Volterra Equations Driven by Rough Signals

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product behaving like a convolution extending the classical tensor product. We show that this convolution product is well defined for a large class of Volterra paths, and we provide an analogue of the extension theorem from the theory of rough paths (which guarantees in particular the existence of a Volterra signature). Moreover the concept of convolution product is essential in the construction of Volterra controlled paths, which is the natural class of processes to be integrated with respect to the driving noise in our situation. This leads to a rough integral given as a functional of the Volterra signature and the Volterra controlled paths, combined through the convolution product. The rough integral is then used in the construction of solutions to Volterra equations driven by H\"older noises with singular kernels. An example concerning Brownian noises and a singular kernel is treated


Introduction and main results
Consider a Volterra equation of the second kind written as where f is some initial condition, b and σ are sufficiently smooth functions, k 1 and k 2 are possibly singular kernels, and x is an irregular signal (typically a (fractional) Brownian motion). Integral equations on this form have several applications to physics, biology or even finance. For example in physics, such equations are used to model viscoelastic materials [5], or in biology these equations may be used to model the spread of epidemics [4]. Volterra equations also play a crucial role in renewal theory [9], with several applications.
From a mathematical point of view, Volterra equations have been studied for a long time. At a heuristic level, in order to obtain existence and uniqueness of (1.1) one is typically confronted with the regularity assumption of b and σ, and the regularity of the initial data f as well as the driving noise x. Additionally one needs some type of regularity on the kernel k. Although the conditions on b, σ and f ensuring existence and uniqueness in (1.1) are generally similar to the case of classical ODEs, the assumption on the noise x and the kernels k 1 and k 2 are more challenging objects to analyse in this context. Typically, one searches for the most general conditions on k 1 , k 2 and x in order to still obtain existence and uniqueness for equation (1.1).
The introduction of irregular controls in terms of a random or irregular path x, as illustrated in the second integral term in (1.1), has been investigated in stochastic analysis for decades. Most of the early analysis in this field has been done under the assumption that x is a semi-martingale (see e.g. [1,25]), and high regularity of k 2 (i.e. non-singular cases). During the 1990's these equations received much attention from the perspective of white noise theory, see for example [24] for the case of linear equations with non-singular kernels k 2 , and [3] for the case of linear equations with singular k 2 .
A new direction in stochastic differential equations, called rough paths theory, has been initiated in the late 1990's by Terry Lyons (see in particular [21]) . In contrast to white noise analysis, the theory of rough paths gives a completely path-wise perspective on differential equations driven by irregular signals. In fact Terry Lyons showed (see [19]) that given an irregular Hölder continuous path x, the construction of a differential calculus with respect to x relied mostly on the ability to define iterated integrals of x. In particular the solution to an ordinary differential equation of the forṁ is obtained as a continuous functional of the noise x, together with its iterated integrals and the initial data ξ. That is, if we let x = (x, x 2 , . . . , x n ) for some n ≥ 1 be the collection of the path x together with its iterated integrals, then the solution y can be viewed as y t = I(x, ξ) t where I is a Lipschitz continuous functional in both arguments. Therefore the theory of rough paths not only opens up the analysis of stochastic differential equations to a vast new class of driving stochastic processes, but it also provides simple stability results with respect to that noise. The cost of the improved analytical tractability of the solutions is that non-linear functions in the diffusion term (corresponding to σ in (1.2)) need to be better behaved than in Itô's theory. Typically one requires coefficients which are at least C 2 and bounded in order to get existence and uniqueness of solutions for rough differential equations driven by Brownian motion. This is in contrast to the Lipschitz and linear growth assumptions well known from classical Itô stochastic analysis.
In order to test the robustness of rough paths theory, a natural endeavor has been to explore more general differential systems than the ordinary differential equation (1.2). One can think for example of delay equations [22] and cases of stochastic PDEs [6,14], culminating in the theory of regularity structures [17]. During the years 2009-2011, A. Deya and the second author of this paper provided in [7] and [8] a rough path perspective on Volterra equations driven by irregular signals x. In particular they proved that existence and uniqueness hold whenever k 2 is sufficiently regular (i.e non-singular) and the driving rough path is Hölder continuous with Hölder exponent greater than 1/3. Notice that in these papers, the authors also discussed the challenges of extending the theory of rough paths in order to include singular kernels in equation (1.1).
This remained an open question until late 2018, when Prömel and Trabs [27] gave a para-controlled perspective on Volterra equations driven by irregular signals. Highly influenced by the theory of rough paths, the theory of para-controlled distributions developed by Gubinelli, Imkeller and Perkowski [15] gives a path wise perspective on SDEs and SPDEs through Paley-Littlewood para-controlled calculus, and Bony's paraproduct. Although the result of Prömel and Trabs is very interesting in itself, it seems to be currently limited in the same way as for the theory of para-controlled calculus. Namely one has to assume that the regularity of the noise x, minus the order of the singularity of the kernel k 2 must be greater than 1 3 . Thus a full rough path "picture" in terms of Lyons' theory is not available at this time through the paracontrolled methodology. It should also be mentioned that paracontrolled distributions are mostly expressed through Fourier modes, which is usually not the natural way to handle nonlinear Volterra type equations.
With the above preliminary considerations in mind, this article is devoted to a complete and comprehensive picture of the theory of rough paths in a Volterra setting with singular kernels. The main idea in order to achieve this goal is to extend the concept of a path t → z t to a two variable object (t, τ ) → z τ t for (t, τ ) ∈ ∆ 2 , where ∆ 2 is a simplex of two variables. This extension of the notion of path is motivated from the generic form of a Volterra integral for some (possibly singular) kernel k and a Hölder continuous function x. Note that by considering the mapping t → z t t we recover the classical well known Volterra integral. However, the main advantage with the splitting of the variables into one variable coming from the kernel and the other coming from the integration limit is the following: the regularity of the mapping τ → z τ t is then completely determined by the regularity/singularity of the kernel k, while on the other hand the mapping t → z τ t is completely determined by the regularity/singularity of the driving noise. While it is the composition of these regularities which yields the regularity of t → z t t , the separation of the two arguments allows us to give a framework for Volterra rough paths, similar to the classical rough path framework. More specifically, consider a two parameter E-valued path z as defined in (1.3). Our main assumption will be the existence of a n−tuple of the form z = (z, z 2 , . . . , z n ) : satisfying a modified Chen type relation z τ ts = z τ tu * z · us . (1.5) Notice that in (1.4) the classical tensor product ⊗ used in rough paths theory is replaced by a bi-linear convolution operation * . We will go back to this convolution product (which is one of our main ingredients) below. For the time being, let us just notice that it can be defined as a component-wise operation similarly to the classical tensor algebra, i.e. With the Volterra structure for (t, τ ) → z τ t and the proper definition of the convolution product * , we will argue that the solution to a V -valued Volterra equation can be viewed as a continuous functional of the noise z and the initial data ξ ∈ V . That is, the solution y is given by y = I(z, ξ) where I is Lipschitz continuous in both arguments. It is worth noting that for k = 1 the element z ∈ n i=1 E ⊗i given as in (1.4) is fundamentally different from the classical iterated integrals in the theory of rough paths, both algebraically and analytically.
Let us go back to our first goal, namely the path-wise construction of the Volterra paths in (1.3) as well as the algebraic and analytical properties of the the associated Volterra-signature (as generalized from the concept of signatures in the theory of rough paths). We begin to show that given an α−Hölder continuous path x and a singular kernel k such that |k(t, s)| |t − s| −γ and α − γ > 0, then the path (t, τ ) → z τ t is well defined and is contained in a space of two-variable Volterra-Hölder paths which will be specified later. Starting from this object, we will prove that the convolution product given in (1.5) is well defined for any two Volterra paths z andz built from Volterra kernels k andk and driving noise x andx respectively. In fact, intuitively one can think of this operation between z andz as where the increment z τ tu is defined by z τ tu = z τ t − z τ u . In (1.8), note that the integration is done with respect to the upper parameter inz (corresponding to a regularity coming from the kernelk) and the lower variable in z (representing the regularity coming from the driving noise x). This operation will be extended to any two Volterra type objects in the n−tuple z, and leads naturally to the algebraic relation in (1.5). Let us also mention at this stage that the Hölder type norm under consideration in this paper, taking into account both the regularity coming from the kernel k and the noise x, will be given in the following way for the component z i of z (below we have α, γ ∈ (0, 1)), where we omit some of the other regularities to be considered for sake of clarity. As mentioned above, expression (1.9) is thus separating a singularity of order γ on the diagonal t = τ from the α-Hölder regularity in t − s. The object z satisfying (1.8) and (1.9) is called a Volterra rough path. In order to provide a full picture of the construction of these objects, we include in this article a generalization of the sewing lemma [13,Proposition 1], as well as of the rough path extension theorem (see e.g. [19,Theorem 3.7]) in the Volterra context.
Once the construction of a Volterra rough path is secured, our second goal is concerned with the construction of solutions to (1.7). To this end, we will extend the theory of controlled rough paths, as described by Gubinelli in [13], to the Volterra-rough path setting. Observe that this extension also relies upon the convolution product * introduced in (1.5). In particular, a Volterra path (t, τ ) → y τ t controlled by the Volterra noise (t, τ ) → z τ t given as in (1.3) satisfies y τ ts = z τ ts * y ′,τ,· s + R τ ts , (1.10) where we recall the notation y τ ts = y τ t − y τ s , and where R τ ts is a sufficiently regular remainder term. Processes of the form (1.10) are the ones which can be naturally integrated with respect to x in the rough Volterra sense. Furthermore, once a rough integral is defined for a large enough class of processes and one can prove the stability of the structure (1.10) under composition with a nonlinear mapping, equations like (1.7) are solved thanks to a standard fixed point argument.
Let us now say a few words about the regularity of x and the singularity of k on the diagonal in equation (1.7). We believe that, provided they can be pushed to arbitrary orders, expansions like (1.10) yield a proper notion of Volterra rough type integral as long as k is a singular kernel of order −γ and x is a α-Hölder continuous noise with α − γ > 0. However, for sake of conciseness, this article is restricted to the case α − γ > 1 3 . In this situation one only needs to assume the existence of the second step Volterra iterated integral z 2 , and the first order controlled path structure (1.10) is enough for our purposes. We defer rougher situations and more singular kernels to a further publication. For the construction of a Volterra rough path, one should also be aware of the fact that the concept of geometric rough paths is not directly transferable to the Volterra setting. Simply put, if k is a singular kernel one cannot expect to have a satisfying integration by parts formula (at least not in a classical sense) when integrating against k. Therefore the Volterra rough path z defined by (1.3) and (1.4) is in general not a continuous function of the classical rough path above x. We thus expect some of the algebraic considerations related to the Volterra case to be different from the classical rough path theory, possibly requiring the regularity structures techniques of [17].
The basic example we have chosen in order to apply our abstract theory is given by a rough Volterra path (1.3) constructed from a driving noise x given as a Brownian motion B and a Volterra kernel k of order −γ with γ < 1 4 . We give an explicit construction of the second order term in the Volterra signature, and show that this object satisfies the Volterra-Chen relation (1.5), as well as certain regularity results measured with respect to L p (Ω)-norms. In order to keep the current article to a reasonable size, we also defer the almost sure analysis of the Hölder continuity for z to a subsequent project. This almost sure analysis will also require a new Garsia type lemma adapted to our parametrization in the simplex. The construction of a Volterra rough path for a general Gaussian process is another challenging problem which will be tackled in a forthcoming publication.
Below we give a brief outline of the sections in the paper.
(i) Section 2 provides the elementary tools of rough paths theory and fractional calculus needed in order to develop our framework in the sequel.
(ii) Section 3 gives an introduction to the concept of Volterra iterated integrals and Volterra signatures in the case of smooth driving noise x, possibly involving a singular kernel k. In this section we will encounter the convolution product * for the first time and give a detailed description of this product. We will also provide a working hypothesis on the regularity of the kernel k which will be used throughout the rest of the text.
(iii) In Section 4 we move to the case when the driving noise x of a Volterra path is only α-Hölder continuous with α ∈ (0, 1). We construct a generalized space of Volterra-Hölder paths, and give a pathwise construction of the Volterra process given by (1.3) sitting in this Volterra-Hölder space. Furthermore, we prove that the convolution product is well defined for any Volterra path. This results in the definition of a convolutional path (obtained as an extension of Lyon's concept of multiplicative paths) and then the creation of the Volterra signature from such paths. Both algebraic and analytic aspects of these objects are discussed.
(iv) Section 5.1 deals with the extension of the rough path theory to the Volterra equations case, through the introduction of the Volterra signature and the convolution product defined in Section 4. To this end we define a class of Volterra controlled paths, and prove that the Volterra integral and the operation of composition with regular functions are continuous operations on this class of functions. This is then used to show existence and uniqueness of Volterra integral equations on the form of (1.7) with singular kernel k and rough driving noise x.
(v) At last, in Section 6 we consider the canonical example of choosing the driving noise x in (1.3) to be a Brownian motion, and k to be a singular kernel of regularity −γ with γ < 1 4 . We prove the existence of a second order iterated integral with respect to this Volterra path, satisfying the Volterra-Chen relation in (1.5), as well as an analytic regularity statement aimed towards the regularity requirement of a Volterra rough path.

Preliminary notions
This section is devoted to some preliminary notations and notions of classical rough paths, which will help to understand our considerations in the Volterra case. We start with some general notation in Section 2.1, and recall some notions of rough paths analysis in Section 2.2.
2.1. General notation. We will frequently use Banach spaces E, V and H, and write | · | = · E as long as this does not leave any confusion. Throughout we will write a b meaning that there exists a constant C > 0 such that a ≤ Cb. We will denote by ∆ n ([a, b]) the n-simplex over [a, b] defined by and when the set [a, b] is clear from context we will just write ∆ n . The kernels involved in equations like (1.1) are closely related to fractional integral operators. We will mostly use the operator I α : , which is defined for a given α > 0 and (u, t) ∈ ∆ 2 as follows: where Γ denotes the Gamma function. Fractional integrals have been widely studied in the literature, and we refer to [26] for a thorough account on the topic. However, we mention here a few properties of the operators I α which will be frequently used. Most important is the convolution property; for α, β > 0 and (u, t) ∈ ∆ 2 : We will also use the following action of I α u+ on elementary functions where α, β > 0. Throughout the article we will rely on partitions of intervals. A partition over an interval [a, b] is usually denoted by P[a, b] or D[a, b]. If the interval [a, b] is clear from the context we may write P or D. Throughout this article we will work with increments of functions, which for (s, t) ∈ ∆ 2 will be denoted by We ask the reader to note that the order of t and s in f ts is changed from the traditional notation used in rough path theory. This is to accommodate the algebraic side of the Volterra specific setting we will encounter in later sections. For α ∈ (0, 1), we will denote by C α (I; E)) the space of Hölder continuous functions from an interval I to a Banach space E. If I is reduced to a singleton {t} then where V t stands for a neighbourhood of {t}. Furthermore, we will frequently use an operator δ well known in the theory rough paths, given by 2.2. Short introduction to rough path theory. In this section we recall some basic notions about signatures of paths and related geometric structures, which will make the generalization to Volterra type objects more natural.
2.2.1. Signatures. One natural way to introduce signatures of paths is to see how they arise from expansions of linear differential equations. Namely assume first the path x : [0, T ] → E is smooth, where E is a given Banach space. Let V be another Banach space and consider the V -valued ODĖ where A is a linear operator, namely A ∈ L (E, L (V )). Whenever x is smooth, a Picard type iteration yields the following expansion: where A •i is the i-th composition of the linear operator A which is given as a linear operator on E ⊗i defined from the action The expansion (2.8) reveals that y can be seen as a continuous function of the collection { 0<r 1 ,..,<r i <t dx r 1 ⊗ · · · ⊗ dx r i ; i ≥ 1}, which is called the signature of x.
In order to describe the algebraic structures behind the expansion (2.8), let us first give some definitions. Definition 1. Let E be a real Banach space. For l ∈ N, the truncated algebra T (l) is defined by T (l) = l n=0 E ⊗n , with the convention E ⊗0 = R. The set T (l) is equipped with a straightforward vector space structure, plus an operation ⊗ defined by where g n designates the projection on the n-th tensor level for n ≤ l.
Notice that T (l) should be denoted T (l) (E). We have dropped the dependence on E for notational sake. Also observe that with Definition 1 in hand, (T (l) , +, ⊗) is an associative algebra with unit element 1 ∈ E ⊗0 . The polynomial terms in the expansions which will be considered later on are contained in a subspace of T (l) that we proceed to define now.
Definition 2. The free nilpotent Lie algebra g (l) of order l is defined to be graded sum Here L k is the space of homogeneous Lie polynomials of degree k given inductively by where the Lie bracket is defined to be the commutator of the tensor product.
We now define some groups related to the algebras given in Definitions 1 and 2. To this aim, introduce the subspace T (l) 0 ⊆ T (l) of tensors whose scalar component is zero and recall that 1 ∆ = (1, 0, · · · , 0). For u ∈ T (l) 0 , one can define the inverse (1 + u) −1 , the exponential exp(u) and the logarithm log(1 + u) in T (l) by using the standard Taylor expansion formula with respect to the tensor product. For instance, where the sum is indeed locally finite and hence well-defined. We can now introduce the following group.
Definition 3. The free nilpotent Lie group G (l) of order l is defined by The exponential function is a diffeomorphism under which g (l) in Definition 2 is the Lie algebra of G (l) .
As mentioned above, the link between free groups and differential equations like (2.7) is made through the notion of signature. Namely a continuous map x : ∆ 2 → T (l) is called a multiplicative functional if for s < u < t one has x ts = x tu ⊗ x us , where ⊗ is the operation introduced in Definition 1. A particular occurrence of this kind of map is given when one considers a smooth path w and sets for (s, t) ∈ ∆ 2 , w n ts = t>rn>···>r 1 >s dw rn ⊗ · · · ⊗ dw r 1 . (2.11) Then the so-called signature of w is the following object: It is worth mentioning that S l (w) will be our typical example of multiplicative functional. In addition, signatures of paths belong to the group G (l) introduced in Definition 3 and in fact any element in G (l) can be written as the signature of a smooth path.
Another important property in the theory of signatures, originally proved by Chen [2], relates the multiplicative property to the signature of the concatenation of two paths. That is, if x : [0, s] → E and y : [s, t] → E we can define their concatenation x ⋆ y : . (2.13) Then if S l is the truncated signature of a path as described in (2.12), we get the following relation, whose proof can be found e.g in [19, Theorem 2.9]: S l (x ⋆ y) = S l (y) ⊗ S l (x) . (2.14) One can now go back to the the expansion (2.8), and realize that it can be expressed in terms of the signature of the path x. Whenever x is smooth, the terms x n exhibit a factorial decay, which kill the possibly exponential growth from A ⊗n . This fact is not obvious anymore in case of an irregular path x, which motivates the notion of rough path introduced below.

2.2.2.
Rough path lift of a Hölder path. Let us now assume that the path x driving (2.7) is only α-Hölder continuous with α ∈ (0, 1). Then the iterated integrals appearing in the expansion in equation (2.8) are possibly not well defined. In particular when the continuity of the driving signal is of order α ≤ 1 2 , there is no canonical way of constructing such integrals. The seminal idea put forward by T. Lyons is that one can construct those iterated integrals by means of probabilistic tools, and then build a differential calculus with respect to x starting from the iterated integrals. Those considerations motivate the introduction of Hölder continuous multiplicative functionals.
Here Γ is the Gamma function. Then we call x a rough path above x and we denote the space of all α-Hölder rough paths by C α ([0, T ] ; E).
Note that C α is not a vector space. Indeed, C α is not a linear space due to the fact that G (n) is not a linear space. However, we can equip C α with the following metric: One can also consider a subspace of this space called the space of geometric rough paths and denoted by C α g , which is defined as the closure of all smooth rough paths with respect to the metric d α given by (2.16). Otherwise stated, x ∈ C α is a geometric rough path if there exists a sequence of smooth paths {x n } : The next theorem will give us a canonical extension of the rough path from the truncated space T (n) (E) to all the space T (E) . This extension is crucial in order to ensure the existence and uniqueness of linear differential equations controlled by irregular noise. The theorem and its proof can be found in [19,Theorem 3.7].
Theorem 5. Let x ∈ C α be a rough path of order α ∈ (0, 1) and let n = 1 α . Then there exists a unique extension of x to the space T (E) which satisfies the multiplicative and analytic property. That is, for all m ≥ n + 1 there exists an object x m : ∆ 2 → E ⊗m such that and for all (s, t) ∈ ∆ 2 we have Notice that Theorem 5 tells us that in order to construct the solution to a rough differential equation in terms of its signature, we just need to give a probabilistic construction of the first n = 1 α iterated integrals. Then we know that the all higher order iterated integrals have a canonical (and deterministic) construction only depending on the lower order integrals. We will try to reproduce this mechanism in the Volterra context.

Volterra Signatures
3.1. Definition and first properties. In this section we will define precisely what we mean by a Volterra signature over a smooth path. In this way the Volterra type integrals will be trivially defined and we can focus on their algebraic and analytic properties. This gives some insight on what can be expected in more irregular cases. First we need to present an elementary inequality we will use later (see e.g [7, Lemma 4.4] for more details).
Our constructions will rely on specific assumptions about the power type singularity of the kernel k appearing in (1.1). The main hypothesis we shall use can be summarized as follows.
Here all the inequalities are independent of the parameters γ, β and η. In the sequel a kernel fulfilling condition (H) will be called Volterra kernel of order −γ.
Remark 7. If a kernel k satisfies (H) then by the interpolation inequality a ∧ b ≤ a θ b 1−θ for any θ ∈ [0, 1] applied to the minimum of (3.4) and (3.5) it follows that for any β, η ∈ [0, 1] we have With those assumptions in hand we can now introduce the notion of iterated Volterra integral and Volterra signature, which parallel (2.11) and (2.12).
Definition 8. Let us consider a path x ∈ C 1 ([0, T ] ; E) and a Volterra kernel k : ∆ 2 → R satisfying (H). The iterated Volterra integral of order n is a mapping z n : ∆ 3 → E ⊗n given by We also consider the collection of iterated Volterra integrals as an element of the free algebra. Specifically, we define the element z τ ts ∈ T (∞) (E) as follows: z τ ts = 1, z 1,τ ts , . . . , z n,τ ts , . . . , where we recall that the spaces T (∞) (E) are introduced in Definition 3.
Remark 9. As already highlighted in the introduction, notice that in the definition (3.7) the variable τ is considered as an additional parameter indexing z. While we might be mostly interested in the case τ = t, this extra freedom will play an essential role in our considerations.
Remark 10. Observe that the Volterra integrals are denoted by (s, t, τ ) → z n,τ ts as opposed to (s, t) → z n st in the regular rough path setting. This small modification will ease our notation when one has to deal with integrals of the form t 0 k(t, r)f u du.
Remark 11. A particularly important note is that the collection of Volterra iterated integrals z = (1, z 1 , . . .) is not contained in the free nilpotent Lie group of G given in Definition 3. We expect that one needs a different algebraic approach to these integrals due to the kernels k involved in the integrals. Especially in the singular case it is quite intuitive that Volterra iterated integrals does not lie in the free nilpotent lie group, as there is no concept of integration by parts. That is, let x i and x j be two real valued smooth paths, and consider the second level z 2 . Then observe that a simple integration by parts would yield However, since k is singular we have k(r, r) = ∞. This additional singularity prevents us to exhibit a bracket defined as the commutator of the tensor product in Definition 2 (here considered for the second level term). Therefore a deeper investigation into the algebraic properties of the Volterra iterated integrals given in (3.7) would be highly interesting, and we hope to tell more on this aspect in the future.
When x is a smooth function, iterated Volterra integrals enjoy a regularity property which is similar to the analytic property in Definition 4. This is labelled in the following proposition.
Proposition 12. Let k : ∆ 2 ([0, T ]) → R be a Volterra kernel which satisfies (H) with γ < 1, and assume x is a continuously differentiable function. For n ≥ 1, consider the path z n,τ defined by (3.7). Then for (s, t) ∈ ∆ 2 ([0, T ]) we have that |z n,τ ts | ≤ where the C 1 norm of x is defined by Proof. Starting from Definition (3.7) and invoking the fact that x is a C 1 function, we directly get Therefore hypothesis (H) on the kernel k entails where we have used the convolution property (2.2) of the Riemann-Liouville integral operator I α described in Section 2.1. Furthermore, it follows from the identities in where we have used the notation c n,γ = [Γ((n − 1)(1 − γ) + 1)] −1 and the substitution r = s + θ (t − s). In addition, since τ ≥ t, it is clear that where B is the Beta function. Observe that classical identities for Gamma and Beta functions, yields that plugging relation (3.11) into (3.10) and then (3.9) we have then obtained which is our claim.
3.2. Convolution product. We will now try to get an equivalent to the multiplicative property of the signature (2.15) in a Volterra context. Unfortunately this property does not hold directly for a Volterra rough path, due to the interaction between variables in the kernel k. However, we will show that if we modify the tensor product to be a type of convolution product, then we still get a concatenation type property under this product.
Proposition 13. Let (s, u, t) ∈ ∆ 3 . Consider two C 1 functions x : [s, u] → E and y : [u, t] → E, and denote by q = x ⋆ y their concatenation. Let z n be the n-th Volterra integral of q on (s, t) as defined in (3.7), namely for all (s, t, τ ) ∈ ∆ 3 set z n,τ ts := with the convention that r n+1 = τ . Then for (s, u, t, τ ) ∈ ∆ 4 we have where the convolution product * is defined as follows for all 0 ≤ i ≤ n Here we have used the convention z 0 ≡ 1 and z n * 1 = 1 * z n = z n .
Proof. This proof is left to the patient reader. The result is easily checked by splitting the domain Remark 14. In order to make formula (3.13) more concrete, let us explicitly compute the integrals we obtain for n = 2. In this case relation (3.12) reads and we observe that where we note the common integration variable r 2 in the above product. In relation (3.13), we also notice that since the kernel k is smooth except on the diagonal, the function inherits the smoothness of k. Therefore the integral which features in (3.15), can be interpreted as a Riemann-Stieltjes integral. One of our main task will then be to control possible singularities arising from k when x is no longer assumed to be smooth, but rather a Hölder path. We refer to Section 4 for a further analysis of this point.
Next we will present a technical lemma which will become useful in later analysis of the Volterra signature. It states that the convolution product * behaves similarly to the tensor product ⊗ on small scales.
Lemma 15. Let D be a partition of [s, t] such that |D| → 0, and consider z j for j = 1, . . . , p as constructed in Equation (3.7) with a continuously differentiable driving noise and a kernel k satisfying (H) with singularity of order γ < 1 2 . Then for n, p ≥ 1 Proof. In order to study the left hand side of (3.16), let us set for Then according to Definition (3.13) it is readily checked that where we have written r n+1 = τ for the sake of readability. We now proceed along the same lines as for Proposition 12. Namely if we assume that ẋ ∞ ≤ M, we get Thus restricting β ∈ (0, 1 − γ) , we get Hence, integrating the outside integral over the simplex u > r n > · · · > r 1 > s, we end up with and where we have used the convolution property (2.2) of the Riemann-Liouville fractional integral. Now we can do a change of variables where c γ,τ,u,v is a function bounded by the Beta function, i.e.
Plugging this identity into (3.18) and writing C = C β,γ,p,n for constants which may change from line to line, we get Therefore it is readily checked that where |D| denotes the size of the mesh of D. Taking into account the definition (3.17) of D(u, v), this finishes the proof.

Volterra Rough Paths
To begin the study of Volterra rough paths, we need understand the structure and regularity which may be extracted from a Volterra path. As we have already seen, a Volterra path is really a two parameter function on a simplex ∆ 2 taking values in some space E . A simple example of a function of this form could be the singular kernel defined for t ≤ τ and γ ∈ (0, 1). Note that for a function f given as in (4.1) and This tells us that as long as t < τ then we have a Lipschitz bound on f τ , i.e. for any . Similarly one can consider the function g τ t = (τ − t) α , for some α ∈ (0, 1) and t ≤ τ . Then it is easy to see that globally, g is α-Hölder continuous in both variables. However, for any small In the sequel we will generalize the above considerations to processes of the form where x is an α-Hölder path and k a possibly singular kernel of order −γ. This section is devoted to a definition and analysis of generic Volterra type rough paths like in (4.2).

4.1.
Definition and sewing lemma. Let us go back for a moment to the increment defined in (4.2). One way to define the term t s k(τ, r)dx r is to split the integral in the right hand side of (4. Then for each [u, v] ∈ P we have some regularity of v u k (t, r) dx r coming from the difference v − u which is contributed by the driving noise, and some (possibly singular) regularity coming from the difference t − v. Much of the difficulty in the analysis of Volterra rough paths will be due to such considerations. In order to capture the different regularities discussed above, we will make use of three different quantities, which will later be used in the definition of various classes of Volterra Hölder functions. For two parameters (α, γ) ∈ (0, 1) 2 we will consider the semi-norms defined by with the convention z τ ts = z τ t − z τ s and z τ τ ′ s = z τ s − z τ ′ s With these quantities at hand, let us define a space of functions which we will call Volterra paths.
We denote this space by V (α,γ) (∆ 2 ; E). In addition, under the mapping Remark 17. Conventionally, we will use the notation y τ ts to signify both functions with three arguments, and the increment of functions with two arguments, i.e. y τ ts = y τ (s, t) and y τ ts = y τ t − y τ s . We hope the specific meaning will always be clear from the context. Moreover, we will use the same norms as those defined in (4.6) for three variable functions y : is of ρ-Hölder regularity in both variables, where ρ = γ −α. However, at any point off the diagonal we have α-regularity in the lower variable and 1-regularity in the upper variable. The space could have therefore be defined more generally to capture three different regularities. However, for our purposes, under the assumption (H) and the fact that a Volterra path is of the form z τ t = t 0 k (τ, r) dx r , we easily get the 1−regularity in the upper argument. This will play a central role throughout the analysis of such paths.
Remark 19. The norms and spaces in Definition 16 can be easily generalized to increments of two variables, which yields the definition of a space V (α,γ) 2 (4.7) Those spaces will be used for the definition of convolutional controlled paths in Section 5.1.
We are now ready to state our Sewing Lemma adapted to Volterra integrands.
(ii) For all (s, t, τ ) ∈ ∆ 3 we have that while for (s, t, τ ′ , τ ) ∈ ∆ 4 we get Proof. This is an elaboration of [11, Lemma 4.2] and we give some details here for the sake of completeness. Specifically, we will focus on the convergence of Riemann type sums [u,v]∈P Ξ τ vu along dyadic partitions. Referring to [11,Lemma 4.2], we leave to the patient reader the task of checking the convergence of [u,v]∈P Ξ τ vu along a general partition whose mesh converges to 0, as well as the relation δI (Ξ) = 0.
With those preliminaries in mind, let us consider the n-th order dyadic partition P n of [s, t] where each set [u, v] ⊂ P n is of length 2 −n |t − s|. We define the n-th order Riemann sum of Ξ τ , denoted I n (Ξ) ts , as follows Our aim is to show that the sequence {I n (Ξ τ ) ; n ≥ 1} converges to an element I(Ξ) which fulfils relation (4.12). To this aim we will analyse differences I n+1 (Ξ τ ) − I n (Ξ τ ) and prove the following bound (4.14) In order to prove (4.14), observe that where we recall that δ is given by relation (4.9) and where we have set m = u+v 2 . Plugging relation (4.10) into (4.15), it is thus readily checked that We will now upper bound the right hand side above. Invoking the fact that β > 1 and |v − u| = 2 −n |t − s| for u, v ∈ P n we write In addition, some elementary calculations similar to those in Remark 7 show that for κ ∈ (0, 1) we have where we have used the fact that the integral t s |τ − r| −κ dr is converging for κ < 1. Putting this inequality into (4.18) we get Inserting (4.19) into (4.17) and then into (4.16), our claim (4.14) is thus easily obtained.
With relation (4.14) in hand, one immediately gets that the sequence {I n (Ξ τ ) ts } n≥0 is a Cauchy sequence. It thus converges to a quantity I (Ξ τ ) ts which satisfies (4.12). As mentioned above, the remainder of the proof goes along the same lines as [11,Lemma 4.2]. We leave it to the patient reader for the sake of conciseness. This proves that the element I (Ξ τ ) has finite · (β,κ),1 norm and that (4.12) holds. The next step will be to show that also the integral I(Ξ τ τ ′ ) of the increment in the upper variable Ξ τ τ ′ ts is finite in the · (β,κ),1,2 norm. Following the lines for the proof above, we can just change the integrand Ξ τ ts with Ξ τ τ ′ ts and the norms accordingly. Thus, using exactly the same arguments as before, Inequality (4.13) holds as well. This concludes the proof.
In order to test the compatibility of our first definitions with the sewing lemma, we will show that one can construct a Volterra path of the form z τ ts = t s k (τ, r) dx r in terms of Lemma 21.
(iii) For any η ∈ [0, 1] there exists a strictly positive constant c such that for any (s, t, q, p) ∈ ∆ 4 we have where z pq ts = z p t − z q t − z p s + z q s . Remark 23. According to the standard rules of algebraic integration we would be naturally prone to set Ξ τ ts = k(τ, t)x ts . Here we have chosen to take Ξ τ ts = k(τ, s)x ts , which will ease the treatment of the singularity of k on the diagonal. This small twist on the usual theory does not affect the fact that we are generalizing Volterra equations from the smooth to the rough case.
In order to check that δΞ (β,κ),1 < ∞, we start by noting that the increment δ m Ξ τ ts can be written as δ m Ξ τ ts = [k(τ, s) − k(τ, m)] x tm , which stems from elementary algebraic manipulations. Therefore, according to Hypothesis (H) we have and a direct application of Lemma 6 with an additional parameter ν > 0 yields Next we pick our parameter ν > 0 such that the condition is satisfied. As far as the singularity at τ is concerned, relation (4.21) asserts that in order to apply Lemma 21 item (ii) we get the restriction Note that if we put conditions (4.23) and (4.24) together, we get 1 − α < ν < 1 − γ which can be fulfilled as long as α > γ. Furthermore, it is immediate that β −κ = α−γ.
Remark 24. Owing to Theorem 22, we now know that a typical example of a Volterra path in V (α,γ) is given by processes of the form t s k (τ, r) dx r . Having this large class of objects in hand, we will mostly focus on computations for general elements in V (α,γ) whenever it is not needed to explicitly state the kernel k or the driving noise x.

4.2.
Convolution product in the rough case. As we have seen in Section 3.2, the equivalent of Chen's relation in our Volterra context involves convolution type integrals. In order to clarify this point, let us go back to Remark 14 concerning second order iterated integrals. One way to rephrase relation (3.14) with the operator δ introduced in (4.9) is the following In the right hand side of (4.27) we point out that the limits of the integration with respect to x r 1 are fixed; the only thing that is connecting the two integrals is the dependence on r 2 through the kernels. Thus the integral s 0 k (r 2 , r 1 ) dx r 1 can really be thought of as a re-scaling of the path x as r 2 moves from s to t. Our next step is to show that this operation is indeed valid for two generic Volterra paths y, z. Theorem 25. We consider two Volterra paths z ∈ V (α,γ) and y ∈ V (α ′ ,γ ′ ) as given in Definition 16, where we recall that α, γ, α ′ , γ ′ ∈ (0, 1), and define ρ ≡ α − γ > 0 and ρ ′ ≡ α ′ − γ ′ > 0. Then the convolution product is a bilinear operation on V (α,γ) given by The integral is understood as a Volterra-Young integral for all (s, u, t, τ ) ∈ ∆ 4 . Moreover, the following inequality holds true, Proof. Define Ξ τ r ′ r := z τ r ′ r ⊗ y r us , for 0 ≤ s < u ≤ r ≤ m ≤ r ′ ≤ t. In spirit of Lemma 21, we will show that Following the strategy outlined in the proof of Lemma 21, we know from (4.16) that we must show that the sum [r,r ′ ]∈P n [u,t] |δ m Ξ τ r ′ r | is converging (here P n is the dyadic partition used in the proof of Lemma 21). Let us therefore consider the action of δ on Ξ. By simple algebraic manipulations we see that Let us now analyse the right hand side of (4.30). The term z τ r ′ m can be bounded thanks to assumption (4.3). We get |z τ r ′ m | ≤ z (α,γ),1 |τ − r ′ | −γ |r ′ − m| α . (4.31) As for the term y mr us we can use assumption (4.5) to write |y mr us | ≤ y (α ′ ,γ ′ ),1,2 |m − r| η |r − u| −η |r − s| ρ ′ , (4.32) for an arbitrary η ∈ [0, 1]. Hence gathering (4.31) and (4.32) we bound (4.30) by where we have used the fact that |r ′ − m| |r ′ − r| and |m − r| |r ′ − r|. Combining (4.33) with (4.30) and summing over the points of the dyadic partition P n , we end up with Note that we have two separate possible singular points above, both when r → u and r ′ → τ . However, taking limits in the Riemann sums on the right hand side of (4.34), we know that we obtain a converging integral as long as η + α > 1 and η < 1. Indeed, the right hand side of (4.34) is bounded (up to a multiplicative constant) by the integral |P n | α+η−1 t u (τ − a) −γ (a − u) −η (a − s) ρ ′ da, and by doing a change of variables a = u + θ(t − u) as well as applying the inequality where c η,γ = B(1 − γ, 1 − η) and we recall that B stands for the Beta function as in the proof of Proposition 12. It follows that Since we must choose η > 1 − α, let us choose η = 1 − α + ǫ for some small ǫ > 0 satisfying ρ − ǫ > 0. Then inequality (4.36) reads Note that for the dyadic partition P n we have |P n | ǫ = 2 −nǫ (t − u) ǫ , and observe that (4.37) is the equivalent of (4.19) in our current setting. Therefore, one one can follow the same steps as in Lemma 21 in order to get the following relation, which is the analog of (4.14): where we recall that 2α − γ = 2ρ + γ. We also let the patient reader check from (4.38) that 39) Putting together (4.38) and (4.39) and reasoning exactly as in Lemma 21 after (4.19), we obtain that I n (Ξ τ ) converges to an element I(Ξ τ ) verifying We therefore define z τ tu * y · us := I (Ξ τ ) tu , and one can directly see from (4.40) that z τ tu * y · us satisfies the relation This completes the proof.
Our next step is to mimick Proposition 13 in a rough Volterra context. Specifically we would like to extend Theorem 25 in order to get a proper definition of the n-th order convolution products for Volterra rough paths (where we recall that Volterra rough paths are introduced in Definition 16). For those n-th order convolution rough paths, we also wish to get a multiplicative property similar to Proposition 13. Observe that in order to properly define the aforementioned n-th order convolution product, we will need to extend the domain of the definition of our convolution product * . Namely, we would like to define products of the form z 2,τ ts * f ·,· s for a generic function (s, τ 1 , τ 2 ) → f τ 2 ,τ 1 s . Let us first explain how a product of this form behaves in case of a smooth path x with a Volterra kernel k. Namely in this situation, consider a smooth three variable function f : ∆ 3 → L(E, L(E)). Then a natural way to define z 2,τ ts * f ·,· s is the following (the reason we assume f has two upper arguments will be discussed in detail in Section 5.1).
Definition 26. Let x be a continuously differentiable function and consider a Volterra kernel k which fulfills (H) with γ < 1. Let also f : ∆ 3 → L (E, L (E)) be a smooth function. Then for τ ≥ t > s ≥ v the convolution z 2,τ ts * f · 1 ,· 2 v is defined by where the notation f · 1 ,· 2 v is introduced to prevent ambiguities about the order of integration.
We now state an algebraic type lemma which will be useful in order to extend Definition 26 to rougher contexts. This yields a decomposition of z 2,τ ts * f · 1 ,· 2 s of the form where A and B are respectively given by Now we immediately recognize the term A τ vu as the expression z 2,τ vu * f · 1 ,· 2 s given by (4.41). Moreover, it is also readily checked that B τ vu = z 1,τ vu * (z 1,· 1 us * f · 1 ,· 2 s ). Hence thanks to relation (3.12) for smooth paths we can also write Plugging this relation into (4.43) and gathering the information we have on the term A τ , our proof is complete.
Remark 28. The identity (4.42) makes sense as long as one can define z 2,τ * f · 1 ,· 2 and if z 2 verifies (3.12). This opens the way to a generalization to rougher situations, having Theorem 25 in mind for the equivalent of (3.12). These considerations motivate the definition in Theorem 30.
We now take another step towards a proper definition of general convolution products. To this aim, we will assume for a moment that our generic Volterra path z τ gives raise to a stack {z j,τ ; j ≤ n} of iterated integrals. Specifically our standing assumption is the following: H2: Let z ∈ V (α,γ) be a Volterra path, as introduced in Definition 16. For n such that (n + 1)ρ + γ > 1, we assume that there exists a family {z j,τ ; j ≤ n} with z 1 = z satisfying where the convolution product is defined by the right hand side of (4.28). In addition, we suppose that for j = 1, . . . , n we have z j ∈ V (jρ+γ,γ) (∆ 3 , E).
Let us also specify the kind of norm we shall consider for processes with 2 upper variables of the form y · 1 ,· 2 .
Assuming Hypothesis (H2), and having Definition 29 in mind, we now state a general convolution result for functions defined on ∆ 3 .
Theorem 30. Let z ∈ V (α,γ) with α, γ ∈ (0, 1) satisfying ρ = α − γ > 0, as given in Definition 16. We assume that z fulfills hypothesis (H2). Consider a function y : ∆ 3 → L(E, V ) such that y is in the space V · 1 ,· 2 (α,γ) given in Definition 29. Then we have for all fixed (s, t, τ ) ∈ ∆ 3 that is a well defined Volterra-Young integral. It follows that * is a well defined bi-linear operation between the three parameters Volterra function z 2 and a 3-parameter path y. Moreover, we have that Remark 31. Our definition (4.48) for z 2,τ ts * y · 1 ,· 2 s is obviously motivated by (4.42), which had been obtained for smooth Volterra paths. We are now extending this identity to a generic path in V (α,γ) .
Remark 32. The term (δ u z 2,τ vs ) * y · 1 ,· 2 s in (4.41) is defined in the following way: observe that according to (4.44) we have Therefore we get (δ u z 2,τ vs ) * y · 1 ,· 2 s = z 1,τ vu * z 1,· us * y · 1 ,· 2 s , which is well defined from a successive application of Theorem 25. Indeed, the convolution z 1,p ts * y r,· s for p ≥ r can be constructed in the exact same way as we constructed z 1,τ vu * z 1,· us . Namely, y · 1 ,· 2 s has to be considered as a constant in the lower variable. Hence, the y (α,γ),1,2 norm invoked in (4.29) can be reduced to the regularity required in (4.47).
In order to treat the remaining terms in (4.52) and (4.53), observe that formula (4.28) trivially yields (recall again that y u,u s has to be considered as a constant in the lower variable) z 1,τ ts * y u,u s = z 1,τ ts ⊗ y u,u s . Therefore we can gather our two remaining terms into Now in the spirit of Theorem 25, Inequality (4.29) and using condition (4.45) as well as relation (4.54), we have Notice that the regularity obtained in (4.56) is the same as for (4.54). Hence repeating the same arguments as after (4.54) and recalling (4.52) and (4.53), we have obtained that |δ r Ξ τ vs | c y,z |τ − v| −γ |u − s| −η |v − u| µ , where η < 1 and µ = 2ρ + γ + η > 1, and where the constant c y,z is the same as in the right hand side of (4.56).
We are now in a situation which is similar to the one we had encountered in the proof of Theorem 25 (see inequality (4.34) in particular). Thus along the same lines as Theorem 25, resorting to a slight modification of the Sewing lemma 21 involving two possible singularities, we get that the Riemann sums defined by (4.51) converge as |P| → 0, and we define z 2,τ ts * y · 1 ,· 2 := lim |P|→0 I P . (4.57) In order to check (4.49), let us apply inequality (4.12) to the increment Ξ τ defined in Equation (4.51). To this aim, observe that taking v = t and u = s in the definition of Ξ τ we get δ s z 2,τ ts = 0, and thus Ξ τ ts = z 2,τ ts ⊗ y s,s s . In addition, we have just seen in (4.57) that I (Ξ τ ) ts = z 2,τ ts * y · 1 ,· 2 , and thus I (Ξ τ ) ts − Ξ τ ts = z 2,τ ts * y · 1 ,· 2 − z 2,τ ts ⊗ y s,s s . Our claim (4.49) is then a direct application of Lemma 21, together with the inequality estimates (4.54) and (4.56).
Remark 34. As a special case of Remark 33, we can define the convolution z 2,τ tu * z 1,· us by setting y r u = z 1,r us . Then y trivially satisfies y (α,γ),2 < ∞ if z 1 ∈ V (α,γ) , which ensures a proper definition of z 2,τ tu * z 1,· us . Moreover, a direct application of Theorem 30 yields |z 2,τ tu * z 1,· us | |τ − t| −γ |t − s| 3ρ+γ ∧ |τ − s| 3ρ . Remark 35. In our applications to rough Volterra equations we will consider the case ρ = α − γ ∈ (1/3, 1/2], and therefore it is sufficient to show that the convolution product * can be performed on the first and second level of a Volterra rough path. Indeed, whenever ρ > 1/3, the convolution product for third or higher order terms in the Volterra rough path are of regularity 3ρ which is greater than 1. Therefore the higher order convolutions z n,τ introduced in (H2) may be constructed as a classical Riemann integral. For a general ρ ∈ (0, 1), it is easily conceived that one could extend the construction of the convolution product given in Theorem 30 to any order Volterra rough path z n satisfying (4.44). This can be done by induction on n, and one first need to give a proper definition of the convolution product up to order k = [1/ρ]. The convolution product between elements z K of order K ≥ k + 1 is then constructed canonically through Riemann integration, together with (4.44). We defer this extension to a subsequent publication.

Volterra convolutional functionals.
With the preliminary notions of Section 4.2 in hand, we are now ready to generalize the notion of multiplicative functional (as introduced by Lyons et. al. in [19]) to a Volterra context. The basic definition of Volterra convolutional functional is the following.
We call this mapping a Volterra convolutional functional if for all (s, u, t, τ ) ∈ ∆ 3 it satisfies z τ ts = z τ tu * z · us , where for all 1 ≤ p ≤ n the convolution product (z τ tu * z · us ) p is defined by and where the convolution in the right hand side of (4.58) is understood as in (4.28) or (4.48).
Remark 37. In order to define (4.58), we need in fact an extension (4.48) to higher order integrals of the form z j,τ . As mentioned in Remark 35, we defer this extension to a future article. Notice however that, thanks to our restriction to ρ > 1 3 , we mostly need p − i and i ≤ 2 in (4.58). This case is covered by (4.28) or (4.48).
Proceeding as in [19], we will now define some Hölder type norms adapted to to our Volterra multiplicative functionals.
Remark 39. All rough paths (in the classical framework recalled in Section 2.2) are also Volterra rough paths with Volterra kernel k = 1, i.e x 1 t := t 0 1dx r . Thus the definition of Volterra rough paths is truly extending the definition of a rough path, and the convolutional product * is extending the usual truncated tensor product by coupling the product through the integration of kernels.
By definition we can see that a Volterra rough path is a continuous mapping from ∆ 3 ([0, T ]) to T (⌊ρ −1 ⌋) (E). We will also find it useful to equip the space with a metric generalizing (2.16). Let us therefore define a metric for two Volterra rough paths z and y in V (α,γ) where ρ = α − γ by Definition 40. We define the space of geometric Volterra paths as the closure of smooth Volterra paths (i.e. paths in V (1,γ) ) in the rough path metric from equation (4.60). The space of all geometric Volterra rough paths is denoted by G V (α,γ) .
Remark 41. Note that the geometric Volterra paths are not contained in a free-nilpotent Lie group, as is the case for regular rough paths. Indeed, there exists no concept of integration by parts in general for Volterra paths due to the possible singularities, and thus the notion of geometric Volterra paths can not be seen as an object in the space G (l) given in Definition 3.
The following is an equivalent of the extension theorem for multiplicative functionals to a Volterra context. It can also be seen as an extension of Proposition 12 and Proposition 13 to a rough context. Theorem 42. Let n = ⌊ρ −1 ⌋ for ρ = α − γ > 0 and assume that z ∈ V (α,γ) is an n-th order Volterra rough path with values in T (n) (E) according to Definition 38. Then there exists a unique extension of z to T (E). In particular, for all m ≥ n + 1 there exists a unique element z m ∈ E ⊗m such that for any u ∈ [s, t] the following algebraic property is satisfied where we have used the convention z 0 ≡ 1 and z j * 1 = 1 * z j = z j . In addition the bound (4.59) can be extended to z. Namely for m ≥ n+ 1 we have for a constant M > 0 such that z 1 (α,γ) ≤ M the following properties , and z m (mρ+γ,γ),1,2 M m Γ (mρ + 1) , for any β ∈ [0, 1]. It follows that there exists a unique Volterra signature with respect to the n-th order Volterra rough path.
Proof. We will divide the proof into several steps.
Step 1: Uniqueness. The uniqueness problem will be addressed by induction. Indeed, for m = n + 1 relation (4.61) reads The right hand side of (4.63) only depends on the stack {z j | 1 ≤ j ≤ n}, and is therefore uniquely defined thanks to our assumptions. Now considerz m andz m two candidates for z m with m = n + 1, and define ψ τ ts =z m,τ ts −z m,τ ts . Then according to (4.63) and relation (4.62) we have δψ τ = 0, and |ψ τ ts | |τ − s| −γ |t − s| (n+1)ρ+γ . (4.64) In particular ψ is an additive functional with regularity greater than 1. It is thus readily seen that ψ = 0, which proves the uniqueness for m = n + 1. Once the uniqueness is shown for the levels k = n + 1, . . . , m, an induction procedure similar to what lead to (4.64) also shows uniqueness for k = m + 1.
Step 2: Existence. The existence will be proved again based on induction. We will first show that an (m = n + 1)-th order Volterra rough path can be constructed purely based on the information of z n . In addition z m,τ is required to satisfy (4.61). Thus recalling that m = n + 1 we have Our strategy is now to prove that L τ ts exists by applying the Sewing Lemma 21 to the increment Ξ. The main assumption to check in order to apply Lemma 21 concerns δΞ, and thus we obtain where the first identity is obtained thanks to an elementary computation of δ r (z m−i,τ vu * z i,· us ). Also note that the second inequality in (4.68) directly stems from the assumption (4.59), which stipulates that One can improve (4.68) in the following way: applying the neo-classical inequality from [19, Lemma 3.8], we know that there exists a C > 0 such that Plugging this information into (4.68), we conclude that δΞ satisfies With (4.70) in hand, we can apply Lemma 21 to the increment Ξ. We get that the limit L τ ts defined by (4.66) exists, and we set I (Ξ τ ) ts = L τ ts = z m,τ ts for m = n + 1. Moreover, a direct application of (4.12) together with the fact that Ξ τ ts = 0 yield It now follows that z m  We also let the patient reader check that a simple induction procedure allows to generalize all our considerations until (4.72) for a generic m ≥ n + 1.
We will now prove that To this aim, we need to repeat the procedure of Steps 1-2 for z m,τ τ ′ ts = z m,τ ts − z m,τ ′ ts . In particular, the equivalent of the incremental Ξ τ defined in (4.67) will be With this increment in hand, relation (4.73) is proved along the same lines as (4.72). Details are omitted for the sake of conciseness. The norm z m (mρ+γ,γ),2 can also be estimated with the same kind of argument. Hence gathering (4.72) and (4.73), we have obtained that z m ∈ V (mρ+γ,γ) where V (α,γ) is given in Definition 16.
Step 3: Convolutional property. It remains to be proven that z m is a convolutional functional in terms of Definition 36, i.e. that for m ≥ n + 1 and (s, r, t, τ ) ∈ ∆ 4 it satisfies In order to prove identity (4.74), recall that (4.67) can be read as Let us now divide a typical partition P into P ∩ where we have invoked (4.67) again for the second identity and where we have set As in the previous steps we now proceed by induction. Namely assume that (4.61) holds for k = 1, . . . , m − 1, and let us propagate the relation until k = m. Then applying the identity z i,τ ts = i j=0 z i−j,τ tu * z j,· us , which is valid for all l < m, we get where we defineL Next, another application of (4.67) enables us to obtain directlŷ L 1,τ trs = z m,τ tr . (4.77) In order to handle the termL 2,τ trs , let us change the order of the sums with respect to i, j and invoke the associativity of the convolution product * . We get Plugging this information into (4.78) and invoking (4.65), we end up witĥ This concludes our induction procedure, and thus (4.61) holds for all m ≥ 1.
Remark 43. Theorem 42 tells us that the Volterra signature associated to a Volterra path is uniquely determined from the Volterra rough path introduced in Definition 38. That is, once we have constructed a truncated Volterra rough path (remember that this object is by no means unique) then there exists a unique extension with respect to the full Volterra rough path.

Non-linear Volterra integral equations driven by rough noise
In this section we will see how we can substitute the conventional tensor product from rough path theory with the convolution product defined in Section 4 in order to show existence and uniqueness of Volterra equations with singular kernels. Similarly to the theory of controlled rough path introduced by Gubinelli in [13], we define a class Volterra controlled paths. The composition of the Volterra controlled paths with the Volterra rough path from Definition 38 gives an abstract Riemann integrand such that we may construct a Volterra integral by application of the Volterra sewing Lemma 21. This abstract integration step is then the key in order to define and solve Volterra type equations. 5.1. Volterra controlled processes and rough Volterra integration. As many of the results here are extensions of classical texts on rough path such as [11] or [13], we will try to keep the proofs as concise as possible. The reader is sent to the aforementioned references for further information on the results and properties of controlled rough paths and solutions to non-linear differential equations driven by rough paths. We will first give a definition of another modification of the Volterra-Hölder spaces given in Definition 16 in order to give a precise analysis of Volterra-controlled paths.
Remark 45. Note in particular that the definition of the space W (α,γ) 2 does not involve a norm similar to (4.5). Although the definition of u · 1 ,· 2 (α,γ) is a slight abuse of notation, we believe that it will be clear from the superscripts of u what norm we apply.
We now turn to the definition of controlled Volterra paths, which is crucial for a proper definition of rough Volterra equations.
Remark 47. We call the function y ′ the Volterra-Gubinelli derivative, and emphasize that this function is evaluated on ∆ 3 , where it has two upper arguments. This is denoted by ∆ 3 ∋ (s, p, q) → y q,p s as opposed to the increment of a path y in the upper variable denoted by ∆ 3 ∋ (s, p, q) → y qp s .
Remark 48. For a controlled Volterra path the regularity of y in the upper argument is inherited from the regularity of the upper argument of the driving noise z, Gubinelli derivative and remainder term R y . That is, it is implied from relation (5.3) that for (y, y ′ ) ∈ D (α,γ) z (∆ 2 ; V ) we have y qp ts = z qp ts * y ′,p,· 2 s + z q ts * y ′,qp,· s + R qp ts .
(5.5) Our next step is to show that we may construct the Volterra rough integral in a very similar way to the classical rough path integral, but changing ⊗ for * as well as applying the Volterra sewing lemma 21. It follows that the Volterra integral of a controlled path with respect to a driving Hölder noise x ∈ C α is again a controlled Volterra path.
We shall analyse the two terms in the right hand side of (5.20) separately. We start with the derivative ϕ ′ , for which we will bound the two norms given by (5.2) and (4.45). Specifically, observe first that the difference in the lower variable for the derivative ϕ ′ is given by , and thus by addition and subtraction of y ′,p s f (y q t ) it is readily checked that the following bound is satisfied By a similar argument on the upper variables, we bound the quantity f (y · 1 ) y ′,· 2 (ρ+γ,γ),2 in the same way, and get a bound which is similar to (5.21). Therefore we have proved that the (α, γ)-norm defined by (5.1) can be estimated as Let us now handle the term R f (y) (2ρ+γ,γ) in equation (5.20). More precisely recalling that the norm · (2ρ+γ,γ) is given by (4.7), let us first bound the quantity R f (y) (2ρ+γ,γ),1 . Towards this aim, we go back to the definition (5.18) of R f (y) and apply Taylor's expansion in a standard way. This yields the existence of a c τ ts = ay τ s +(1−a)y τ t for some a ∈ [0, 1] such that The regularity of R f (y) for the · (2ρ+γ,γ),1 norm now follows from the boundedness of the second derivative of f , the squared regularity of the increment of y and the regularity of R y . Next, we will compute the regularity in the upper argument in R f (y) , which corresponds to the semi-norm · (2ρ+γ,γ),1,2 in (4.5). In particular, we will consider the increment R qp,f (y) ts = R p,y ts f ′ (y s ) qp + R qp,y ts f ′ (y p s ) ), it follows from a combination of (5.23) and (5.22) that We now use the fact that the regularity of the controlled Volterra path is inherited by the noise, as discussed in Remark 48 and see that y (α,γ) ≤ |y ′ 0 | + y ′,· 2 (α,γ) ( z (α,γ) + R y (2ρ+γ,γ) ).

(5.24)
We now chooseT satisfying C z (α,γ)T α−β = 1 2 , and we obtain that (w, w ′ ) is an element of BT . Summarizing our considerations so far, we end up with the relation We will mainly focus on the term F ′,· 1 ,· 2 (β,γ) , the remainder R F being treated similarly. Now recall from (5.34) that F ′,· 1 ,· 2 (y, y ′ ) = f ′ (y · 2 ) f (y · 1 ) − f ′ (ỹ · 2 ) f (ỹ · 1 ). To be able to treat the fact that we have two upper variables to take care of, we do a simple addition and subtraction to see that By invoking the fact that f ∈ C 4 b , let g and h denote the remainders from a first order Taylor expansion of the differences f (y) − f (ỹ) and f ′ (y) − f ′ (ỹ). Note in particular that this implies that g ∈ C 3 b and h ∈ C 2 b , and we have that Then it follows from (5.41) that for any t ∈ [0,T ] we have for some a ≥ 0. All these solutions are connected on the boundaries, and thus we use that a function which is Hölder on any subinterval [a, aT ] ⊂ [0, T ] of lengthT is also Hölder continuous on [0, T ] (see e.g. [11], exercise 4.24), which applies to the Hölder continuity in both variables. Here notice that the time stepT can be made constant thanks to the fact that f is a bounded function (see relation (5.31)).
We can conclude that there exists a unique global solution to Equation (5.25) in the space D (β,γ) z 1 (∆ 2 ; V ) for β < α. Actually, by (5.5) it is clear that the solution inherits the regularity of the controlling noise, and thus, the solution is in D Remark 55. We would like to point out that the existence and uniqueness of Equation (5.25) requires one more degree of regularity on the diffusion coefficient f than what is standard for regular Rough differential equations (see e.g. [11] section 8). This higher regularity requirement comes from the fact that we need control of the Hölder regularity of the upper argument when composing a function with a controlled Volterra path, as seen in (5.23). This is in contrast to [27] where the authors only need a C 3 b diffusion coefficients. However, [27] is restricted to the case of a coefficient f such that f (0) = 0 and to Volterra equations with kernels which can be written as k(t, s) = k(t − s).
Remark 56. Although Equation (5.25) is a two parameter object, we can study the solution on the diagonal of ∆ 2 to obtain the classical type of one parameter Volterra equations. The Hölder continuity on the diagonal is already guaranteed by the Hölder topologies used on the space of controlled paths. In particular, there exists a unique solution to the equation y t ≡ y t t = y 0 + t 0 k(t, r)f (y r r )dx r , y 0 ∈ V.
One can easily check that t → y t ∈ C ρ for ρ = α − γ, where C ρ denotes the classical Hölder spaces of order ρ.

5.3.
Discussion. Theorem 54 tells us that for any T > 0 there exists a solution to Equation (5.25) on [0, T ] for any singular Volterra kernel satisfying (H) (in particular, as mentioned in Remark 55, we do not require a convolutional type of kernel like in [27]). Furthermore, since the extension developed here is fully based on the framework of classical rough path, one can also construct solutions to equations driven by lower regularity noise (i.e. with ρ = α − γ positive but lower than 1/3). In fact, let n be the whole number part of 1/ρ. One can extend Definition 46 to any regularity α by considering a formal expansion of a path to degree n such that the j-th Volterra-Gubinelli derivative is convoluted with the (j + 1)-th term in the Volterra rough path, namely y τ ts = n−1 j=1 z j,τ ts * y j,τ,· j ,...· 1 s + R τ ts , where R ∈ V (n−1)ρ+γ,γ 2 , and each derivative y j ∈ V α,γ for j = 1, . . . , n − 1. We will perform this construction more explicitly in a forthcoming paper.

Volterra rough path driven by Brownian motion
So far we have assumed the existence of a Volterra rough path z ∈ V (α,γ) driven by some Hölder noise x ∈ C α , where ρ = α − γ > 0 and k is a possibly singular Volterra kernel of order −γ satisfying (H). In this section we will give a non trivial example for which one can construct such a Volterra rough path, focusing on the case of a path x given as the realization of a d-dimensional Brownian motion.
Remark 57. The reader might argue that the Brownian case is already covered by [25]. However, the application of our general result to a Brownian situation brings some interesting continuity results for equation (5.25), which are not accessible to the Itô type theory developed in [25] (see [20] for applications to diffusion processes). It would obviously be satisfying to construct Volterra rough paths in the fractional Brownian case or for more general Gaussian processes verifying the assumptions of [10,16]. We postpone this construction to a further publication for sake of conciseness.
Let us now show that a realization of a Brownian motion gives rise to a Volterra path.
Corollary 58. Let B : [0, T ] → R d be a Brownian motion defined on a probability space (Ω, F , P). It is well known that there exists a set N c ⊂ Ω of full measure, such that for all paths ω ∈ N c the path t → B t (ω) ∈ C α for any α ∈ (0, 1 2 ). Consider now a Volterra kernel k : ∆ 2 → L(R d ) satisfying hypothesis (H) for some γ < α. Then we construct the path z 1 : ∆ 2 → R d given by where the integral is constructed as in Theorem 22. It follows that z 1 ∈ V (α,γ) for any α < 1 2 and γ < α.
Notice that in Corollary 58, the fact that the Brownian motion B has Hölder continuous trajectories of order α < 1 2 is a classical result and can be found in most text books on stochastic calculus (see for example [18]).
Our goal is to show the existence of the second level of the Volterra rough path related to z 1 . Namely we shall construct a d × d-dimensional process z 2,τ formally given by Since B is a d−dimensional vector, the product dB (u)⊗dB r forms a matrix of dimension d × d. We will therefore consider a component-wise version of the process z 2 defined by (6.2), i.e. We will show that for i = j we can use techniques which are similar to the ones used in the Volterra Extension Theorem 5, however in a probabilistic setting under L p (Ω) norm for p ≥ 2. On the diagonal (i = j) we will define the integral in the Itô sense z τ,2,i,i ts = t s k (τ, r) r s k (r, u) dB i u dB i r . (6.3) In the classical rough paths context when the kernel k is simply given by k = 1, the canonical choice for (6.3) is given by the Stratonovich integral. Indeed, the Stratonovich integral is obtained as the limit under approximations from smooth paths, and one thus obtain a geometric rough path from this choice. However, since we don't use the concept of geometric rough path for our construction leading to Theorem 54, we have chosen below to work with Itô integrals for sake of conciseness.
where P is a partition of [s, t]. In the end, we will show that the limit |P| → 0 exists, and is independent of the choice of partition. To this end we will consider the L 2 (Ω) norm of the approximating integral z 2 P , i.e. observe that E z 2,τ,i,j ts;P In order to analyze the right hand side of (6.8), denote by Q:[0, T ] 4 → R the co-variance function associated to the process given in (6.1), that is k (τ, r) k (τ ′ , r) dr.
Without loss of generality we can assume P refines P ′ , and also that there exists a ǫ > 0 such that P| ∨ |P ′ | = |P ′ | = ǫ. In particular, note that we can write the partition P in the following way Considering then the difference of approximating integrals z 2 P and z 2 P ′ , we invoke the bi-linearity of the convolution product * and observe that z τ,i,j ts;P ′ − z 2,τ,i,j ts;P = Since we have assumed γ < 1 4 , it follows that 1 − 4γ > 0. Therefore we can bound the right hand side of (6.14) in the following way where the integral converges whenever γ < 1 4 . Inserting relation (6.15) into (6.14) we thus get that z τ,2,i,j ts;P − z τ,2,i,j ts;P ′ L 2 (Ω) We conclude that z 2 as defined in (6.4) exists P − a.s. and the limit in L 2 (Ω) is independent of the choice of partition P. Notice that a classical argument based on dyadic partitions and Borel-Cantelli's lemma would also bring an almost sure convergence. The algebraic and analytic properties of z 2 given in (6.6) and (6.7) follows now from previously proven statements. Indeed, the Chen type relation (6.6) can be proven exactly in the same way as we proved the Chen relation for higher order iterated integrals in Theorem 42 (see Step 3 of the proof). For conciseness we encourage the patient reader to adapt this proof to the current problem. The regularity inequality in (6.7) follows directly from (6.12). This concludes the proof.
Remark 61. Although Theorem 59 gives us the correct algebraic condition for the second step of the Volterra signature in (6.6), the regularity condition in (6.7) is only the first step in providing the required regularity statement in order to guarantee that the tuple (z 1 , z 2 ) is a Volterra rough path according to Definition 38. In order to find a set N c ⊂ Ω of full measure such that for all ω ∈ N c the component z 2 (ω) satisfies the Hölder condition in Definition 38, one needs to extend the classical Kolmogorov continuity criterion to the Volterra-Hölder spaces introduced in Definition 16. We are currently working in this direction, together with a and extension of the well known Garsia -Rudemich -Rumsey inequality to the Volterra case. This will produce a simple criterion for the verification of the Volterra-Hölder continuity. For conciseness of the current paper, we will present these results in a future article.