Curvilinear Integrals Along Enriched Paths

1 Inspired by the fundamental work of T.J. Lyons (loc.cit), we develop a theory of curvilinear integrals along a new kind of enriched paths in IR d . We apply these methods to the fractional Brownian Motion, and prove a support theorem for SDE driven by the Skorohod fBM of Hurst parameter H > 1 / 4.


I. Introduction
In a remarkable work, T. Lyons [20,21,24] defined a differential calculus along continuous rough paths X(t). He used the notion of multiplicative functional which are bounded in p-variation. He then developped a calculus for geometric rough path for any p. In the case p < 3, the calculus also holds for the non-geometric case.
Here we extend a calculus even for the non-geometric case p < 4, with application to the Skorohod-fractional Brownian motion. We are led to a new notion of enriched paths, and the calculus works for Lyons enhanced path, if it is also enriched.
In fact we only use the α-Hölder continuity, this is not a restriction (cf. below).
Section I is the Introduction. In Section II, we prove a fundamental lemma (independant of the theory of rough paths, the so-called sewing lemma), which allows us to prove the convergence of very general Riemann-type sums. The proof uses a Hölder control, but it also holds with any control function as explained in corollary 2.3. Examples are given: existence of the Young integral, stochastic integral (Ito or Stratonovich), fractional Brownian motion, and also a very simple proof of the theorem of Lyons concerning the almost rough paths.
In Section III, we introduce the notion of an enriched path, which allows us to define curvilinear integrals (α ∈]0, 1] arbitrary). Notice that for α > 1/3, this notion is equivalent to the notion of p-rough path (p < 3). The equivalence does not hold for α ≤ 1/3. An application is given to enrich the Peano curve.
In Section IV, we deal with the case α > 1/3 for to solve the differential equation naturally associated to the theory of enriched paths. Here, an important tool is the Schauder fixed point theorem which applies thanks to the linearity of the enrichement (α > 1/3). For the unicity, we use the Banach fixed point theorem (of course with stronger hypotheses).
In this Section is also included a precise comparison with the theory of Lyons. In Section V, in view of application to the fBM, we deeply study the case α > 1/4. It turns out that for to solve a differential equation by the Picard iteration procedure, the good hypothesis is to have an enriched and enhanced path. Then the procedure works even in the non-geometric case.
In Section VI, we deal with the fBM with Hurst parameter H > 1/4. We first approximate the geometric rough path by C ∞ functions thanks to a martingale procedure. In a second time, this allows us to approximate the Skorohod fBM. Using the results of Section V, we solve SDE driven by the fBM.
In Section VII, we prove a support theorem for general stochastic enriched paths including the geometric or the Skorohod fBM.
Section VIII is an appendix: a technical lemma extending the Kolmogorov lemma, in relation with the sewing lemma.
We also show how to reparametrize a continuous bounded p-variation function F (s, t) to obtain a 1/p-Hölder control function.
Let k be an integer, and put v(a, b) Then v is semi-additive and has |v(a, b) − µ(a, b)| ≤ Cst |b − a| 1+ε . Hence v = u by uniqueness. It then follows that u(a, b) = u(a, c) + u(c, b) for every rational barycenter c ∈ [a, b]. The same holds for every c ∈ [a, b] by continuity of u. Putting ϕ(t) = u(0, t) gives the result. The uniqueness of ϕ follows in the same way as u.
Applications Example 1. The Young integral: Let x and y be α and β-Hölder continuous functions with α + β > 1, and put µ(a, b) = x a (y b − y a ). Then the sewing lemma yields a function ϕ, and the difference ϕ(b) − ϕ(a) is called the Young integral b a x t dy t Another way to define the Young integral is to take µ (a, b) = [a,b] x t dy t where [a, b] is the oriented line segment from the point (x a , y a ) to the point (x b , y b ). The resulting function ϕ is the same as above, as follows from the estimate |µ (a, b) − µ(a, b)| ≤ Cst |b − a| α+β .
It should be remarked that this extends to Banach valued functions x and y with a bilinear continuous map B(x, y) by putting µ(a, b) = B(x a , y b − y a ). The Young integral obtained can be denoted b a B(x t , dy t ) Example 2. The stochastic integral: Let X t be the IR d -valued standard Brownian motion. If f is a C 2 -vector field, consider the scalar product with the Ito integral in the right member. The sewing lemma applies thanks to N p (δµ(a, b, c)) ≤ Cst |b − a| 3/2 As easily verified, the obtained difference we would have obtained a similar estimate, and the obtained ϕ(b) − ϕ(a) would have been the Stratonovich integral of f (X t )• dX t . Moreover, observing that we recover the well known difference 1 2 b a div f (X t ) dt between the two integrals.
Example 3. The fractional Brownian motion: Let X H t be the IR d -valued fractional Brownian motion with the Hurst parameter H > 0 defined by dX H t has an obvious meaning for every polynomial P . We proved in [10] that this integral has an L p -valued analytic continuation for H > 1/4. Moreover we have the estimates for i = 0, 1, 2 For H = 1/2 we recover the Stratonovich integral. As we shall see below, we can also extend the "Ito integral" with respect to dX H t for H > 1/4. where X k ab belongs to T k (V ) the k-tensor product of a linear space V , indexed by the pairs {a, b} ⊂ [0, S] ⊂ IR, and satisfying the multiplicative relation X ab = X ac X cb for a ≤ c ≤ b, that is for every k ≤ n (for more clarity we dropped the symbol ⊗) It is continuous if every function (a, b) → X k ab is continuous (we suppose V of finite dimension). An n-truncated series is an n-almost multiplicative functional if for every a ≤ c ≤ b, we have only |X k ab − (X ac X cb ) k | ≤ ω(a, b) 1+ε with a control ω and an ε > 0. Hence we have the following, substantially due to Lyons [24, theorem 3.2.1]: 2.5 Theorem: Let X be an n-truncated continuous multiplicative functional, and let Y n+1 ab ∈ T n+1 (V ) be continuous and such that Y = (1, X 1 , X 2 , . . . , X n , Y n+1 ) is an n + 1-almost multiplicative functional. Then there exists a unique X n+1 ab such that Z = (1, X 1 , X 2 , . . . , X n , X n+1 ) is an n + 1-multiplicative functional with the condition The least constant is at most θ(ε) = (1 − 2 −ε ) −1 .
Let P m be the space of all real polynomials on IR d with degree ≤ m. We assume that we are given a linear map denoted P → I(P ) with values in the space C α ([0, T ], IR d ). We shall also denote We suppose the following property: for every ξ belonging to the convex hull of the image y[a, b] and every linear function f on IR d we have for k ∈ {0, . . . , m} The least constant in (3.1) is denoted I α .
If we take P = 1, we get a C α -function x with values in IR d .
We can introduce the notation 3.1 Definition: The pair (y, I) is called an enriched path, I is the enrichment of the path y.

Remark:
Even if y or x is C ∞ , the definition does not necessarily agree with the usual path integral.

3.3
Theorem: Let f be a C m+1 function on IR d . There exists a function ϕ(t) unique up to an additive constant, such that for every a, b ∈ [0, 1] where T (y a , y) is the Taylor polynomial of f at the point y a of degree m and Q ab the uniform norm on the image y[a, b].
Proof : Put Put ξ s = y a + s(y c − y a ) for s ∈ [0, 1], and ψ(s) = T (ξ s , y t ). We get where the dots stand for contracted tensor products. Hence where y α is the norm of y in C α . As (m+2)α > 1 the result follows from theorem 2.1. Besides, In particular, if f is a polynomial of degree m, we recover the integral b a f (y t ) dx t , since f coincides with every of its Taylor polynomial of degree m.
The hypotheses are the same as in section III, with m = 1.

Proposition:
Put We then have Conversely, let x and z be C α -functions with values respectively in IR d and IR d ⊗ IR d such that (4.2) holds. Then we can define an enrichment of y by putting Proof : Obvious.

Lemma :
Assume that x ∈ C α , α ≤ 1 and 0 < H ≤ T . On the space C α [0, T ], the following semi-norms are equivalent This is a non void convex set, and it is compact in the topology induced by Proof : Left to the reader.

Iteration
Let x t and y t be two enriched paths, such that I ab (1) = J ab (1) = x b − x a . We shall use the notations Let σ be a matrix valued C 2 function, put (with the dot for the matrix product)

Proposition:
There exists a C α function z t such that where the dot is a natural contracted tensor product. We have Applying the sewing lemma yields the expected function z t .
We shall denote Denote C 2 b the space of functions which are bounded with its derivatives up to order 2.

Theorem:
Suppose that σ ∈ C 2 b . The set K H (R) is invariant under the map (y, z) → (y, z) for R large enough and H sufficiantly small.
Proof : Put σ a = σ(y a ), σ a = ∇σ(y a ) and as above Proof : K H (R) is convex and compact in C β × C β for β < α by proposition 4.3, moreover the map is obviously continuous in this topology. So the Schauder theorem applies.

Corollary:
Assume that σ is C 2 b . Then the "differential equation" Then µ t = t 0 ν s ds has the same property, and the resulting function ϕ t is derivable in t.
Proof : Let ψ t be the unique function such that ψ t (0) = 0 and It is continuous with respect to (t, a, b). We get the result by putting Then (y, z) is weakly derivable in the direction of (u, v). Moreover the map (y, z) → (y, z) is a contraction in the set K H (R) for R large enough, H and T small enough, in the the topology of C α × C α .
Proof : Write again Denote D the derivative at t = 0, and assume that u α ≤ ρ, |U 2 ab | ≤ ρ|b − a| 2α and RT α ≤ 1 Denote M a constant only depending on X and σ. We have As RT α ≤ 1, each term in the right hand side is majorized by M Rρ|b − a| 3α , so that |δDµ 1 | ≤ 4M Rρ|b − a| 3α . By proposition 4.8, the variation u = ∆y satisfies Now compute the variation of Y 2 , denoted U 2 . Let ∆µ 2 be the variation of µ 2 . We have where µ 2 n is defined as in the proof of the sewing lemma. We have Next we have to estimate δ∆µ 2 . As above, we estimate δDµ 2 = Dδµ 2 . Recall that with dots as contracted tensor products. We get As seen above, the first term is majorized by M Rρ|b − a| 4α ≤ M ρ|b − a| 3α , the other ones are also majorized by M ρ|b − a| 3α . We then get Finally we obtain |U 4.10 Corollary: The differential equation (4.4) has a unique solution.
Proof : The Picard sequence converges in K R (T ), so that the fixed point is unique.

Remark:
In the frame of enhanced paths, this theorem is due to Lyons [24, corollary 6.2.2].

Comparison with the Lyons rough paths
First observe that X = (1, X 1 , X 2 ) defined in formulae (4.3) is a rough path in the sense of Lyons, which is easy to verify. Conversely, if we are given a rough path X = (1, X 1 , X 2 ), we can put We then get an enriched path (x, v).
If we are given an enriched path y with Conversely if we are given the couple (Y 1 , Y 2 ), satisfying the preceding relation, putting is an enriched path (y, z) in the sense of proposition 4.4.

Proposition:
Let (x t , z t ) be an enriched path, then is a symmetric tensor of class C 2α . Moreover, if we put We then say that z t is a normal enrichment of x t , it is the normal part of z t .
Proof : Left to the reader.

Remark:
We also have 4.14 Theorem: Let f be of class C 3 . We then have an "Ito formula" where the last integral is Young, or in differential form Proof : Obvious.

Remark:
Taking the normal part z t of z t we get We then have

An example: The enriched Peano's curve
There is a description of the curve γ(t) = z t = (x t , y t ) in Favard [7]. Recall that we have a sequence of piecewise linear curves t → z n t = (x n t , y n t ) ∈ IR 2 with vertices for t = k.9 −n , 0 ≤ k < 9 n . The sequence uniformly converges to a continuous curve z t , which is the Peano curve.
Let ∆ n be the set of numbers k.9 −n , ∆ = n≥0 ∆ n . The sequence z n is stationnary at each point of ∆. One then has |z t − z t | ≤ √ 2 |t − t| 1/2 for every t ∈ ∆ n , and t = t + 9 −n . Hence, there exists by the the combinational lemma of the appendix, a constant M 1 such that |z b − z a | ≤ M 1 |b − a| 1/2 for every a, b, so that z t is 1/2-Hölder continuous.
We first have A k 01 = 1 2 z 1 ⊗ z 1 for every k (easy computation), so that by the similarities of the Peano curve for t ∈ ∆ n , t = t + 9 −n and k ≥ 0. Next, if t = t + 9 −n we have A n+k tt = A n+k tt + A n+k t t + z tt ⊗ z t t Hence for every a, b ∈ ∆, the sequence A n ab converges stationarily to a limit A ab . Now, for t ∈ ∆ n and t = t + 9 −n , we have |A tt | ≤ |t − t|, and for a ≤ c ≤ b ∈ ∆ From the combinational lemma we first deduce an M 2 such that |A ab | ≤ M 2 |b − a| for every n ≥ 0, a, b ∈ ∆, next that A extends continuously on [0, 1] 2 . Finally (1, z, A) is a rough path, and we have an α-enriched curve for α = 1/2 > 1/3.
V. The case α > 1/4 Suppose that x and y are two enriched paths in the sense of section III, with I(1) ab = J(1) ab = where S is the symmetry of the tensor product defined by Proof : Straightforward.
Let σ be a C 3 matrix function on IR d . We want to define a triple (Y 1 , Y 2 , Z) with the same properties as the triple (Y 1 , Y 2 , Z). First define Hypothesis: there exists X 3 such that (1, X 1 , X 2 , X 3 ) is a rough path, and Y 3 such that for and |Y 3 ab | ≤ Cst |b − a| 3α (5.4)

Proposition: Put
where the dots are obvious contracted tensor products. There exists a function U such that Proof : Put σ a = σ(y a ), σ a = ∇σ(y a ). We get for a ≤ c ≤ b Every term is majorized by Cst |b − a| 4α , so that the sewing lemma applies, and the function U exists. The last claims are straightforward.

Proposition:
Put Every term is majorized by Cst |b − a| 4α , so that the sewing lemma applies, and the function V exists. The last claims are straightforward.

Proposition:
Put Then ν satisfies the conditions of the sewing lemma. Let W be a function such that satisfies the following relation Proof : If a function u(a, b) is such that u(a, b)/|b − a| 4α is bounded, then we write u ≈ 4α 0. In the same way we write u ≈ we infer that the first term in (5.5) is ≈ 4α 0.
As we have It remains to prove the last claims, this is straightforward.
Let (1, X 1 , X 2 , X 3 ) be a rough path. Let Y = (Y 1 , Y 2 , Y 3 , Z) as above. As we have seen, there exists Y = (Y 1 , Y 2 , Y 3 , Z) with the same properties as Y . By induction we get a sequence Y n = (Y 1 n , Y 2 n , Y 3 n , Z n ) of enriched paths with respect to X. The problem is to know if this sequence converges as for α > 1/3. For σ ∈ C 3 b , a proof analogous to the case α > 1/3 could be possible, by the use of many tedious computations, which are left to an upcoming paper.
In the particular case of geometric rough paths, this problem was solved by Lyons [24, theorem 6.3.1].

VI. The Fractional Brownian Motion
Many authors have studied the fBM with Hurst index H > 1/4, recent publications are [3,4,5,6,8,9,10,12,13]. Let X t be the IR d -valued linear standard Brownian motion. For H > 0, put We get a fractional Brownian motion with Hurst parameter H. Recall that we have for s, t ≥ 0 We get a Hilbert basis of the fist Wiener chaos. One has Hence the linear fractional Brownian motion writes If n is an integer, put Then the process x H,n has C ∞ paths. Moreover x H,n is a L p -martingale with values in the separable Banach space C H 0 which is the closure of C ∞ in C H . (Recall that H < H). Hence x H,n converges to x H in the space L p (C H 0 ) for every p ≥ 1, and almost everywhere.
Then we have for r < a so that the sequence J n ab is bounded. The convergence and (6.1) follow from the Pythagoras theorem. Then K n converges as n → ∞ to a limit K and Proof : Observe that y 2 belongs to L 2 so that For r > a, one has W r = U r + V r with Hence the sequence J n ab is bounded, the convergence and (6.2) follow as above.
All of these expressions are analytic functions of H > 1/4, and are all majorized by Cst |b − a| 2H for the three first, and by Cst |b − a| 3H for the last ones.
6.5 Proposition: Let X H t an IR d -valued fBM as defined in the beginning of this section. Denote X H ab = X H b − X H a as usual, and put all these expressions are defined, they are analytic functions of H > 1/4. For a ≤ c ≤ b, one has Moreover we have the estimates Proof : The three expressions are combinations of coordinates as defined above. The three equalities are obvious for H > 1, they follow for H > 1/4 by analyticity. The three inequalities follow from (6.1) and (6.2), as seen above.
6.6 Remark: The triple (X H , X H,2 , Z H ) is an enriched path in the sense of Section III, the quadruple (1, X H , X H,2 , X H,3 ) is a vector valued geometric rough path in the sense of Lyons [24].

Path regularity
In the sequel, we assume that 1/4 < H < H.

Theorem:
There exists a function F which belongs to every L p , such that Proof : The first inequality follows from the classical Kolmogorov lemma. The other three depend on the following lemma.

Lemma :
Let ∆ n be the subset of numbers k.2 −n for k ∈ {0, . . . , 2 n − 1}. If H < H, there exists a function F which belongs to every L p such that for every n ≥ 0 and every t, t ∈ ∆ n such that t − t = 2 −n , one has Proof : For any s, t one has with ε = 2H − 2H , p > 1/ε The function F p belongs to L p . Putting F = Inf p>1/ε F p yields the same inequality (the first in (6.7)) and F belongs to every L p . The same argument holds for the other formulae (6.7).
Proof of theorem 6.7: We first deal with X H,2 ab . Put where F belongs to every L p . The first conclusion follows by the combinational lemma of the appendix.
Next take µ(a, b) = Z H ab . We get by formula (6.4) where F belongs to every L p .
Besides for H = 1/2 (standard Brownian motion), Z t coincides with the Stratonovich integral (it suffices to take f ∈ C 2 b ).
Proof : The only point is to verify that Z t is the the Stratonovich integral for H = 1/2. This holds by definition if f is an affine polynomial. In general, for to define the integral of f , we put pathwise µ(a, b) = f (X a ).(X b − X a ) + ∇f (X a ).X H,2 ab We have (with dots as contracted tensor products) There exists F ∈ p L p such that |δµ| ≤ |b − a| 3H F , so that we get N 2 (δµ) ≤ Cst |b − a| 3H . As we have seen in Section II, the Stratonovich integral satisfies so that it coincides with the pathwise integral.
6.11 Remark: Many authors use a fractional Brownian motion which is a centered Gaussian process G H t with the covariance function |t − s| 2H . Thanks to [5], it can be represented in terms of the standard Brownian motion X t by the formula where is the tensorization of the Skorohod integral, as defined in [10]. Recall that we have where I is the Kronecker tensor. For H = 1/2, the Skorohod integral coincides with the Ito integral, as well known. It is straightforward to check that (1, X H , X H,2 ) is a rough path and also an enrichment in the sense of section III. Notice that this is a pathwise analytic function of H > 1/3, so that the Ito-Skorohod formula which was proved in [10] holds for every real polynomial F (here we have again a contracted tensor product).
For H > 1/4, we must define an enrichment (X H , X H,2 , Z H ). We shall also define X H,3 such that (1, X H , X H,2 , X H,3 ) be a rough path. Put For b > a > 0, it is easy to check that there exists F ∈ p L p such that In the same way, put for b > a > 0 for a vector field f of class C 3 on IR d , where div is the Euclidean divergence.
6.12 Remarks: a) For H > 1/4, t 2H is not C 3H until the origin. Nevertheless, as a → 0, Z H ab and X H,3 ab converge to limits which satisfy inequalities (6.7) on all of [0, 1]. b) In fact, we can replace I η t by any other tensor of class C 3α with α > 3/4.

VII. Support theorem for the fBM
This theorem was established for the ordinary Brownian motion in [18,25,26] and for the geometric fBM in [11,12].
where the integrals are taken in the sense of X and in the sense of Young. It remains to define X h,3 .

Corollary:
There exists a unique X h,3 such that (1, X h , X h,2 , X h,3 ) is a rough path.
We then have a canonical system X h .

Proposition:
The map T h is continuous (even locally Lipschitz) on E α , and we have the group property T h •T k = T h+k , so that (T h ) −1 = T −h .
Proof : Left to the reader.

Definition:
The skeleton S(X ) of E α is the subset constituted with the C ∞ 0 elements of E α (where the integrals are taken in the ordinary sense).
Observe that the group T h transitively operates on the skeleton.
We denote J α the closure of the skeleton, we have T h (J α ) = J α .

Remark:
For every X ∈ J α , (1, X, X 2 , X 3 ) is a geometric rough path. 7.6 Proposition: Let θ be a measure on J α which is quasi-invariant under T h , that is T h (θ) is absolutely continuous with respect to θ, for every h ∈ C ∞ 0 . If (0, 0) belongs to the support of θ, then J α is the support of θ.
Proof : Let X which belongs to J α . Let V be a neighbourhood of X in J α . There exists h such that T h (V ) is a neighbourhood of (0, 0), so that V is of positive θ-measure.

Application to the fBM
Let µ H be the law of the fBM X H as defined in section VI. This is a Gaussian measure, so that it is quasi-invariant by the Cameron-Martin translations. Observe that C ∞ 0 is included in the Cameron-Martin space. Hence Proof : Consider the approximation X n of X defined with the help of the martingale X H,n t in section VI. As X n belongs to J H and converges in L p (E H ) and almost everywhere as proved in section VI, then X belongs to J H , so that θ H is supported by J H . The quasi-invariance follows from the quasi-invariance of µ H .

Lemma :
For θ H -almost every X , the following property holds For H > 1/4, a similar proof yields the result.
7.9 Theorem: (Support theorem) The support of θ H in E H is exactly the closure of the skeleton.
Proof : By proposition 7.7, we only have to prove that every neighbourhood of (0, 0) has a positive θ H -measure. Let V be such a neighbourhood, and let X a point in the support of θ H for which the property of the lemma holds. Let n be such T −X H,n (X ) ∈ V . We have θ H [T X H,n (V )] > 0, so that θ H (V ) > by the quasi-invariance.
7.10 Corollary: Let σ ∈ C 3 b be matrix valued. Put Then the support of the law of y is the closure in C α of the More generally, let η t be a tensor of class C ∞ 0 . Consider the map X → X as defined in section VI X = (x, x 2 , x 3 , z) The support of the law of the η-peturbated fBM is exactly J H , that is the closure of the η-skeleton (the elements X which are C ∞ 0 ). The only point is that integration along the C ∞ -skeleton is not taken in the usual sense.
7.13 Remark: In the case of the Skorohod fBM, we have η t = c(H) 2 I.t 2H /4H which is not C ∞ until the origin. Nevertheless, analogous results hold (corollaries and proposition) with less regular skeleton, thanks to the remark 6.12 a).

VIII. Appendix
The combinational lemma Let ∆ n the set of dyadic numbers k.2 −n for 0 ≤ k < 2 n . : Let µ(a, b) be a real or Banach valued function defined on ∆ × ∆, such that for n ≥ 0, t ∈ ∆ n , t = t + 2 −n one has |µ(t, t )| ≤ k|t − t| 2α where n i > n 0 is an increasing sequence of integers (finite since b ∈ ∆). Put
Hölder norms, and p-variation If p = 1/α, we have the obvious inequality There is a kind of converse property. Suppose that V p (F ) is finite. Define V (s, t) as the pvariation of F restricted to the segment [s, t], and put w(s, t) = V (s, t) p . For s < t < u we have w(s, u) ≥ w(s, t) + w(t, u).
Proof : Put v(t) = w(a, t), u(t) = w(t, b). As F is continuous, v is a non-decreasing l.s.c. function, then a left continuous function. In the same way, u is right continuous.
We have w(s, t) + v(s) ≤ v(t). As s ↑ t, we get Lim Sup s↑t w(s, t) + v(t) ≤ v(t) and w(s, t) converges to 0. By the same argument, we have w(s, t) + u(t) ≤ u(s). As t ↓ s, we get Lim Sup t↓s w(s, t) + u(s) ≤ u(s) and w(s, t) converges to 0.

Proposition:
The function v(t) = w(a, t) is continuous.
Proof : Let a ≤ s ≤ t. One has v(t) = Sup (A(t), B(t)), where where Σ t s denotes the set of subdivisions σ of [a, t] such that s is a point of subdivision. First A(t) = v(s) + w(s, t) converges to v(s) as t ↓ s.
Next, suppose that t > s converges to s along a sequence such that B(t) > M . For each t there exists σ ∈ Σ t s such that s ∈]t i , t i+1 [ and Then t i+1 ∈ [s, t] converges to s, and |F (t i , s)| p − |F (t i , t i+1 )| p converges to 0 (uniform continuity), hence M ≤ v(s). It follows that b) Take F (s, t) = f (t) − f (s) where f is a continuous function with finite p-variation, then f is 1/p-Hölder continuous for a suitable reparametrization.