (4.3) as δ u z ,τts = 2 z ,τtu ∗ z , · us + z ,τtu ∗ z ,τts . (4.6)Assuming Hypothesis 4.3 holds, similarly to Theorem 3.8, we now give a convolution result for z ,τ . Theorem 4.6.
Let z ∈ V ( α,γ ) with α, γ ∈ (0 , satisfying ρ = α − γ > . We assume that theVolterra rough path z over z fulfills Hypothesis 4.3. Consider a function y : ∆ → L (( R m ) ⊗ , R m ) as given in Notation 3.3 such that k y , , k ( α,γ ) , , , < ∞ and y , , = y , where k y , , k ( α,γ ) , , , isdefined by (3.16) . Then with Notation 2.13 in mind, we have for all fixed ( s, t, τ ) ∈ ∆ that z ,τts ∗ y , , s = lim |P|→ X [ u,v ] ∈P z ,τvu y u,u,us + (cid:16) δ u z ,τvs (cid:17) ∗ y , , s . (4.7) is a well defined Volterra-Young integral. It follows that ∗ is a well defined bi-linear operationbetween the three parameters Volterra function z and a -parameter path y . Moreover, we havethat (cid:12)(cid:12)(cid:12) z ,τts ∗ y , , s − z ,τts y s,s,ss (cid:12)(cid:12)(cid:12) . (cid:13)(cid:13) y , , (cid:13)(cid:13) ( α,γ ) , , , ||| z ||| α,γ ) (cid:0)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ (cid:1) . (4.8) Proof.
The proof goes along the same lines as the proof of Theorem 3.8, and is omitted for sakeof conciseness. (cid:3)
We are now ready to introduce the natural class of processes one can integrate with respect to z , called Volterra controlled processes. Definition 4.7.
Let z ∈ V ( α,γ ) for some ρ = α − γ > , and consider a Volterra path y : ∆ → R d .We assume that there exists a family { y σ ; σ ∈ F } , with F as in Notation 4.1, such that thefollowing holds true. (i) y σ is a function from ∆ | σ | +2 to L (( R d ) ⊗| σ | , R d ) , and y σ has | σ | + 1 upper arguments. The initialconditions are respectively given by y ,p,q = y , y ,p,q,r = y , y ,p,q,r = y , for all ( r, q, p ) ∈ ∆ . (ii) The family { y σ ; σ ∈ F } is related to the increments of y τ in the following way: for ( s, t, τ ) ∈ ∆ we have y τts = z ,τts ∗ y ,τ, · s + z ,τts ∗ y ,τ, · , · s + z ,τts ∗ y ,τ, · , · s + R y,τts , (4.9) and y ,τ,pts = z ,τts ∗ ( y ,τ,p, · s + 2 y ,τ,p, · s ) + R ,τ,pts , (4.10) where y , y , ∈ V ( α,γ ) , R ∈ W (2 ρ +2 γ, γ )2 ( L ( R d )) and R y ∈ V (3 ρ +3 γ, γ ) ( R d ) (recall that V ( α,γ ) and W (2 ρ +2 γ, γ )2 are introduced in Definition 2.3 and Definition 2.17 respectively).Whenever y ≡ ( y, y , y , y ) satisfies relation (4.9) - (4.10) , we say that y is a Volterra path con-trolled by z (or simply controlled Volterra path) and we write y ∈ D ( α,γ ) z (∆ ; R m ) . We equip thisspace with a semi-norm k · k z , ( α,γ ) given by k y k z , ( α,γ ) = (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) = k y k ( α,γ ) + k y k ( α,γ ) + k R y k (3 ρ +3 γ, γ ) + k R k (2 ρ +2 γ, γ ) . (4.11) where the norms in (4.11) are respectively defined by (2.1) and (2.20) . Equipped with the norm y = (cid:16) y, y , y , y (cid:17)
7→ | y | + | y | + | y | + | y | + (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) , the space D ( α,γ ) z is a Banach space.Remark . It is easily seen from (4.9) and (4.10) that if y ∈ D ( α,γ ) z , then y, y ∈ V ( α,γ ) . Indeed,we observe directly from (4.10) that k y k ( α,γ ) . k z k ( α,γ ) ( | y | + | y , | + k y k ( α,γ ) + k y , k ( α,γ ) ) + k R k (2 ρ +2 γ, γ ) , where the quantities on the right hand side are finite by assumption. Furthermore, by relation(4.9) we then have that k y k ( α,γ ) ≤ k z k ( α,γ ) (cid:16) | y | + | y | + | y , | + k y k ( α,γ ) + k y k ( α,γ ) + k y , k ( α,γ ) + k R y k (3 ρ +3 γ, γ ) (cid:17) . (4.12) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 21
Remark . According to Definition 4.7, the function y is defined on ∆ and has two uppervariables, while y and y are defined on ∆ and have three upper arguments. Therefore in (4.11)the norm k y k ( α,γ ) has to be understood as a norm in W ( α,γ )2 , while the norms k y k ( α,γ ) and k y k ( α,γ ) have to be considered as norms in W ( α,γ )3 . The readers is referred to Definition 2.17 and 3.6 forthe definition of W ( α,γ )2 and W ( α,γ )3 respectively. We stick to the notation k · k ( α,γ ) for the norm onthose different spaces, for notational ease.4.1.3. Integration of controlled processes.
Our next step is to show that we may construct aVolterra rough integral in the rough case α − γ > , and then prove that the Volterra roughintegral of a controlled path with respect to a driving H¨older noise x ∈ C α is again a controlledVolterra path. Theorem 4.10.
For α ∈ (0 , , let x ∈ C α ([0 , T ]; R d ) and k be a Volterra kernel satisfyingHypothesis 2.1 with a parameter γ such that ρ = α − γ > . Define z τt = R t k ( τ, r ) dx r andassume there exists a tree indexed rough path z = { z σ,τ ; σ ∈ T } above z satisfying Hypothesis 4.3.Let M > be a constant such that ||| z ||| ( α,γ ) ≤ M . We now consider a controlled Volterra path y ∈ D ( α,γ ) z ( L ( R d )) , as introduced in Definition 4.7. Then the following holds true: (i) Define Ξ τvu := z ,τvu ∗ y · u + z ,τvu ∗ y , · , · u + z ,τvu ∗ y , · , · , · u + z ,τvu ∗ y , · , · , · u . The following limit exists for all ( s, t, τ ) ∈ ∆ , w τts = Z ts k ( τ, r ) dx r y rr := lim |P|→ X [ u,v ] ∈P Ξ τvu . (4.13)(ii) Let w be defined by (4.13) . There exists a positive constant C = C M,α,γ such that for all ( s, t, τ ) ∈ ∆ we have | w τts − Ξ τts | ≤ C (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) ||| z ||| ( α,γ ) (cid:0)(cid:2) | τ − t | − γ | t − s | ρ + γ (cid:3) ∧ | τ − s | ρ (cid:1) . (4.14)(iii) There exists a positive constant C = C M,α,γ such that for all ( s, t, p, q ) ∈ ∆ , η ∈ [0 , and ζ ∈ [0 , ρ ) we have | w qpts − Ξ qpvu | ≤ C (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) ||| z ||| ( α,γ ) × | p − q | η | q − t | − η + ζ (cid:16)h | q − t | − γ − ζ | t − s | ρ + γ i ∧ | q − s | ρ − ζ (cid:17) . (4.15)(iv) The triple w = ( w, w , w , is a controlled Volterra path in D ( α,γ ) z (∆ , R m ) , where we recallthat w is defined by (4.13) , and where w , w are respectively given by w ,τ,pt = y pt , and w ,τ,q,pt = y ,q,pt . Remark . From Theorem 4.10, we also can find a bound for k R w k (3 α, γ ) and k R w k (2 α, γ ) .Specifically, according to Theorem 4.10 (ii) we have k R w k (3 α, γ ) . (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) ||| z ||| ( α,γ ) (4.16) Moreover, thanks to Theorem 4.10 (iv) we have w ,τ,pt = y pt . Recalling (4.9) together with (4.16),we obtain (cid:13)(cid:13) R w (cid:13)(cid:13) (2 α, γ ) . (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) ||| z ||| ( α,γ ) (4.17) Proof of Theorem 4.10.
Let Ξ be given as in (i) . Thanks to Proposition 2.14, Theorem 2.19,Theorem 3.8, and Theorem 4.6, Ξ is well-defined. Our global strategy is to show that the Volterrasewing lemma can be applied to Ξ . In order to do so, let us compute δ m Ξ τvu for ( u, m, v, τ ) ∈ ∆ .Owing to (2.19), as well as some elementary properties of the operator δ , we get δ m ( Ξ τvu ) := A τvmu + B τvmu . (4.18)where the quantities A τvmu and B τvmu are given by A τvmu = − z ,τvm ∗ y · mu + z ,τvm ∗ y , · , · mu + z ,τvm ∗ y , · , · , · mu + z ,τvm ∗ y , · , · , · mu ! (4.19)and B τvmu = δ m z ,τvu ∗ y , · , · u + δ m z ,τvu ∗ y , · , · , · u + δ m z ,τvu ∗ y , · , · , · u . (4.20)Due to the assumption that ( y, y , y , y ) ∈ D ( α,γ ) z , we have that for any ( s, t, τ ) ∈ ∆ y · ts = z , · ts ∗ y , · , · s + z , · ts ∗ y , · , · , · s + z , · ts ∗ y , · , · , · s + R y, · ts , and y , · , · ts = z , · ts ∗ (cid:16) y , · , · , · s + 2 y , · , · , · s (cid:17) + R , · , · ts . Plugging the above two relations into (4.19), we obtain A τvmu = − z ,τvm ∗ z , · mu ∗ y , · , · u − z ,τvm ∗ z , · mu ∗ y , · , · , · u − z ,τvm ∗ z , · mu ∗ y , · , · , · u − z ,τvm ∗ R y, · mu − z ,τvm ∗ z , · mu ∗ y , · , · , · u − z ,τvm ∗ z , · mu ∗ y , · , · , · u − z ,τvm ∗ R , · , · mu (4.21) − z ,τvm ∗ y , · , · , · mu − z ,τvm ∗ y , · , · , · mu . Thanks to Hypothesis 2.15 and Hypothesis 4.3, plugging in the algebraic relations from (2.19) and(4.3) into (4.20), we have B τvum = z ,τvm ∗ z , · mu ∗ y , · , · u + z ,τvm ∗ z , · mu ∗ y , · , · , · u + z ,τvm ∗ z , · mu ∗ y , · , · , · u + 2 z ,τvm ∗ z , · mu ∗ y , · , · , · u + z ,τvm ∗ ( z , · mu ) ⊗ ∗ y , · , · , · u . (4.22)We now insert (4.21) and (4.22) into (4.18). Let us also recall that z , · mu = ( z , · mu ) ⊗ according to(4.5). Then some elementary algebraic manipulations and cancellations show that δ m ( Ξ τvu ) = − z ,τvm ∗ y , · , · , · mu − z ,τvm ∗ y , · , · , · mu − z ,τvm ∗ R , · , · mu − z ,τvm ∗ R y, · mu . (4.23)We now bound successively the 4 terms in the right hand side of (4.23). First we apply a smallvariant of (3.23) and (4.8), which takes into account the fact that increments of the form y mu and OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 23 y mu are considered. We also bound the terms involving y ,u,u,umu , y ,u,u,umu properly in (3.23) and (4.8).Resorting to (4.11), we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ,τvm ∗ y , · , · , · mu + z ,τvm ∗ y , · , · , · mu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (cid:16) k y k ( α,γ ) , , , + k y k ( α,γ ) , , , (cid:17) ||| z ||| ( α,γ ) | u − m | ρ | τ − m | − γ | v − m | ρ + γ . (4.24)where we recall that ||| z ||| ( α,γ ) was defined in (4.4). Next invoking the fact that R y ∈ V (3 ρ +3 γ, γ ) and Proposition 2.14, we obtain | z ,τvm ∗ R y, · mu | ≤ k R y k (3 ρ +3 γ, γ ) k z k ( α,γ ) || τ − v | − γ | u − m | ρ + γ . (4.25)Eventually, resorting to Theorem 2.19 and owing to the fact that R ∈ W (2 ρ +2 γ, γ )2 , we can checkthat (cid:12)(cid:12)(cid:12) z ,τvm ∗ R , · , · mu (cid:12)(cid:12)(cid:12) ≤ k R k (2 ρ +2 γ, γ ) , , ||| z ||| α,γ ) | τ − v | − γ | v − u | ρ + γ . (4.26)Plugging (4.24), (4.25) and (4.26) into (4.23), we have thus obtained | δ m Ξ τvu | . (cid:13)(cid:13)(cid:13)(cid:16) y, y , y , y (cid:17)(cid:13)(cid:13)(cid:13) z , ( α,γ ) ||| z ||| α,γ ) | τ − v | − γ | v − u | ρ + γ . (4.27)Recall that by assumption, ρ > , and therefore β ≡ ρ + γ >
1. We have thus obtainedthat k δΞ k ( β,γ ) , < ∞ . Following along the same lines above, it is readily checked that also k δΞ k ( β,γ ) , , < ∞ . Therefore we apply directly the Volterra sewing Lemma 2.10 in order toachieve the claims in (4.13), (4.14) and (4.15).We now proceed to prove the last claim, (iv) . To this aim, observe that the bound in (4.14)together with the fact that z , z ∈ V (3 ρ +3 γ, γ ) ( R d ), implies the existence of a function R ∈V (3 ρ +3 γ, γ ) ( R m ) such that w τts = z ,τts ∗ y · s + z ,τts ∗ y , · , · s + R τts . (4.28)From (4.28) it is readily seen that w τ can be decomposed as a controlled Volterra path in D ( α,γ ) z (∆ , R m ) with w ,τ,pt = y pt , w ,τ,q,pt = y ,q,pt , w ,τ,q,pt = 0. This finishes our proof. (cid:3) Remark . From Theorem 4.10 ( d ), we know that the process w defined by (4.13) satisfies w ,τ,pt = y pt = w ,pt . Therefore w depends on two variables instead of 3 variables in the general definition (4.9). In thesame way, we have w ,τ,q,pt = y ,q,pt = w ,q,pt , that is, w depends on three variables (vs 4 variables in the general definition (4.9)). Thereforewe can refine Theorem 4.10 and state that the Volterra rough integration sends ( y, y , y , y ) ∈D ( α,γ ) z ( R d ) to a controlled process ( w, w , w , ∈ ˆ D ( α,γ ) z ( R d ), where the space ˆ D ( α,γ ) z ( R d ) is definedby ˆ D ( α,γ ) z ( R d ) := n (cid:16) w, w , w , (cid:17) ∈ D ( α,γ ) z (∆ ; R d ) (cid:12)(cid:12) w ,τ,ps = w ,ps ,w ,τ,q,ps = w ,q,ps , and w ,τ,q,ps = 0 o . (4.29) The composition of a Volterra controlled processes with a smooth function.
With Remark4.12 in mind, we will now prove that one can compose processes in ˆ D ( α,γ ) z and still get a controlledprocess. Proposition 4.13.
Let f ∈ C b ( R d ; R m ) and assume ( y, y , y , ∈ ˆ D ( α,γ ) z ( R d ) as given in Re-mark 4.12. Also recall our Notation 2.13 for matrix products. Then the composition f ( y ) can beseen as a controlled path ( φ, φ , φ , φ ) , where φ = f ( y ) and where in the decomposition (4.9) wehave φ ,q,pt = y ,pt f ′ ( y qt ) , (4.30) φ ,r,q,pt = y ,q,pt f ′ ( y rt ) , and φ ,r,q,pt = 12 ( y ,qt ) ⊗ ( y ,pt ) f ′′ ( y rt ) . (4.31) Moreover, there exists a constant C = C α,γ, k f k C b > such that k ( φ, φ , φ , φ ) k z ;( α,γ ) ≤ C (1 + ||| z ||| ( α,γ ) ) h (cid:16) | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) ∨ (cid:16) | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) i . (4.32) Proof.
We separate this proof into two parts: in the first step we will find the appropriate expres-sion for φ , φ and φ ( namely (4.30) and (4.31)), as well as proving that ( φ, φ , φ , φ ) ∈ D ( α,γ ) z .In the second step, we will prove relation (4.32). Without loss of generality, we do the below anal-ysis component-wise for f ( y ) = ( f ( y ) , . . . , f m ( y )), where each f i : R d → R for i = 1 , . . . , m . Witha slight abuse of notation, we drop the subscript notation, and still just write f ( y ) representingeach component. Step 1: Expression for φ , φ and φ . An elementary application of Taylor’s formula enables usto decompose the increment f ( y τ ) ts into f ( y τ ) ts = y τts f ′ ( y τs ) + 12 ( y τts ) ⊗ f ′′ ( y τs ) + r τts , (4.33)where r τts = ( y τts ) ⊗ R f (3) ( c τts ( θ )) dθ , where c τts ( θ ) = θy τs + (1 − θ ) y τt . It is readily checkedfrom (4.33) that r ∈ V (3 α, γ ) . Indeed, it follows directly that k r k (3 α, γ ) , . k y k α,γ ) , k f k C b . Furthermore, for ( s, t, τ, τ ′ ) ∈ ∆ , we have | r τ ′ τts | ≤ | y τ ′ τts || ( y τ ′ ts ) ⊗ + ( y τts ) ⊗ |k f k C b + k y k α,γ ) , k f k C b ( | y τ ′ τs | + | y τ ′ τt | )( | τ − t | − γ | t − s | α ∧ | τ − s | ρ ) . It is simply checked that the following inequality hold: | y τ ′ τs | . | y | + k y k ( α,γ ) , , | τ ′ − τ | η [ | τ − s | η − ζ | s α | ∧ | s | ρ − ζ ] , and thus it follows that k r k (3 α, γ ) , , . ( k y k ( α,γ ) , , k y k α,γ ) , + k y k α,γ ) , ( | y | + k y k ( α,γ ) , , )) k f k C b . Combining the above estimates, we get k r k (3 α, γ ) . ( | y | + k y k ( α,γ ) ) k f k C b . (4.34) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 25
Now observe that ( y, y , y , ∈ ˆ D ( α,γ ) z where ˆ D ( α,γ ) z is the subset of the space D ( α,γ ) z as de-fined in Remark 4.12. In particular, y, y , y satisfy relation (4.9). Then taking squares in therelation (4.9), we end up with ( y τts ) ⊗ = ( z ,τts ∗ y ,τ, · s ) ⊗ + ˜ r τts , (4.35)where the reminder term ˜ r τts is defined by˜ r τts = ( z ,τts ∗ y ,τ, · , · s ) ⊗ + ( R y,τts ) ⊗ + ( z ,τts ∗ y ,τ, · , · s ) ⊗ ( z ,τts ∗ y ,τ, · s ) + ( z ,τts ∗ y ,τ, · s ) ⊗ ( z ,τts ∗ y ,τ, · , · s )+ ( z ,τts ∗ y ,τ, · s ) ⊗ R y,τts + R y,τts ⊗ ( z ,τts ∗ y ,τ, · s ) + ( z ,τts ∗ y ,τ, · , · s ) ⊗ R y,τts + R y,τts ⊗ ( z ,τts ∗ y ,τ, · , · s ) . (4.36)Plugging (4.35) into (4.33), we get f ( y τ ) ts = y τts f ′ ( y τs ) + 12 ( z ,τts ∗ y , · s ) ⊗ f ′′ ( y τs ) + 12 ˜ r τts f ′′ ( y τs ) + r τts , We now invoke (4.9) in order to further decompose y τts above. We end up with f ( y τ ) ts = z ,τts ∗ y , · s f ′ ( y τs ) + z ,τts ∗ y , · , · s f ′ ( y τs ) + 12 ( z ,τts ∗ y , · s ) ⊗ f ′′ ( y τs ) + R φ,τts , (4.37)where the reminder R φ is defined by R φ,τts = R y,τts f ′ ( y τs ) + 12 ˜ r τts f ′′ ( y τs ) + r τts . (4.38)Thanks to (4.37) and the definition of the relations in (4.30)-(4.31), the proof of ( φ, φ , φ , φ ) ∈D ( α,γ ) z are now reduced to proving the following two claims: • Claim 1: The remainder term R φ,τts in (4.38) is of order 3. Specifically, due to (4.33) andthe fact that ( y, y , y , ∈ ˆ D ( α,γ ) , relation (4.38) this is reduced to the following claim:˜ r ∈ V (3 ρ +3 γ, γ ) . (4.39) • Claim 2: φ fulfills relation (4.10), which can be written as φ , · , · − z , · ∗ ( φ , · , · , · + 2 φ , · , · , · ) ∈ W (2 ρ +2 γ, γ )2 , (4.40)where we recall that φ , φ , φ are defined by (4.30)-(4.31).In the following, we will prove those two Claims separately. Proof of Claim 1.
According to relation (4.36), there are eight terms to evaluate in ˜ r . Forconciseness, we can consider one of these terms, say the increment I τts defined by I τts = ( z ,τts ∗ y ,τ, · , · s ) ⊗ ( z ,τts ∗ y ,τ, · s ), and the remaining terms will follow directly from similar considerations. Tothis aim, a first observation is that since ( y, y , y , ∈ ˆ D ( α,γ ) z as given in (4.29), then both y and y don’t dependent on τ and we have I τts = ( z ,τts ∗ y , · , · s ) ⊗ ( z ,τts ∗ y , · s ). Moreover, due to the factthat I is part of the reminder ˜ r , we have to evaluate k I k (3 ρ +2 γ, γ ) . Owing to Definition 2.3, this isequivalent to evaluate k I k (3 ρ +2 γ, γ ) = k I k (3 ρ +2 γ, γ ) , + k I k (3 ρ +2 γ, γ ) , , . (4.41) In order to upper bound the right hand side of (4.41), it suffices to estimate | I τts | and | I qpts | . Someelementary computations reveal that | I τts | = (cid:12)(cid:12)(cid:12)(cid:16) z ,τts ∗ y , · , · s (cid:17) ⊗ ( z ,τts ∗ y , · s ) (cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12) z ,τts ∗ y , · , · s (cid:12)(cid:12)(cid:12) | z ,τts ∗ y , · s | , (4.42)and | I qpts | = | I qts − I pts | = (cid:12)(cid:12)(cid:12)(cid:16) z ,qts ∗ y , · , · s (cid:17) ⊗ ( z ,qts ∗ y , · s ) − (cid:16) z ,pts ∗ y , · , · s (cid:17) ⊗ ( z ,pts ∗ y , · s ) (cid:12)(cid:12)(cid:12) . | z ,qpts ∗ y , · s | (cid:12)(cid:12)(cid:12) z ,qts ∗ y , · , · s (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) z ,qpts ∗ y , · , · s (cid:12)(cid:12)(cid:12) | z ,qts ∗ y , · s | . (4.43)To bound the right hand side of (4.42), thanks to a slight variation of Proposition 2.14 andTheorem 2.19, we have | I τts | . ( | y | + | y | + k ( y, y , y , k z , ( α,γ ) ) ||| z ||| α,γ ) (cid:0)(cid:2) | τ − t | − γ | t − s | ρ +2 γ (cid:3) ∧ | τ − s | ρ (cid:1) . Similarly, we can bound | I qpts | is the following way for all β ∈ [0 ,
1] and ζ ∈ [0 , ρ ): | I qpts | . ( | y | + | y | + k ( y, y , y , k z , ( α,γ ) ) ||| z ||| α,γ ) × | q − p | β | p − t | − β + ζ (cid:16)h | p − t | − γ − ζ | t − s | ρ +2 γ i ∧ | p − s | ρ − ζ (cid:17) . (4.44)It follows by definition of the quantities k I k (3 ρ +2 γ, γ ) , and k I k (3 ρ +2 γ, γ ) , , as given in Definition 2.3,that k I k (3 ρ +2 γ, γ ) , ∨ k I k (3 ρ +2 γ, γ ) , , . ( | y | + | y | + k ( y, y , y , k ( α,γ ) ) ||| z ||| z , ( α,γ ) (4.45)which implies I ∈ V (3 ρ +3 γ, γ ) according to Lemma 2.7. Similarly, we let the patient reader checkthat( z ,τts ∗ y ,τ, · s ) ⊗ ( z ,τts ∗ y ,τ, · , · s ) ∈ V (3 ρ +2 γ, γ ) , ( z ,τts ∗ y , · , · s ) ⊗ ∈ V (4 ρ +2 γ, γ ) , ( R yts ) ⊗ ∈ V (6 ρ +6 γ, γ ) , (4.46)as well as ( z ,τts ∗ y , · s ) ⊗ R yts ∈ V (4 ρ +4 γ, γ ) , ( z ,τts ∗ y , · , · s ) ⊗ R yts ∈ V (5 ρ +4 γ, γ ) . (4.47)In fact the appropriate norm for each of these terms is easily seen to be bounded by the product( ||| z ||| ( α,γ ) + ||| z ||| α,γ ) )( | y | + | y | + k ( y, y , y , k ( α,γ ) ) . Combining (4.45), (4.46) and (4.47), we havethus obtained that ˜ r ∈ V (3 ρ +3 γ, γ ) , and it follows that k ˜ r k (3 ρ +3 γ,γ ) . ( ||| z ||| ( α,γ ) + ||| z ||| α,γ ) )( | y | + | y | + k ( y, y , y , k z , ( α,γ ) ) (4.48) Proof of Claim 2.
Before proving relation (4.40), we will give some algebraic insight on the termsof φ σ for σ ∈ F . Indeed, resorting to (4.37), we can safely set φ ,q,pt = y ,pt f ′ ( y qt ) , (4.49)as stated in (4.30). According to (4.37), we also let φ ,r,q,pt = y ,q,pt f ′ ( y rt ) , φ ,r,q,pt = 12 ( y ,qt ) ⊗ ( y ,pt ) f ′′ ( y rt ) . (4.50)With relation (4.9) in mind, we can rewrite (4.37) as f ( y τt ) − f ( y τs ) = z ,τts ∗ φ ,τ, · s + z ,τts ∗ φ ,τ, · , · s + ( z ,τts ) ⊗ ∗ φ ,τ, · , · s + R φ,τts (4.51) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 27
Let us briefly give a few details regarding the expressions on the right hand side of (4.51). Specif-ically, we will explain how to compute z ,τts ∗ φ ,τ, · s = z ,τts ∗ y , · s f ′ ( y τs ). Referring to Notation 2.13,the expression z ,τts ∗ y , · s f ′ ( y τs ) can be rewritten as [( z ,τts ) ⊺ ∗ ( y , · s f ′ ( y τs )) ⊺ ] ⊺ = f ′ ( y τs ) y , · s ∗ z ,τts , wherewe have used also that y , · s f ′ ( y τs ) can be rewritten as [( y , · s ) ⊺ ( f ′ ( y τs )) ⊺ ] ⊺ = f ′ ( y τs ) y , · s . In addition,notice that f ′ ( y s ) ∈ R d , y s ∈ L ( R d , R d ) and z ,τts ∈ R d . Therefore the quantity z ,τts ∗ y , · s f ′ ( y τs ) hasto be interpreted as an inner product, and we let the patient reader perform the same kind ofmanipulation for the term z ,τts ∗ y f ′ ( y τs ). In the end we get that both the left hand side and theright hand side of (4.51) are real-valued.Now we are ready to prove (4.40). To this aim we set J τ, · ts := φ ,τ, · ts − z ,τts ∗ ( φ ,τ, · , · s + 2 φ ,τ, · , · s ) . (4.52)Our claim (4.40) amounts to show that J ∈ W (2 ρ +2 γ,γ )2 , with W (2 ρ +2 γ, γ )2 given in Definition 2.17.Thanks to (4.49) and (4.50), we first write J τ, · ts = y , · t ( f ′ ( y τt ) − f ′ ( y τs )) + y , · ts f ′ ( y τs ) − z ,τts ∗ y , · , · s f ′ ( y τs ) − z ,τts ∗ y , · s ⊗ y , · s f ′′ ( y τs ) . (4.53)We now invoke (4.10), recalling that y = 0 since we have assumed that y ∈ ˆ D ( α,γ ) z . Plugging thisinformation in (4.53), we end up with J τ,τts = y ,τt ( f ′ ( y τt ) − f ′ ( y τs )) + R ,τts f ′ ( y τs ) − z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs ) . (4.54)Let us apply a Taylor expansion to the first term of right hand side of (4.54). Specifically we write f ′ ( y τt ) − f ′ ( y τs ) − y τts f ′′ ( y τs ) = F (2) ,τts , where the term F (2) ,τts = ( F (2) ,τ, ts , . . . , F (2) ,τ,dts ) is defined as a reminder in a Taylor expansion.Namely consider multi-indices β = ( β , . . . , β d ) with β i ∈ { , , } . We set | β | = P dj =1 β j and | β | ! = Π dj =1 β j !. Then for i = 1 , . . . , d , F (2) ,τ,its is given by F (2) ,τ,its = 2 X | β | =2 ( y τts ) ⊗| β | β ! Z (1 − r ) ∂ β ( ∂ i f ( y τs + ry τst )) dr. (4.55)With expression (4.55) in hand and recalling (4.54), we thus get J τ,τts = y ,τt ( f ′ ( y τt ) − f ′ ( y τs ) − y τts f ′′ ( y τs )) + y ,τt y τts f ′′ ( y τs ) − z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs ) + R ,τts f ′ ( y τs )= y ,τt F (2) ,τts + R ,τts f ′ ( y τs ) + y ,τt y τts f ′′ ( y τs ) − z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs ) , (4.56)Furthermore, we plug in identity (4.9) in the above expansion in order to expand the term y ,τt y τts f ′′ ( y τs ) in (4.56). This yields J τ,τts = y ,τt F (2) ,τts + R ,τts f ′ ( y τs ) + y ,τt z ,τts ∗ y , · , · s f ′′ ( y τs ) + y ,τt R τts f ′′ ( y τs )+ y ,τt z ,τts ∗ y , · s f ′′ ( y τs ) − z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs ) . (4.57)Next we resort to the forthcoming identity (4.70) in order to handle the term z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs )above. One obtains that the last two terms in (4.57) combine into one term y ,τts z ,τts ∗ y , · s f ′′ ( y τs ).We end up with J τ,τts = y ,τt F (2) ,τts + R ,τts f ′ ( y τs ) + y ,τt z ,τts ∗ y , · , · s f ′′ ( y τs )+ y ,τt R τts f ′′ ( y τs ) + y ,τts z ,τts ∗ y , · s f ′′ ( y τs ) . (4.58) In the same way, we let the patient reader check that we can rewrite J q,qts − J p,pts as J q,qts − J p,pts = J qp + J qp + J qp + J qp + J qp , (4.59)where the terms J qp , J qp , J qp , J qp , and J qp are defined respectively by J qp = y ,qpt F (2) ,qts + y ,pt (cid:16) F (2) ,qts − F (2) ,pts (cid:17) J qp = (cid:16) y ,pt z ,qpts ∗ y , · , · s + y ,qpt z ,qts ∗ y , · , · (cid:17) f ′′ ( y qs ) + (cid:16) y ,pt z ,pts ∗ y , · , · s (cid:17) ( f ′′ ( y qs ) − f ′′ ( y ps )) J qp = ( y ,pt R qpts + y ,qpt R qts ) f ′′ ( y qs ) + y ,pt R pts ( f ′′ ( y qs ) − f ′′ ( y ps )) , (4.60) J qp = ( y ,qpts z ,qts ∗ y , · s + y ,pts z ,qpts ∗ y , · s ) f ′′ ( y qs ) + y ,pts z ,pts ∗ y , · s ( f ′′ ( y qs ) − f ′′ ( y ps )) J qp = R ,qpts f ′ ( y qs ) + R ,pts ( f ′ ( y qs ) − f ′ ( y ps )) . With (4.58)-(4.60) at hand, and recalling Definition 2.17 for the spaces W , it is readily checked,using the information of the regularities in the different terms of J i for i = 1 , . . . , J ∈W (2 ρ +2 γ, γ )2 . We omit further details, as the arguments follows directly along the same lines as inprevious computations in the proof of claim 1.Summarizing our analysis so far, we have now proved both Claim 1 and Claim 2 above. Thereforewe obtain that ( φ, φ , φ , φ ) is an element of D ( α,γ ) z . Step 2: Proof of relation (4.32) . According to the definition (4.11) for the norm in D ( α,γ ) z , we have k ( φ, φ , φ , φ ) k z ;( α,γ ) = k φ k ( α,γ ) + k φ k ( α,γ ) + k R φ k (3 ρ +3 γ, γ ) + k R φ k (2 ρ +2 γ, γ ) . (4.61)In the following, we will bound four terms in the right hand side of (4.61) separately.We begin to handle the term k φ k ( α,γ ) in (4.61). We recall that φ is given by (4.31), and its( α, γ )-norm is introduced in Definition 3.6. According to this definition, it is thus enough to bound k φ k ( α,γ ) , and k φ k ( α,γ ) , , , . Towards this aim, we write (cid:12)(cid:12)(cid:12) φ ,τ,τ,τts (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) y ,τ,τt f ′ ( y τt ) − y ,τ,τs f ′ ( y τs ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) y ,τ,τt ( f ′ ( y τt ) − f ′ ( y τs )) + y ,τ,τts f ′ ( y τs ) (cid:12)(cid:12)(cid:12) . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) (cid:0)(cid:2) | τ − t | − γ | t − s | α (cid:3) ∧ | τ − s | ρ (cid:1) . (4.62)This yields k φ k ( α,γ ) , . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) . (4.63)We now wish to handle the norm k φ k ( α,γ ) , , , in (3.14). Otherwise stated, we wish to bound theterms in the right hand side of (3.16) for φ . For the term k φ k ( α,γ ) , , ,> , we thus write (cid:12)(cid:12)(cid:12) φ ,p ′ ,p ,pts − φ ,p ′ ,p ,pts (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) y ,p ,pt f ′ ( y p ′ t ) − y ,p ,ps f ′ ( y p ′ s ) − y ,p ,pt f ′ ( y p ′ t ) + y ,p ,ps f ′ ( y p ′ s ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:16) y ,p ,pts − y ,p ,pts (cid:17) f ′ ( y p ′ t ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) y ,p ,ps − y ,p ,ps (cid:17) (cid:16) f ′ ( y p ′ t ) − f ′ ( y p ′ s ) (cid:17)(cid:12)(cid:12)(cid:12) . In addition, owing to Remark 4.9 and (3.19) and since y ∈ ˆ D ( α,γ ) z , we have y ∈ W ( α,γ )2 . Due tothe fact that y is also an element of V ( α,γ ) according to Remark 4.8, for any η ∈ [0 ,
1] and ζ < ρ
OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 29 we get (cid:12)(cid:12)(cid:12) φ ,p ′ ,p ,pts − φ ,p ′ ,p ,pts (cid:12)(cid:12)(cid:12) . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) × | p − p | η | p − t | − η + ζ (cid:16)h | p − t | − γ − ζ | t − s | α i ∧ | p − s | ρ − ζ (cid:17) , (4.64)and thus k φ k ( α,γ ) , , . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) . (4.65)Moreover, it is easily seen that k φ k ( α,γ ) , , and k φ k ( α,γ ) , , are bounded exactly in the same wayas (4.65). Hence we get the following bound for k φ k ( α,γ ) , , , : k φ k ( α,γ ) , , , . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) . (4.66)Eventually, plugging (4.63) and (4.66) into (3.14), we obtain the desired bound for k φ k ( α,γ ) : k φ k ( α,γ ) . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) . (4.67)We let the reader check that the term k φ k ( α,γ ) in (4.61) can be treated in a similar way. Indeed, φ has to be considered as a process in W , exactly like φ . Therefore owing to the definition(4.31) of φ and to the definition (3.16) of the (1 , , W , we get the following boundalong the same lines as (4.62)-(4.67): k φ k ( α,γ ) . k f k C b ( | y | + k y k ( α,γ ) , + k y k ( α,γ ) , , ) . (4.68)We are now ready to bound the fourth term k R φ k (2 ρ +2 γ, γ ) in the right hand side of (4.61). Tothis aim, recall that according to (4.10) we have R ,τ,pts = y ,τ,pts − z ,τts ∗ (cid:16) y ,τ,p, · s + 2 y ,τ,p, · s (cid:17) . Comparing this expression to (4.52), we get R φ = J . Now recall that J has been analyzedthrough a decomposition in (4.58)-(4.60). Note that all the terms appearing in the decompositionare directly bounded due to the fact that ( y, y , y , ∈ ˆ D ( α,γ ) z . It is therefore readily checked that k R φ k (2 ρ +2 γ, γ ) ≤ C (1 + ||| z ||| ( α,γ ) ) h (cid:16) | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) ∨ (cid:16) | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) i . Eventually, we handle the term k R φ k (3 ρ +3 γ, γ ) in (4.61). Recall that R φ is given by (4.38), andthat we have already bounded the term r and ˜ r in (4.34) and (4.48) respectively. Furthermore, itfollows directly that k R y k (3 ρ +3 γ, γ ) ≤ k ( y, y , y , k z , ( α,γ ) . Combining the above considerations, wesee that k R φ k (3 ρ +3 γ, γ ) . (1 + ||| z ||| ( α,γ ) ) h (cid:16) | y | + | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) ∨ (cid:16) | y | + | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) i . Gathering the bounds found above, it is now evident that k ( φ, φ , φ , φ ) k z ;( α,γ ) . (cid:16) ||| z ||| ( α,γ ) (cid:17) (cid:20) (cid:16) | y | + | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) ∨ (cid:16) | y | + | y | + | y | + k ( y, y , y , k z , ( α,γ ) (cid:17) (cid:21) , where the hidden constant depends on k f k C b , α , and γ . The above relation is exactly (4.32),which concludes our proof. (cid:3) Remark . In Proposition 4.13 we have obtained useful bounds on the composition map fromˆ D ( α,γ ) z (∆ ([0 , T ]); R d ) to D ( α,γ ) z (∆ ([0 , T ]); R m ). Let us now choose a parameter β such that β < α and we still have β − γ > . We will in the next section onsider the composition map fromˆ D ( β,γ ) z (∆ ([0 , T ]); R d ) to D ( β,γ ) z (∆ ([0 , T ]); R m ). Due to Remark 2.6, it is readily checked thatthere exists a constant C = C M,α,β,γ, k f k C b such that, k ( φ, φ , φ , φ ) k z ;( β,γ ) ≤ C (cid:0) k z k ( α,γ ) (cid:1) (cid:16)h | y | + | y | + k ( y, y , y , k z , ( β,γ ) i(cid:17) ∨ (cid:18)h | y | + | y | + k ( y, y , y , k z , ( β,γ ) i (cid:19) T α − β . (4.69)We close this section by presenting a technical result which leads to some useful cancellationsin the rough path expansion (4.57). Lemma 4.15.
Let f ∈ C b ( R d ) and assume ( y, y , y , ∈ ˆ D ( α,γ ) z ( R d ) as given in Remark 4.12.Also recall our Notation 2.13 for matrix products. Then for any ( s, t, τ ) ∈ ∆ , we have y ,τs z ,τts ∗ y , · s f ′′ ( y τs ) = z ,τts ∗ y , · s ⊗ y ,τs f ′′ ( y τs ) (4.70) Proof.
Let L (respectively M) be the left hand side (respectively right hand side) of (4.70) . Re-calling the dimension considerations after equation (4.51), notice that both L and M are elementsof R d × d . For a ∈ R d , we consider the matrix products aL and aM in the sense of Notation 2.13.In particular, our Notation 2.13 implies that aL has to be interpreted as f ′′ ( y τs ) y , · s ∗ z ,τts y ,τs a .Expressing this in coordinates we get aL = m X i,j =1 f ′′ ( y τs ) ij m X i =1 y , · ,ii s z ,τ,i ts m X j =1 y ,τ,jj s a j = m X i,j,i ,j =1 f ′′ ( y τs ) ij y , · ,ii s z ,τ,i ts y ,τ,jj s a j . (4.71)Similarly, the product aM can be expressed as aM = f ′′ ( y τs ) ij y ,τs ⊗ y ,τs ∗ z , · ts · a = m X i,j,i ,j =1 f ′′ ( y τs ) ij y , · ,ii s y ,τ,jj s z ,τ,i ts a j . (4.72)Comparing (4.71) and (4.72), it is clear that aL = aM for any a ∈ R d . Thus L = M , whichfinishes the proof. (cid:3) OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 31
Rough Volterra Equations.
In this section we gather all the element of stochastic calculusput forward in Sections 3.2-4.1, in order to achieve one of main goals in this paper. Namely wewill solve Volterra type equations in a very rough setting.We start by introducing a new piece of notation.
Notation 4.16.
Let us define a new space D ( β,γ ) z ; y (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R d (cid:1) , where y is of the form ( y , y , y , y ) . For ≤ a < b ≤ T we define a simplex type set ∆ T ([ a, b ]) as follows, ∆ T ([ a, b ]) = (cid:8) ( s, τ ) ∈ [ a, b ] × [0 , T ] (cid:12)(cid:12) a ≤ s ≤ τ ≤ T (cid:9) . (4.73) Note that the first component of ( s, τ ) ∈ ∆ T ([ a, b ]) is restricted to [ a, b ] while the second componentis allowed to vary in the whole interval [0 , T ] . Without loss of generality, we assume that k z k ( α,γ ) ≤ M ∈ R + . As in Remark 4.14, we choose a parameter β < α but still satisfying β − γ > / . Letus also consider a time horizon ¯ T ≤ T (this ¯ T will be made small enough to perform a contractionargument later on). We will work on a space D ( β,γ ) z , y (∆ T ([0 , ¯ T ]); R m ) defined by D ( β,γ ) z , y (∆ T ([0 , ¯ T ]); R m ) = n (cid:16) y, y , y , y (cid:17) ∈ D ( β,γ ) z (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R m (cid:1) (cid:12)(cid:12)(cid:12) y = { y τ , y ,τ , y ,τ,τ , y ,τ,τ } = { y , y , y , y } o . (4.74) Notice that the norm on D ( β,γ ) z , y is still defined by (4.11) . The only difference between D ( β,γ ) z , y and D ( β,γ ) z in Definition 4.7 is that D ( β,γ ) z , y has an affine space structure, in contrast with the Banachspace nature of D ( β,γ ) z . We are now ready to solve Volterra type equations in the rough case α − γ > . Theorem 4.17.
Consider a path x ∈ C α ([0 , T ]; R d ) , and let k : ∆ → R be a Volterra kernel oforder γ , with α − γ > . Define z ∈ V ( α,γ ) (∆ ; R d ) by z τt = R t k ( τ, r ) dx r and assume there exists atree indexed Volterra rough path z = { z σ,τ ; σ ∈ T } above z satisfying Hypothesis 4.3. Additionally,suppose f ∈ C b ( R m ; L ( R d ; R m )) . Then there exists a unique solution in D ( α,γ ) z ( R m ) to the Volterraequation y τt = y + Z t k ( τ, r ) dx r f ( y rr ) , ( t, τ ) ∈ ∆ ([0 , T ]) , y ∈ R m , (4.75) where the integral is understood as a rough Volterra integral according to Theorem 4.10.Proof. We will proceed in a classical way by (i) Establishing a fixed point argument on a smallinterval. (ii) Patching the solutions obtained on the small intervals. Since this procedure isstandard, we will skip some details.We wish to solve (4.75) in a class of controlled processes. This means that the right hand side of(4.75) has to be understood according to Theorem 4.10. In particular referring to Theorem 4.10 (iv) , the controlled process y will be of the form y = { y, y , y , } . In the remainder of the proof, wewill consider a controlled path y ∈ D ( β,γ ) z ; y (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R m (cid:1) as given in (4.74), that is a controlledprocesses y starting from an initial value y = ( y , f ( y ) , f ( y ) f ′ ( y ) , β such that β < α, and β − γ > . (4.76) In addition, we introduce a mapping M ¯ T : D ( β,γ ) z , y (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R m (cid:1) → D ( β,γ ) z , y (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R m (cid:1) , (4.77)such that for all (cid:16) y, y , y , (cid:17) ∈ D ( β,γ ) z , y ( R m ), we have M ¯ T (cid:16) y, y , y , (cid:17) τt = (cid:26)(cid:18) y + Z t k ( τ, r ) dx r f ( y rr ) , f ( y τt ) , f ( y τt )) f ′ ( y τt ) , (cid:19) (cid:12)(cid:12)(cid:12) ( t, τ ) ∈ ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1)(cid:27) . (4.78)We are now ready to implement the first piece (i) of the general strategy described above. Step 1: Invariant ball on a small interval.
In this step, our goal is to show that there exists aball of radius 1 in D ( β,γ ) z , y (∆ T ([0 , ¯ T ]); R m ) which is left invariant by M ¯ T provided that ¯ T is smallenough. To this aim, we introduce some additional notation. Namely for y as in (4.78) we definea controlled process w in the following way:( s, t, τ ) w τts = (cid:16) w τts , w ,τts , w ,τts , (cid:17) = M ¯ T (cid:16) y, y , y , (cid:17) τts , (4.79)where we recall that M ¯ T is defined by (4.78). Next consider the unit ball B ¯ T within the space D ( β,γ ) z , y (∆ T ([0 , ¯ T ]); R m ), defined by B ¯ T = n(cid:16) y, y , y , (cid:17) ∈ D ( β,γ ) z , y (cid:0) ∆ T (cid:0)(cid:2) , ¯ T (cid:3)(cid:1) ; R m (cid:1) (cid:12)(cid:12)(cid:12) k ( y, y , y , k z , ( β,γ ) ≤ o . (4.80)In order to bound the process defined by (4.79), notice that M ¯ T is given as the Volterra typeintegral of φ = f ( y ). Hence according to (4.69) there exists a constant C such that k ( φ, φ , φ , φ ) k z , ( β,γ ) . (cid:0) k z k ( α,γ ) (cid:1) (cid:0) Q (cid:1) ¯ T α − β , (4.81)where we have set Q = | f ( y ) | + | f ( y ) f ′ ( y ) | + k ( y, y , y , k z , ( β,γ ) . (4.82)In addition, our process w is defined in (4.79) as w τts = Z ts k ( τ, r ) dx r φ rr . Thus an easy extension of (4.14)-(4.17) to a process φ ∈ D ( β,γ ) z with β satisfying (4.76) yields k w k z , ( β,γ ) ≤ C k ( φ, φ , φ , φ ) k z , ( β,γ ) k z k ( α,γ ) ≤ C (cid:0) k z k ( α,γ ) (cid:1) (cid:0) Q (cid:1) ¯ T α − β , (4.83)for a universal constant which can change from line to line. Furthermore, since we have assumedthat k z k ( α,γ ) ≤ M , one can recast (4.83) as k w k z , ( β,γ ) ≤ C (cid:0) M (cid:1) (cid:0) Q (cid:1) ¯ T α − β . (4.84)Considering ¯ T ≤ ( C (1 + M ) (1 + Q )) α − β and back to our definition (4.79), it is now easily seenthat B ¯ T in (4.80) is left invariant by the map M ¯ T . This completes the proof of step 1.Next, we handle the second piece (ii) of the general strategy described above. Step 2: M ¯ T is contractive. The aim of this step is to prove that M ¯ T is a contraction mappingon D ( α,γ ) z , y (∆ T ([0 , ¯ T ]); R m ). That is, we will show that there exists a small ˆ T ≤ ¯ T and a constant OLTERRA EQUATIONS DRIVEN BY ROUGH SIGNALS 33 < q < y = ( y, y , y ,
0) and ˜y = (˜ y, ˜ y , ˜ y ,
0) in D ( β,γ ) z , y (∆ T ([0 , ˆ T ]); R m )we have (cid:13)(cid:13)(cid:13) M ¯ T (cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ≤ q (cid:13)(cid:13)(cid:13)(cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) . (4.85)To this aim, we set F = f ( y ) − f (˜ y ), and consider the controlled path F = ( F, F , F , F ) ∈D ( β,γ ) z (∆ T ([0 , ˆ T ]); R m ) defined through Proposition 4.13. According to expression (4.78), we have M ˆ T (cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17) τts = (cid:26)(cid:18)Z ts k ( τ, r ) dx r F rr , F τst , F ,τ,τts , (cid:19) (cid:12)(cid:12)(cid:12) ( s, t, τ ) ∈ ∆ T (cid:16) [0 , ˆ T ] (cid:17)(cid:27) . (4.86)Hence in order to prove (4.85), it is sufficient to bound the right hand side of (4.86). Now similarlyto Step 1, thanks to Remark 4.11 and upper bounds (4.14)-(4.15), we obtain (cid:13)(cid:13)(cid:13) M ˆ T (cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ≤ C k z k ( α,γ ) (cid:13)(cid:13)(cid:13)(cid:16) F, F , F , F (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ˆ T α − β . (4.87)In the following, we will bound k ( F, F , F , F ) k z , ( β,γ ) , that is, we need to find a bound for k ( F, F , F , F ) k z , ( β,γ ) with respect to k ( y − ˜ y, y − ˜ y , y − ˜ y , k z , ( β,γ ) . Recalling that F = f ( y ) − f (˜ y ) and the definition (4.30)-(4.31), we can rewrite F as F = (cid:16) f ( y ) − f (˜ y ) , f ( y ) f ′ ( y ) − f (˜ y ) f ′ (˜ y ) , f ( y ) f ′ ( y ) f ′ ( y ) − f (˜ y ) f ′ (˜ y ) f ′ (˜ y ) , f ( y ) f ( y ) f ′′ ( y ) − f (˜ y ) f (˜ y ) f ′′ (˜ y ) (cid:17) . (4.88)The strategy to bound k F k z , ( β,γ ) = k ( F, F , F , F ) k z , ( β,γ ) as given in (4.88) is very similar to theclassical rough path case as explained in [11]. Due to the fact that both y and ˜y sit in the ball B T defined by (4.80), we let the patient reader to check that there exists a constant ˜ C = ˜ C M,α,γ, k f k C b such that (cid:13)(cid:13)(cid:13)(cid:16) F, F , F , F (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ≤ ˜ C (cid:13)(cid:13)(cid:13)(cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) . (4.89)Reparting (4.89) into (4.87), we thus get the existence of a constant C such that (cid:13)(cid:13)(cid:13) M ˆ T (cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ≤ CM (cid:13)(cid:13)(cid:13)(cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ˆ T α − β . (4.90)By choosing ˆ T small enough such that q ≡ CM ˆ T α − β <
1, we can recast (4.90) as (cid:13)(cid:13)(cid:13) M ˆ T (cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) ≤ q (cid:13)(cid:13)(cid:13)(cid:16) y − ˜ y, y − ˜ y , y − ˜ y , (cid:17)(cid:13)(cid:13)(cid:13) z , ( β,γ ) . It follows that M ¯ T is contractive on D ( β,γ ) z (∆ T ([0 , ¯ T ]); R m ), which completes the proof of Step 2.Combining Step 1 and Step 2, we have proved that if a small enough ˆ T is chosen then M ˆ T admitsa unique fixed point y = ( y, y , y ,
0) in the ball B ˆ T defined by (4.80). This fixed point is theunique solution to (4.75) in B ˆ T . In addition, owing to (4.82) plus the fact that f, f ′ are uniformlybounded, it is easily proved that the choice of ˆ T can again be done uniformly in the starting point y . Hence the solution on [0 , T ] is constructed iteratively on intervals [ k ˆ T , ( k + 1) ˆ T ]. The proofof Theorem 4.17 is now finished. (cid:3) References [1] F. Harang, S. Tindel,
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Samy Tindel, Xiaohua Wang: Department of Mathematics, Purdue University, 150 N. Univer-sity Street, W. Lafayette, IN 47907, USA.
Email address : [email protected], [email protected] Fabian A. Harang: Department of Mathematics, University of Oslo, P.O. box 1053, Blindern,0316, Oslo, Norway.
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