Singular paths spaces and applications

Abstract Motivated by recent applications in rough volatility and regularity structures, notably the notion of singular modeled distribution, we study paths, rough paths and related objects with a quantified singularity at zero. In a pure path setting, this allows us to leverage on existing SLE Besov estimates to see that SLE traces takes values in a singular Hölder space, which quantifies a well-known boundary effect in the regime We then consider the integration theory against singular rough paths and some extensions thereof. This gives a method to reconcile, from a regularity structure point of view, different singular kernels used to construct (fractional) rough volatility models and an effective reduction to the stationary case which is crucial to apply general renormalization methods.


Introduction
In classical analysis, the improper integral 1 0+ f (t)dt := lim ε↓0 1 ε f (t)dt is studied, with application in many parts of pure and applied mathematics.It is a basic observation of this paper that this extends to improper Young and then rough integrals, as well as recent ramifications to rough volatility within regularity structures [2], [9,Ch.14],which provided the original motivation for this article.The notion of singular controlled rough path, in a Hölder setting, is seen to be consistent with Hairer's singular modelled distributions [15].We give a self-contained presentation of these spaces including a new time-change characterisation.Moreover, we will introduce the notion of singular rough paths, which can be translated in the language of regularity structures as a "singular model" defined over the rough path regularity structure, see [9,Ch. 13].
Our first application of such singular spaces comes from the seemingly remote field of SLE theory.More specifically, the estimates recently obtained in [11] are seen to imply also singular Besov regularity (w.r.t. to half-space capacity parametrisation) of (chordal) SLE trace.We briefly recall that the classical theory of Loewner evolution gives a one-to-one correspondence between scalar continuous real-valued functions (ξ t ) t≥0 and families of continuously growing compact hulls (K t ) t≥0 in the complex upper half-plane H = {z ∈ C : Im(z) > 0}.There is much interest in the case where these sets admit a continuous trace: i.e. there exists a continuous curve (γ t ) t≥0 ⊂ H such that, for all t > 0, H \ K t is the unique unbounded component of H \ γ([0, t]) (or even K t = γ([0, t]) with γ a simple curve).The famous Rohde-Schramm Theorem [20] asserts that Brownian motion with diffusivity κ = 8 a.s.gives rise to a continuous trace, which is simple when κ ≤ 4.This process is known as (chordal) SLE trace, denoted by γ k .Even for the smoothest possible driver ξ t ≡ 0, with explicitly know trace 2i √ t, this function is smooth away from zero.This effect remains visible for γ k , for κ ≤ 1, which enjoys Hölder regularity of exponent α > 1/2 on compacts away from zero, but (trivially) of exponent 1/2 on intervals [0, T ].We will refer to this phenomenon as the boundary effect of SLE at t = 0+.Upon extension of classical embeddings to the singular case, we can use singular Hölder spaces to quantify this boundary effect.Using the notion of C α,η ((0, T ]) space in Definition 2.1, we obtain when κ ∈ (0, 1) the following a.s.regularity result 2 ) − ((0, 1]; H) , where α − * and (1/2) − should be read as any 0 < η < 1/2 and any α < α * = α * (κ) is given by See Theorem 4.2.This refines classical results of Lind [18] and Johansson Viklund, Lawler [16] (1.2) By our characterization of C α,η spaces, from (1.1) we also see that, for κ ∈ (0, 1), (1.3) t → γ κ (t 2 ) ∈ C α − * ([0, 1], H) .See Corollary 4.3.Noting that α * (κ) decreases from 1 to 1/2, as κ ranges from 0 to 1, this constitutes a natural SLE extension of know deterministic results ( [10,21] and the references therein) that {t → γ(t 2 )} ∈ C 1 ([0, 1]; H), for traces γ driven by classes of finite variation paths with some additional properties, like finite energy or being locally regular.An additional heuristic reason behind (1.3) is also related to parametrisation of γ κ , the half-plane capacity, which should behave heuristically like the square root of the intrinsic length, see [16].Then the reparametrization t → γ κ (t 2 ) can be interpreted as a "parametrisation by arc length", which absorb the boundary effect.We then see that the SLE trace gives rise to (singular) rough paths, in the spirit of [22,3], but insisting on the finer Hölder scale (instead of p-variation).
Our second application deals with an aspect of fractional stochastic modelling.In particular, we want to reconcile "similar" definitions of a fractional Brownian motion in a rough analysis perspective.Loosely speaking, we consider the process ξ = dW is a real-valued white noise and * is the operation of convolution.To regain analytic stability in the rough path sense, we need to enrich W H with Itô objects like (W H )dW or equivalently W H ξ. This construction has been useful in the analysis of rough volatility models [2,8], where the modelling (e.g.[1]) imposes a Riemann-Liouville interpretation of W H , i.e. as left-hand side of (1.4) However, when one applies general results in the theory of regularity structures, stationarity of the noise is important, as is the assumption of a compactly supported kernel K H with the right fractional behaviour near 0, which points to a different interpretation of W H , namely the right-hand side of (1.4).As a prototypical example, we consider the Wong-Zakai approximation of the random distribution W H ξ, obtained from mollified noise ξ ε = ξ * ̺ ε , where ̺ ε (t) = ε −1 ̺(ε −1 t) for some smooth, compactly supported function ̺ : R → R such that ̺ = 1 and ε > 0. If we interpret W H as a Riemann-Liouville fBM, the convergence in a rough (model) topology of the quantity W ε,H ξ ε , suitably renormalised, was done by hand in [2], also for non-stationary wavelet approximations.On the other hand, the right-hand interpretation of (1.4) allows for the use of the general BPHZ renormalisation theorem [6] which in the present setting asserts that W ε,H ξ ε converges upon subtracting its (constant) mean.
As it turns out, singular path spaces allow regarding the (non-stationary) left-hand side as singularly controlled by the right-hand side, thereby providing an elegant way to reconcile the difference in (1.4).A related construction appeared in recent work in the setting of singular SPDEs with boundary [12].Thanks to singular rough integration (reconstruction), statements like with diverging Itô-Stratonovich correction, are conveniently obtained, see Theorem 6.3.We restrict our analysis to H ∈ (1/4, 1/2).This is only important to the extent that W H ξ is "enough" to make rough analysis work, otherwise branching objects of the form (W H ) k ξ, for k = 1, 2, ..., K(H), are required for robust integration.Since these considerations are well-known and somewhat orthogonal to the "singular" theme of this note, we have not pushed for generality in this direction.Similarly, as noted in [2], (possibly multidimensional) Itô-Volterra equations of the form admit a rough solution theory, by a fixed point arguments in controlled rough path (modelled distribution) space, once the underlying rough paths (model) has been constructed.(This step does not require any stationarity of the noise).Wong-Zakai approximation again requires renormalisation, and for appealing to general results [5,4] stationarity is again crucial.The same approach then applies, with (1.5) solved in a space singular controlled rough path (singular modelled distribution), relative to a stationary model that fits, as before, in the BPHZ setting of [6].For instance, in case H > 1/4 this reasoning shows that the sequence of solutions Y ε converges to Y in probability uniformly on compact sets.Singular spaces are thus seen to provide an approximation theory for Itô-Volterra problems, commonly used in fractional modelling, as bona fide perturbations of stationary noise problems in the setting of regularity structures for which general results are available.

Singular paths and spaces
2.1.Hölder path spaces.We are interested in the following generalisation of the usual Hölder spaces.
Definition 2.1.Let 0 < α < 1 and η ≤ α.We say that a continuous path This is a Banach space with norm given by |Y T | + Y α,η .The classical Hölder space C α ([0, T ]) can be identified with C α,α ((0, T ]).The regime η > α amounts to enforce a certain vanishing at 0 and is of no interest to us.We have the following characterisation of C α,η ((0, T ]) in terms of • α;I the Hölder semi-norm on the interval I.
Proof.Let Y ∈ C α,η ((0, T ]).By definition of C α,η ((0, T ]) and the monotonicity of s η−α , for every 0 < ε < 1 and every ε ≤ s < t < T one has By taking the sup on s and t we obtain Conversely, for any ε ∈ (0, T ], (2.3) Since Y is α-Hölder on [ε, T ] we have trivially By taking the sup on ε and t, one has Y α,η ≤ C and the proof is finished.
Since we are considering the sup over a fixed set (0, T ], we can equivalently modify the underlying set of values s such that the quantities |t − s| and s are comparable.The following equivalence of seminorms will allow us to compare singular Hölder spaces with singular modelled distributions in section 5.
Lemma 2.3.Let α ∈ (0, 1), δ > 0 and η ≤ δ.For any function Y : (0, T ] → R there exists a constant C > 0 depending on α, η, δ such that one has Proof.We use the shorthand notation [Y ] to denote the seminorm where the sup contains the condition |t − s| ≤ s.We fix 0 < s < t and using the shorthand notation N = ⌊log 2 (t/s)⌋, we consider the sequence of values (2.5) Since these values satisfy |s n+1 − s n | ≤ s n for every n ≤ N , we can iterate the triangle inequality obtaining (2.6) When α + η − δ > 0 we can easily estimate the second line of (2.6) obtaining Using the definition of N and the Hölder behaviour of the function t → t α there exists a constant C ′ > 0 depending on α, η, δ such that Therefore we obtain Dividing this expression by s η−δ (t − s) α and taking the sup, we conclude.In case α + η − δ = 0, we can repeat all the calculation in (2.6), obtaining the inequality Using the elementary estimate N ≤ 2 αN − 1 α ln(2) and the definition of N , there exist a constant c > 0 depending on η, δ and α, such that In case α + η − δ < 0 we bound the second line of (2.6) with Using again the definition of N , we see that when N ≥ 1 then s < |t − s| and there exists a constant c ′ > 0 such that thereby obtaining the result.
Proof.We suppose that Y ∈ C α,η ((0, T ]) and we adopt the notation β = α/η.By hypothesis, there exist two constants M, M ′ > 0 such that for any t > s > 0 one has We conclude that the re-parametrisation can be uniquely extended to a α-Hölder function on [0, T ].On the other hand, if the function t → Y (t β ) is α-Hölder there exists a constant C > 0 such that for any 0 < s < t ≤ T one has Diving both sides by s η−α |t − s| α , we conclude characterisation.
2.2.Improper Young integration.For any couple Y ∈ C α1 ((0, T ]) and X ∈ C α2 ((0, T ]) satisfying α 1 + α 2 > 1, we can define for any t > s > 0 the Young integral Following e.g.[9,Chap.4]there exists a constant C > 0 such that for every T > t > s > 0. The following theorem establishes a sufficient criterion to extend this integral to (0, T ] by showing the existence the corresponding improper Young integral. for any 0 < t ≤ T .We call the left-hand side of (2.9) the improper Young integral.As function of t it belongs to C α2,η ((0, T ]), where η := η 1 ∧ 0 + η 2 > 0. Moreover there exists a constant M > 0 depending on T and the previous parameters such that (2.10) Remark 2.6.Before giving the proof, we stress the optimality of the conditions via the example Y : t → t η1 and X : t → t η2 .In this case, T s Y dX = (const)T η1+η2 − s η1+η2 and the lim s→0 + exists iff η 1 + η 2 > 0. On the other hand, taking Y ≡ 1 shows that X must extend continuously from (0, T ] to [0, T ], which is captured by the condition η 2 > 0. We also remark that the condition η 1 + η 2 > 0 allows η 1 to be strictly negative, which is the non-trivial part of this result.Indeed if η 1 > 0 we can use Proposition 2.2 to conclude that X, Y have appropriate p-variation regularity and the result will follow from classical Young integration in the p-variation setting. Proof.It is sufficient to prove the convergence (2.9) when t = T = 1.In order to show the convergence to some value, we will firstly prove that Thanks to Proposition 2.2, we can derive some bounds of each quantity in the right-hand side above.Iterating the triangle inequality, there exists a constant C ′ > 0 such that for every n ≥ 1 Plugging this estimate in (2.11), there exists a constant c > 0 such that (2.12) Then the conditions in the statement guarantee that I n is a Cauchy sequence converging to some limit I such that for every n ≥ 0 Let us prove the convergence of the whole sequence in the right-hand side of (2.9).For any fixed ε ∈ (0, 1], we take N such that 2 −N < ε ≤ 2 −N +1 and applying the estimate (2.8) with s = 2 −N and t = ε, one has Then, we can repeat exactly the calculations written above to obtain the estimate Combining the two inequalities (2.13) and (2.14), we obtain the convergence (2.9).Finally let us show that . The case η = α 2 follows trivially from (2.13).When η < α 2 we use the characterisation (2.2) obtaining By means of the subadditivity of the Hölder semi-norm and the inequality (2.12), we obtain for any n ≥ 1 for some constant c ′ > 0. Therefore • 0 + Y r dX r α2,η is finite and one has the inequality (2.10).The general case when T and t are arbitrary follows from the scaling properties of Hölder semi-norms.
Remark 2.10.We note a notational clash, (classical) Besov Y δ,q vs. singular Hölder Y α,η , but will not try to resolve this for the simple reason that such Besov spaces will not play a central role in this paper.Definition 2.8 is done for real-valued paths and we can trivially extend it when Y takes value in the complex upper half-plane H (see Section 4).

Singular rough path spaces
We extend the previous on Young integration in Theorem 2.5 to usual rough integration with respect to a specific two-level family of rough paths.In what follows all the definitions are given with respect to two fixed values α, β ∈ (0, 1) such that α + 2β > 1.
Definition 3.1.An inhomogeneous rough path is a triple X = (X, X, X) where We denote the set of inhomogeneous rough paths by C ([0, T ]).
For any given inhomogeneous rough path we define the corresponding space of paths which can be integrated against it.
).We denote the set of inhomogeneous controlled rough paths by D γ+β X ([0, T ]).If the remainder R Y is taken with respect to X and R Y ∈ C γ+α ([0, T ] 2 ), we also say that (Y, Y ′ ) is an inhomogeneous controlled rough path and we denote the associated space by D γ+α X ([0, T ]).
We introduce their corresponding singular version.Definition 3.3.Let η ≤ α + β.We say that X = (X, X, X), defined on (0, T ] is a singular inhomogeneous rough path, in symbols X ∈ C η ((0, T ]), if it satisfies If R Y is taken with respect to X and we replace β with α, we denote the related space by D γ+α,η X ((0, T ]).
Using the intrinsic norms of inhomogeneous rough paths and controlled rough paths1 , one can obtain the same type of equivalences given in Proposition 2.2.
Proof.The proof follows in same way as Proposition 2.2.
From (3.2) one has immediately the identity C α+β ((0, T ]) = C ([0, T ]) moreover one has the continuous embedding Similarly This function belongs to C α ((0, T ]) and it satisfies the inequality for some fixed constant C > 0. We describe some sufficient conditions to show the existence of an improper rough integral in this case.
We call the left-hand side of (3.5) the improper rough integral.The path Z belongs to C α,η ((0, T ]) and the couple (Z, Y ) belongs to D β+α,η X ((0, T ]), where η := η 2 − β + (η 2 − α + (η 1 − β) ∧ 0) ∧ 0.Moreover, there exists a constant M > 0 depending on T and the previous parameters such that Proof.The proof is obtained using the same strategy as Theorem 2.5.We fix again t = T = 1 and we introduce the notation Iterating the triangle inequality, there exists a constant C ′ > 0 such that for every n ≥ 1 one has which implies Passing to the first term in the right-hand side of (3.8), we can apply again (3.9) to obtain for any n ≥ 1 (3.11) Iterating recursively this estimate, we obtain that there exists a constant C ′′ > 0 such that Therefore we obtain Coming back to the initial estimate (3.8), there exist two constants c, c ′ > 0 such that (3.12) By hypothesis on the parameters η 1 and η 2 one has η > 0 and the sequence I n is a Cauchy sequence converging to some value I. Following the last part of Theorem (2.5), we can repeat the same steps to deduce the convergence (3.5) and the first inequality (3.6).Let us show that the couple (Z, Y ) belongs to D α+β,η X .Firstly, we apply (3.11) to show that there exists a constant c ′′ > 0 such that (3.13) and since (η 2 − α + (η 1 − β) ∧ 0)) ∧ 0 + α > η we conclude.On the other hand, we deduce from (3.4) and (3.10) that there exists a constant c ′′′ > 0 such that the function We introduce for every n ≥ 1 the sequence By splitting the above sup into two intervals and applying the two previous estimates (3.13) and (3.14), we obtain for every n ≥ 1 Iterating recursively this inequality, we obtain a general estimate on the sequence γ n which implies trivially (3.7) and we conclude.The case of a generic T > 0 is covered using the scaling properties of the norms X η2 and Y, Y ′ γ+β,η1 .
Remark 3.7.The present theorem is a generalisation of the classical rough integration as introduced in [13].Indeed it is sufficient to set α ∈ (1/3, 1/2], α = β and X = X to recover it.However, the proof of our result can be adapted to cover rough integration in a more general context when the underlying rough path is, for example, branched or geometric, see [19,14].Compared to Definition 3.1, these notions assume some additional algebraic conditions in their formulation but keep essentially the same Hölder-type structure.In addition, it is also possible to state an equivalent notion of rough integration for branched and geometric rough paths, which extends (3.4).Therefore, any new definitions, like singular-branched rough paths or singular-geometric rough paths, can be easily transferred to this new context.Moreover, any extension of Theorem 3.6 to these objects will involve only a careful check of more sophisticated dyadic powers without changing the proof strategy.Hence, we trust that the reader will be able to make suitable amendments on their own.

Application to Schramm-Loewner Evolution
For any κ > 0, we consider the family of conformal maps (g t ) that are the maximal solution to Loewner's equation where U t = √ κB t with (B t ) t≥0 a standard Brownian motion.Using the notations f t = (g t ) −1 and ft (z) := f (z + U t ), the SLE trace γ κ (parametrised according to the half-plane capacity) can be defined a.s.for all t ≥ 0 as the limit γ κ (t) := lim u→0 + ft (iu) see [20,17].In [11] the Hölder regularity and p-variation of γ κ were studied.In essence, there exists set I(κ) ⊂ [0, ∞) depending κ ∈ (0, ∞) and a constant C > 0 not depending on κ such that for any r ∈ I(κ) one has the estimate where the parameters are functions q = q(r) and ζ = ζ(r) are given by 2 We recall from [11] that, at least for κ ∈ (0, 8) (our later interest concerns κ ∈ (0, 1) where boundary effects play a role) the set I(κ) is defined introducing the auxiliary sets And then we define I(κ) := I 0 (κ) ∩ I 1 (κ) ∩ I 2 (κ).Since we want to study the trace of SLE using the singular Besov spaces introduced in Definition 2.8, we introduce the set and we restrict the values of r in I(κ) ∩ J 2 (κ).Thanks to [11,Lem 5.1] one has the characterisation (4.2) and one has trivially that the set (1/q, (ζ + q)/2q)) is non empty.
2 Strictly speaking, the exponents must be modified by arbitrarily small ǫ, but for power counting arguments, this is good enough.
Proof.The result comes immediately from the definition of the singular Besov norm and the estimate (4.1).Indeed we obtain Thus we only need to check that the choice of η and r in the statement implies that the integral on the right-hand side of (4.3) is finite.First of all the condition on η prevents the integral to be infinite at s = 0 indeed one has trivially On the other hand to provide the integrability at the diagonal t = s the condition δ < Thereby obtaining the result.
Applying Proposition 2.4 to the SLE trace, we obtain an interesting property of its trajectories, which was summarised in the property (1.3).
Proof.By direct inspection, this is true for γ 0 (t) = 2i √ t, so assume κ ∈ (0, 8).We first treat the case κ ≥ 1 in which case we can ignore boundary effects and singular parameter.In view of the p-variation regularity established in [22,11], namely p = 1 + κ − /8, iterated integrals are well-defined, but this does not quantify any Hölder control.We thus need to use, the control function ω that emerges from the Besov regularity, i.e. the function ω(s, t) , where C is a specific numerical constant, as established in [7] and • δ,q is the Besov norm.Applying this control ω, we obtain immediately the correct Hölder regularity of γ κ , see [11, Theorem 4.2, Theorem 6.1].Moreover the same control ω provides the right estimates to obtain the correct Hölder regularity of the iterated integrals, which identifies γ κ and its ⌊1/α⌋ iterated integrals as α-Hölder rough path.(Since α < α * (κ) ≤ 1/2, it is inconsequential to regard it as α-Hölder rough paths with singularity parameter η).It remains to deal with the case 0 < κ < 1.In this case α * (κ) > 1/2 away from 0, so that the trace is a level-1 (a.k.a.Young) path on [ε, 1], iterated (Hölder) Young integration is valid.On [0, 1], because of (1.2), this argument fails with classical iterated Young integration, but we can reinstall it via improper Young integration of Theorem 2.5 where α 1 , α 2 > 1/2 and η 1 , η 2 > 0. Details are left to the interested reader.

Connection with regularity structures
Inhomogeneous rough paths can be equivalently described using the formalism of regularity structures (We refer to [15,9] and we suppose the reader is familiar with its main concepts).To see this link we introduce the vector space where each subspace T l is one-dimensional.We represent the element of its canonical basis with the notations We introduce G, a group of linear automorphism Γ h : T → T , defined for any The couple (T , G) is a simple example of regularity structure.Following [9,Lemma 13.20], we can rewrite every element X ∈ C ([0, T ]) as a specific model Π(X) over (T , G) restricted over [0, T ], see [15,Definition 2.17] for the general definition.In few words, Π(X) is a couple of elements Π(X) = (Π X , Γ X ).The first one is a map Π X : [0, T ] → L(T , S ′ ([0, T ])) 3 which is given by where Ẋ, Ẋ Ẋs are respectively the distributional derivatives of X, X and r → X s,r and ψ : (0, T ) → R is a generic test function.The second one is a two-parameters map Γ X : [0, T ] 2 → G which is defined by Following this correspondence, we can also describe every element (Y, Y ′ ) ∈ D γ+β X ([0, T ]) or Y ∈ C α ([0, T ]) in terms of modelled distributions over Π X , see [9,Lemma 13.20] and [15,Definition 3.1] for the general definition.It turns out that the same relation can be used to describe singular controlled rough paths in terms of singular modelled distributions, see [15,Chapter 6].To define them, we use the shorthand notation | • | l for the absolute value of the l-th component in T .
Proof.Let us prove the left-hand inequality in (5.6).We decompose Y D γ+β,η as the sum Using the definition of D γ+β,η X , the definition of Γ s,t in (5.2) and the trivial estimate (5.4) one has trivially Y 2 ≤ Y, Y ′ γ+β,η .In addition, for any 0 < s ≤ T we have Thereby obtaining the first part of (5.6).On the other hand, Let Z and Z ′ denote the two component of Z.
Thanks to Lemma 2.3 with α = γ and δ = γ + β there exists a constant D > 0 such that one has sup 0<s<t≤T Combining this inequality with (5.4) and the trivial estimate we obtain the right-hand inequality in (5.6), as long as we are able to estimate R , we can achieve this estimate by repeating the procedure in the proof of Lemma 2.3.Thus for any 0 < s < t ≤ T and η ≤ γ + β we consider N = ⌊log 2 (t/s)⌋ and there exist two constants C ′ , C ′′ > 0 depending on η, γ and β such that Thereby obtaining the second part of bound (5.6).The equivalence among D γ+α,η X ((0, T ]) and D γ+α,η ((0, T ]) follows immediately by replacing β and X as in the statement.
Remark 5.3.Using this equivalence, the general tools of regularity structures can provide an alternative proof of Theorem 2.5 and 3.6 when η 2 is respectively α and α + β.We will give a simple sketch of it in this second case.Starting from a singular controlled rough path (Y, Y ′ ) ∈ D γ+β,η1 X ((0, T ]), we consider Y ∈ D γ+β,η1 ((0, T ]) given in (5.5) and we define the map YΞ : (0, T ] → T as (YΞ) t := Y t Ξ + Y ′ t XΞ .This function is an example of product between two singular modelled distribution and it is an element of D γ+β+α−1,η1+α−1 ((0, T ]) (see [15,Prop 6.12]).Writing down the hyphotesis of Theorem 3.6 when η 2 = α+β, we have that η 1 + α − 1 > −1.This condition, together with the assumption α + β + γ − 1 > 0 allows us to apply the reconstruction theorem for singular modelled distribution [15, Prop.6.9].Loosely speaking, the theorem states that for any Z ∈ D δ,η ((0, T ]) satisfying δ > 0 and η ∧ α − 1 > −1 we can uniquely associate a distribution R(Z) ∈ S ′ ([0, T ]) which satisfies a bound of the type , where ψ λ s = λ −1 ψ((• − s)/λ), ψ is a generic smooth, compactly supported function ψ : (−1, 1) → R and λ is sufficiently small.Writing this general result in the case of YΞ, we obtain then a unique distribution R(Y Ξ) such that ) .Comparing (5.8) with (3.4), we realise that R(Y Ξ) is the distributional derivative of the improper rough integral and one has the identity up to addition of constants (5.9) where the right-hand side of (5.9) is the primitive of R(Y Ξ) that cancels in zero.To define this operation, it sufficient to test R(Y Ξ) against a sequence of smooth functions approximating the indicator of the set [0, t] and to show that there is a unique limit.The additional properties of the improper rough integral can be deduced by studying the analytic regularisation of taking the primitive of a distribution.

Application to Fractional Modelling and Rough Volatility
Recent advances in quantitative finance led to models where (stochastic) volatility runs on "rougher" than diffusive scales, locally described by a Hurst parameter H ∈ (0, 1/2).Following [1,2], moves in log-price then involve stochastic Itô integrals of the form (6.1) where f is a sufficiently smooth map, W is a standard Brownian motion, which we assume two-sided so that ξ = Ẇ is white noise on R. Lévy or Riemann-Liouville fractional Brownian motion is then given by (6.2) with Riemann-Liouville kernel K H (u) = u H−1/2 1 u>0 , and ξ + white noise on R + , given as distributional derivative of W 1 R+ .There is interest e.g. from asymptotic option pricing [2,8]) to have a rough path type stability for this integral.For H < 1/2, this requires an extension of rough paths that constitutes a most instructive example of a non-trivial regularity structures [2].In order to apply the standard results of the general theory developed in [15], it is desirable -and necessary for BPHZ renormalisation à la [6] -to replace the process (6.2) by a stationary process, which we may take of the form (6.3) W H t := ( K H * ξ)(t) , where ξ is a white noise on R and K H is a smooth function K H : R \ {0} → R + satisfying the properties: (1) K H ≡ K H on [0, T ] and supp( K H ) ⊂ [0, 2T ]; (2) there exist a constant C > 0 such that for k = 0, 1, 2 one has The existence of such function is a trivial exercise in case of the Riemann-Liouville kernel.We now show that W H can be regarded as singular controlled rough path when H > 1/4.The reference inhomogeneous rough path W, over stationary noise, is obtained by considering the triplet of functions (6.4) where the integral in dW is an Itô integral.Using some standard tools of stochastic calculus one has that the conditions (6.4) defined an element W which belongs a.s. to C ([0, T ]), where α = 1/2 − and β = H − .Before stating a rigorous result on W H , we recall an elementary property of the function In case when δ ∈ (0, 1), we consider a standard sequence of mollifiers ϕ ε on [0, T ] such that ϕ ε = 1 and defining B ε = ϕ ε * B we can combine (6.7) and (6.8) to obtain