Rama Cont

Purba Das

Quadratic variation and quadratic roughness

Keywords: Quadratic variation, Pathwise integral, Itô calculus, Local time

come

The concept of quadratic variation plays a central role in stochastic analysis and the modern theory of stochastic integration [START_REF] Dellacherie | Probabilities and potential[END_REF][START_REF] Protter | Stochastic integration and differential equations[END_REF]. The quadratic variation of a (real-valued) random process (X(t), t ∈ [0, T ]) with càdlàgsample paths is defined as the limit in the sense of (uniform) convergence in probability, of the sum of squared increments πn (X(t n k+1 ∧ t) -X(t n k ∧ t)) 2 (1) along a sequence of partitions

π n = (0 = t n 0 < t n 1 < • • • < t n N (π n ) = T ) with vanishing step size |π n | = sup i=1..N (π n )
|t n i -t n i-1 | → 0. The relevance of this notion is underlined by the fact that large classes of random processes -such as Brownian motion and diffusion processes-have finite quadratic variation, while at the same time possessing infinite p-variation for p = 2.

Although quadratic variation for a stochastic process X is usually defined as a limit in probability of [START_REF] Ananova | Pathwise integration and functional calculus for paths with finite quadratic variation[END_REF], it is essentially a pathwise property. In his seminal paper Calcul d'Ito sans probabilités [START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF], Hans Föllmer introduced the class of càdlàg paths X ∈ D([0, T ], R) with finite quadratic variation along a sequence of partitions (π n ), for which (1) has a limit with Lebesgue decomposition [X] π (t) = [X] c (t) + 0≤s≤t (∆X s ) 2 and showed that for f ∈ C 2 (R) one can define a pathwise integral . 0 (∇f • X)d π X as a pointwise limit of left Riemann sums along (π n ):

T 0 (∇f • X)d π X = lim n→∞ π n ∇f (X(t)).(X(t n i+1 ) -X(t n i )), (2)
and this integral satisfies a change of variable formula:

f (X(t)) = f (X(0)) + t 0 (∇f • X)d π X + 1 2 t 0 ∇ 2 f (X(s)).d[X] c π + [0,t]
(f (X(s)) -f (X(s-)) -∇f (X(s))∆X(s)) .

This 'pathwise Itô formula' could be potentially used as a starting point for a purely pathwise construction of the Itô calculus but, unlike the analogous theory for Riemann-Stieltjes or Young integrals, the construction in [START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF] seems to depend on the choice of the sequence of partitions (π n ): both the quadratic variation [X] π and the pathwise integral (2) are defined as limits along this sequence of partitions. In fact, as shown by Freedman [16,p. 47], for any continuous function x one can construct a sequence of partitions π such that [x] π = 0. This result was extended by Davis et al. [START_REF] Davis | Pathwise stochastic calculus with local times[END_REF] who showed that given any continuous path x and any increasing function A, one can construct a partition π A,x such that [x] π A,x = A. These negative results seem to suggest that the dependence of [x] π on π leaves no hope for uniqueness of the quantities in [START_REF] Bertoin | Temps locaux et intégration stochastique pour les processus de Dirichlet[END_REF].

On the other hand, as shown by Lévy [START_REF]Le mouvement brownien plan[END_REF][START_REF] Lévy | Processus stochastiques et mouvement brownien[END_REF] and Dudley [START_REF] Dudley | Sample functions of the gaussian process[END_REF], for typical paths of Brownian motion the sums (1) converge to a unique limit along any sequence of partitions which are refining or whose mesh decreases to zero fast enough. Therefore there exists a large set of paths -containing all typical Brownian paths-for which one should be able to define the quantities in (3) independently of the choice of the partition sequence (π n ) n≥1 for a large class of such sequences.

We clarify these issues by investigating in detail the dependence of quadratic variation with respect to the sequence of partitions, and deriving sufficient conditions for the stability of quadratic variation with respect to this choice. These conditions are related to an irregularity property of the path, which we call quadratic roughness (Def. 3.1): this property requires cross-products of increments along the partition to average to zero at certain scales and is different from other notions of roughness such as Hölder roughness [START_REF] Friz | A course on rough paths[END_REF] or ρ-irregularity [START_REF] Catellier | Averaging along irregular curves and regularisation of odes[END_REF]. Importantly, we show that the quadratic roughness property is satisfied almostsurely by Brownian paths (Theorem 3.8).

Our main result is for Hölder-continuous paths satisfying this irregularity condition, the quadratic variation along balanced partitions (Def. 2.1) is invariant with respect to the choice of the partition sequence. This leads to an intrinsic notion of quadratic variation, a robust formulation of the pathwise Itô calculus (Theorem 5.1) and uniqueness of pathwise local time (Theorem 5.2) for such irregular paths.

Our results thus complement previous results on the pathwise approach to Itô calculus [START_REF] Ananova | Pathwise integration with respect to paths of finite quadratic variation[END_REF][START_REF] Chiu | On pathwise quadratic variation for cadlag functions[END_REF][START_REF] Cont | Change of variable formulas for non-anticipative functionals on path space[END_REF][START_REF] Davis | Arbitrage bounds for prices of weighted variance swaps[END_REF][START_REF] Davis | Arbitrage bounds for prices of weighted variance swaps[END_REF][START_REF] Davis | Pathwise stochastic calculus with local times[END_REF][START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF][START_REF] Imkeller | Existence of lévy's area and pathwise integration[END_REF][START_REF] Kim | Local time for continuous paths with arbitrary regularity[END_REF][START_REF] Perkowski | Local times for typical price paths and pathwise Tanaka formulas[END_REF] by identifying a set of paths for which these results are robust to the choice of the sequence of partitions involved in the construction. In contrast to the constructions in [START_REF] Karandikar | On the quadratic variation process of a continuous martingale[END_REF][START_REF]On pathwise stochastic integration[END_REF][START_REF] Karandikar | On quadratic variation of martingales[END_REF], our construction does rely on a specific choice of partitions and does not rely on any probabilistic tools.

Outline Section 1 recalls the definition of quadratic variation along a sequence of partitions, following [START_REF] Chiu | On pathwise quadratic variation for cadlag functions[END_REF][START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF]. Section 2 defines the class of balanced sequences of partitions and discusses asymptotic comparability of such partitions.

Section 3 introduces the concept of quadratic roughness and explores some of its properties. In particular we show that Brownian paths satisfy this property almost-surely (Theorem 3.8). Section 4 shows that quadratic roughness of a path is a sufficient condition for the invariance of quadratic variation with respect to the choice of partitions (Theorem 4.2). This result allows to give an intrinsic definition of quadratic variation without reference to a specific partition sequence (Proposition 4.5).

Section 5.1 builds on these result to arrive at a robust formulation of the Föllmer integral and the pathwise Itô formula. Section 5.2 extends these results to pathwise local time.

Quadratic variation along a sequence of partitions

Let T > 0. We denote D([0, T ], R d ) the space of R d -valued right-continuous functions with left limits (càdlàg functions), C 0 ([0, T ], R d ) the subspace of continuous functions and, for 0 < ν < 1, C ν ([0, T ], R d ) the space of Hölder continuous functions with exponent ν:

C ν ([0, T ], R d ) = {x ∈ C 0 ([0, T ], R d ), sup (t,s)∈[0,T ] 2 ,t =s x(t) -x(s) |t -s| ν < +∞}, and C ν-([0, T ], R d ) = 0≤α<ν C α ([0, T ], R d ).
We denote by Π([0, T ]) the set of all finite partitions of [0, T ]. A sequence of partitions of [0, T ] is a sequence (π n ) n≥1 of elements of Π([0, T ]):

π n = (0 = t n 0 < t n 1 < • • • < t n N (π n ) = T ).
We denote N (π n ) the number of intervals in the partition π n and

|π n | = sup{|t n i -t n i-1 |, i = 1..N (π n )}, π n = inf{|t n i -t n i-1 |, i = 1..N (π n )} (4)
the size of the largest (resp. the smallest) interval of π n .

Definition 1.1 (Quadratic variation of a path along a sequence of partitions).

Let

π n = (0 = t n 0 < t n 1 .. < t n N (π n ) = T ) be a sequence of partitions of [0, T ] with vanishing mesh |π n | = sup i=0•••N (π n )-1 |t n i+1 -t n i | → 0. A càdlàg function x ∈ D([0, T ], R
) is said to have finite quadratic variation along the sequence of partitions (π n ) n≥1 if the sequence of measures

t n j ∈π n (x(t n j+1 ) -x(t n j )) 2 δ t n j converges weakly on [0, T ] to a limit measure µ such that t → [x] c π (t) = µ([0, t]) - 0<s≤t |∆x(s)| 2
is continuous and increasing. The increasing function

[x] π : [0, T ] → R + defined by [x] π (t) = µ([0, t]) = lim n→∞ πn (x(t n k+1 ∧ t) -x(t n k ∧ t)) 2 (5)
is called the quadratic variation of x along the sequence of partitions π. We denote Q π ([0, T ], R) the set of càdlàg paths with these properties.

Q π ([0, T ], R) is not a vector space (see e.g [START_REF] Schied | On a class of generalized Takagi functions with linear pathwise quadratic variation[END_REF]) and the extension to vector-valued paths requires some care [START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF]:

Definition 1.2. A càdlàg path x = (x 1 , ..., x d ) ∈ D([0, T ], R d ) is said to have finite quadratic variation along π = (π n ) n≥1 if for all i, j = 1..d we have x i ∈ Q π ([0, T ], R) and x i + x j ∈ Q π ([0, T ], R). We then denote [x] π ∈ D([0, T ], S + d ) the matrix-valued function defined by [x] i,j π (t) = [x i + x j ] π (t) -[x i ] π (t) -[x j ] π (t) 2 ∈ S + d ,
where S + d is the set of symmetric semidefinite positive matrices. We denote by Q π ([0, T ], R d ) the set of functions satisfying these properties.

For x ∈ Q π ([0, T ], R d ), [x] π is a càdlàgfunction with values in S + d : [x] π ∈ D([0, T ], S + d ).
As shown in [START_REF] Chiu | On pathwise quadratic variation for cadlag functions[END_REF], the above definitions may be more simply expressed in terms of convergence of discrete approximations. In the case of continuous paths, which is our focus in this paper, we have the following characterization [START_REF] Cont | Functional Ito Calculus and functional Kolmogorov equations[END_REF][START_REF] Chiu | On pathwise quadratic variation for cadlag functions[END_REF]:

Proposition 1.3. x ∈ C 0 ([0, T ], R d ) has finite quadratic variation along π = (π n , n ≥ 1) if and only if the sequence of functions [x] π n defined by [x] π n (t) = t n j ∈π n (x(t n j+1 ∧ t) -x(t n j ∧ t)) t (x(t n j+1 ∧ t) -x(t n j ∧ t)) converges uniformly on [0, T ] to a continuous (increasing) function [x] π ∈ C 0 ([0, T ], S + d ).
The notion of quadratic variation along a sequence of partitions is different from the pvariation for p = 2. The p-variation involves taking a supremum of over all partitions, whereas quadratic variation is a limit taken along a given sequence (π n ) n≥1 . In general [x] π given by ( 5) is smaller than the p-variation for p = 2. In fact, for diffusion processes the typical situation is that the p-variation is (almost-surely) infinite for p = 2 [START_REF] Dudley | Concrete functional calculus[END_REF][START_REF] Taylor | Exact asymptotic estimates of Brownian path variation[END_REF] while the quadratic variation is finite for sequences satisfying some mesh size condition. For instance, typical paths of Brownian motion have finite quadratic variation along any sequence of partitions with mesh size o(1/ log n) [START_REF] Dudley | Sample functions of the gaussian process[END_REF][START_REF] De | On almost sure convergence of quadratic Brownian variation[END_REF] while simultaneously having infinite p-variation almost surely for p ≤ 2 [28, p. 190

]: inf π∈Π(0,T ) π |W (t k+1 ) -W (t k )| 2 = 0, while sup π∈Π(0,T ) π |W (t k+1 ) -W (t k )| 2 = ∞ almost-surely.
The quadratic variation of a path along a sequence of partitions strongly depends on the chosen sequence. In fact, as shown by Freedman [16,p. 47], given any continuous functions, one can always construct a sequence of partitions along which the quadratic variation is zero. This result was extended by Davis et al. [START_REF] Davis | Pathwise stochastic calculus with local times[END_REF] who show that, given any continuous path x ∈ C 0 ([0, T ], R) and any increasing function A : [0, T ] → R + one can construct a partition sequence π such that [x] π = A. Notwithstanding these negative results, we shall identify a class of paths x for which [x] π is uniquely defined across the class of balanced partition sequences, which we now define.

Balanced partition sequences

One difficulty in comparing quadratic variation along two different partition sequences is the lack of uniform bounds on the partition intervals. We introduce in this section the class of balanced partition sequences which allow for such bounds.

We will say that two (real

) sequences a = (a n ) n≥1 and b = (b n ) n≥1 are asymptotically comparable, denoted a n b n , if |a n | = O(|b n |) and |b n | = O(|a n |) as n → ∞. If both sequences are strictly positive then a n b n ⇐⇒ lim sup n→∞ |b n | |a n | < ∞ and lim sup n→∞ |a n | |b n | < ∞.

Definition and properties

Definition 2.1 (Balanced partition sequence). Let π n = (0 = t n 0 < t n 1 .. < t n N (π n ) = T ) be a sequence of partitions of [0, T ] and

π n = inf i=0..N (π n )-1 |t n i+1 -t n i |, |π n | = sup i=0..N (π n )-1 |t n i+1 -t n i |.
We say that (π n ) n≥1 is balanced if

∃c > 0, ∀n ≥ 1, |π n | π n ≤ c. (6)
This condition means that all intervals in the partition π n are asymptotically comparable. Note that since π n N (π n ) ≤ T , any balanced sequence of partitions satisfies

|π n | ≤ c π n ≤ cT N (π n ) . (7)
We will denote by B([0, T ]) the set of all balanced partition sequences of [0, T ].

Proposition 2.2. Let π = (π n ) n≥1 be a sequence of partitions of [0, T ]. Then:

(i) π ∈ B([0, T ]) ⇐⇒ lim inf n→∞ N (π n )π n > 0 and lim sup n→∞ N (π n )|π n | < ∞.
(ii) Let N (π n , t 1 , t 2 ) be the number of partition points of

π n in [t 1 , t 2 ]. If π ∈ B([0, T ]) then for any h > 0, lim sup n→∞ sup t∈[0,T -h] N (π n , t, t + h) inf t∈[0,T -h] N (π n , t, t + h) < ∞. (iii) If π = (π n ) n≥1 ∈ B([0, T ]) then lim sup n N (π n+1 ) N (π n ) < ∞ ⇐⇒ lim sup n |π n | |π n+1 | < ∞ ⇐⇒ lim sup n π n π n+1 < ∞. ( 8
) (iv) If g ∈ C 1 ([0, T ], R
) is strictly increasing with inf g > 0 then g maps any balanced partition of [0, T ] to a balanced partition sequence of g([0, T ]).

The proof of this proposition is given in Appendix A.1.

We will say that wo balanced partition sequences τ = (τ n ) n≥1 and σ = (σ n ) n≥1 are (asymptotically) comparable if

0 < lim inf n→∞ |σ n | |τ n | ≤ lim sup n→∞ |σ n | |τ n | < ∞ (9)
which then implies

0 < lim inf n→∞ N (σ n ) N (τ n ) ≤ lim sup n→∞ N (σ n ) N (τ n ) < ∞ (10)
We denote τ σ (or τ n σ n ). Note that for general (not balanced) sequences of partitions [START_REF] Davis | Arbitrage bounds for prices of weighted variance swaps[END_REF] does not imply [START_REF] Davis | Pathwise stochastic calculus with local times[END_REF]: this is a consequence of [START_REF] Cont | Change of variable formulas for non-anticipative functionals on path space[END_REF]. If τ σ then the number of points of τ n in any interval of σ n remains bounded as n → ∞.

The following lemma, whose proof is given in Appendix A.2, shows how one can adjust the rate at which the mesh of a balanced sequence decreases to zero. (i) There exists a subsequence (τ k(n) ) n≥1 of τ such that:

lim sup n |σ n | |τ k(n) | ≥ 1. (ii) If lim sup n |τ n | |τ n+1 | < ∞, there exists a subsequence (τ k(n) ) n≥1 of τ which is asymptotically comparable to σ: τ k(n) σ n .
(iii) There exists r :

N → N such that lim sup n |σ r(n) | |τ n | ≥ 1.
(Note that r : N → N may not be injective i.e. (σ r(n) , n ≥ 1) is not a subsequence of σ.)

Coarsening A partition may be refined by adding points to it. The inverse operation, which we call coarsening, corresponds to removing points i.e. subsampling or grouping of partition points. We will be specifically interested in coarsenings which preserve the balance property but may modify the asymptotic rate of decrease of the mesh size:

Definition 2.4 (Coarsening of a balanced partition sequence). Let π n = (0 = t n 0 < t n 1 • • • < t n N (π n ) = T ) be a balanced sequence of partitions of [0, T ] with vanishing mesh |π n | → 0 and 0 < β < 1. A β-coarsening of π is a sequence of subpartitions of π n : A n = (0 = t n p(n,0) < t n p(n,1) < • • • < t n p(n,N (A n )) = T ) such that (A n ) n≥1 is a balanced partition sequence of [0, T ] and |A n | |π n | β . Note that since β < 1, |A n | |π n | β |π n | as n increases, so the number of points of π n in each interval of A n increases to infinity: inf j=1...N (A n ) p(n, j) -p(n, j -1) n→∞ → ∞.

Quadratic variation along balanced partition sequences

If a path has quadratic variation along a sequence of partitions, then it also has (the same) quadratic variation along any sub-sequence. This simple remark has interesting implications when the partition sequences is balanced: comparing the sum of squared increments along the original sequence with the sum along a sub-sequence (with finer mesh) we obtain that, under some scaling conditions on the mesh, cross-products of increments along the finer partition average to zero across the coarser partition.

Lemma 2.5 (Averaging property of cross-products of increments

). Let x ∈ C α ([0, T ], R d ) for some α > 0 and σ n = {s n 0 = 0, s n 1 , s n 2 , • • • , s n N (σ n )=T } be a balanced sequence of partitions of [0, T ] such that x ∈ Q σ ([0, T ], R d ). Let κ > 1 α and (σ ln ) n≥1 a subsequence of σ n with |σ ln | |σ n | κ . Define p(k, n) = inf{m ≥ 1 : s ln m ∈ (s n k , s n k+1 ]} for k = 1 • • • N (σ n ). Then N (σ n ) k=1 p(k,n)≤i =j<p(k+1,n)-1 x(s ln i+1 ) -x(s ln i ) t x(s ln j+1 ) -x(s ln j ) n→∞ ----→ 0.
We provide the proof for d = 1. The extension to d > 1 is straightforward.

Proof. Let σ ln be a sub-sequence of

σ n satisfying |σ ln | |σ n | κ . Define [x] σ n (t) = N (σ n )-1 k=1 x(s n k+1 ∧ t) -x(s n k ∧ t) 2 , [x] σ ln (t) = s ln k ∈σ ln x(s ln k+1 ∧ t) -x(s ln k ∧ t) 2 . Then [x] σ ln (t) -[x] σ n (t) → 0.
Grouping the points of σ ln along intervals of σ n , we obtain:

[x] σ n -[x] σ ln = σ n (x(s n i+1 ) -x(s n i )) 2 - σ ln (x(s ln i+1 ) -x(s ln i )) 2 = σ n (x(s n i+1 ) -x(s n i )) 2 - p(i+1,n)-1 j=p(i,n) (x(s ln j+1 ) -x(s ln j )) 2 + N (σ n ) k=1 (x(s ln p(k,n) ) -x(s ln p(k,n)-1 )) 2 ≥ σ n (x(s n i+1 ) -x(s n i )) 2 - p(i+1,n)-1 j=p(i,n) (x(s ln j+1 ) -x(s ln j )) 2 - N (σ n ) k=1 (x(s ln p(k,n) ) -x(s ln p(k,n)-1 )) 2 .
Using the Hölder continuity of f , the last term in the above equation is bounded above by

N (σ n ) k=1 C|σ ln | 2α ≤ CN (σ n )|σ ln | 2α
. Now using the balanced property of σ ln , the last term is bounded above by

N (σ n ) i=1 C 1 |σ ln | 2α ≤ N (σ n )C 1 |σ ln | 2α ≤ C 2 N (σ n ) N (σ ln ) 2α ≤ C 2 × N (σ n ) 1-2ακ n→∞ → 0 since 1 -2κα < 0. So finally we obtain lim n→∞ N (σ n ) k=1 p(k,n)≤i =j<p(k+1,n) x(s ln i+1 ) -x(s ln i ) x(s ln j+1 ) -x(s ln j ) ≤ lim n→∞ |[x] σ n -[x] σ ln | = 0.
3 Quadratic roughness 3.1 Quadratic roughness along a sequence of partitions Lemma 2.5 shows that if a function has finite quadratic variation along a balanced partition sequence, then the product of the increments along any subsequence with sufficiently small mesh average to zero if we do the averaging along the original (coarser) sequence. Intuitively, this means that there is enough cancellation across neighboring increments such that their crossproducts average to zero over a coarser grid. We will now introduce a slightly extended version of this property, which we call quadratic roughness, and show that plays a key role in the stability of quadratic variation with respect to the partition.

Definition 3.1 (Quadratic roughness). Let π n = (0 = t n 0 < t n 1 .. < t n N (π n ) = T ) be a balanced sequence of partitions of [0, T ] with |π n | → 0 and 0 < β < 1. We say that x ∈ Q π ([0, T ], R d ) has the quadratic roughness property along π with coarsening index β on [0, T ] if for any β-coarsening A n = (0 = t n p(n,0) < t n p(n,1) < • • • < t n p(n,N (A n )) = T ) of π we have N (A n ) j=1 p(n,j-1)≤i =i <p(n,j) x(t n i+1 ∧ t) -x(t n i ∧ t) t x(t n i +1 ∧ t) -x(t n i ∧ t) n→∞ ----→ 0. We denote by R β π ([0, T ], R d ) ⊂ Q π ([0, T ], R d
) the set of paths satisfying this property. In other words, the quadratic roughness property states that cross-products of increments along π n average to zero over any (β-)coarsening of π n as the mesh is refined. Remark 3.2. Let σ = (σ n ) n≥1 be a balanced partition sequence of [0, T ] with |σ n | → 0. Then the quadratic roughness property along σ implies existence of quadratic variation along any β-coarsening of σ i.e. any sequence of subpartitions of σ satisfying Definition 2.4:

R β σ ([0, T ], R d ) ⊂ π∈C(β,σ) Q π ([0, T ], R d ),
where C(β, σ) denotes the set of all possible β-coarsenings of σ.

Proposition 3.3. Let π = (π n ) n≥1 be a balanced partition sequence of [0, T ] with |π n | → 0 and x ∈ R β π ([0, T ], R d ) with 0 ≤ β < 1. Then:
1. For any interval I ⊂ [0, T ], x has the quadratic roughness property on

I along π I = (π n I ) n≥1 = (π n ∩ I) n≥1 : x ∈ R β π I ([0, T ], R d ).
2. For any subsequence

τ n = π k(n) of π, we have x ∈ R β τ ([0, T ], R d ). 3. For any λ ∈ R, λx ∈ R β π ([0, T ], R d ).
4. If y is a function with bounded variation then

x + y ∈ R β π ([0, T ], R d ).
The proof of this lemma is given in Appendix A.3.

Quadratic roughness of Brownian paths

We will now show that the quadratic roughness property is satisfied by typical sample paths of Brownian motion. Denote by

d(τ n , σ n ) = max max t∈τ n min s∈σ n |t -s|, max s∈σ n min t∈τ n |t -s|
the Hausdorff distance between two partitions σ n and τ n of [0, T ]. For any t ∈ τ n there exists

s ∈ σ n such that |t -s| ≤ d(τ n , σ n ). Lemma 3.4. Let π = (π n ) n≥1 be a balanced sequence of partitions of [0, T ]. Then for all A, B ∈ C(β, π) there exists M (A, B) > 0 such that, for all n ≥ 1, the number of points of A n in each interval of B n is bounded by M (A, B). Proof. Let, A n = (t n p(n,1) , t n p(n,2) , • • • , t n p(n,N (A n )) ) and B n = (t n q(n,1) , t n q(n,2) , • • • , t n q(n,N (B n )) ) be β-coarsenings of π. Since lim sup n→∞ |A n | B n ≤ lim sup n→∞ |A n | |π n | β × |π n | β |B n | × |B n | B n lim sup n→∞ |A n | B n ≤ lim sup n→∞ lim sup n→∞ |A n | |π n | β × lim sup n→∞ |π n | β |B n | × lim sup n→∞ |B n | B n ≤ M < ∞. using the fact that |A n | |π n | β |B n | and B n is a balanced sequence of partitions of [0, T ].
Interchanging the roles of A n and B n we also have lim sup

n→∞ |B n | A n < ∞.
Hence for all A n , B n ∈ C n (β, π), there exists M (A, B) > 0 such that, for all n ≥ 1, the number of points of A n in each interval of B n is bounded by M (A, B).

For a β-coarsening A = (A n ) n≥1 , of π define: ||A|| π = max{lim sup n→∞ |A n | |π n | β , lim sup n→∞ |π n | β |A n | , lim sup n→∞ |A n | A n } < ∞. Denoting G(β, π, k) = {A ∈ C(β, π) : ||A|| ≤ k} ⊂ C(β, π) we have C(β, π) = ∪ k∈N G(β, π, k).
Lemma 3.5. Let π n be a balanced sequence of partitions of [0, T ]. Then:

∀k ≥ 1, ∃L(β, π, k) > 0, ∀n ≥ 1, sup A,B∈G(β,π,k) d(A n , B n ) ≤ L(β, π, k) |π n | β .
Proof. If the conclusion does not hold, then lim sup

n→∞ sup A,B∈C(β,π) d(A n , B n ) |π n | β = ∞.
Then, given any

K > 0 there exists A, B ∈ C(β, π) such that lim sup n→∞ d(A n , B n ) |π n | β > K.
Since A, B ∈ C(β, π), there exists C > 0 such that for all n ≥ 1,

|A n | ≤ C|π n | β , |B n | ≤ C|π n | β .
From Lemma 3.4 and the definition of G(β, π, k), there exists M (A, B) > 0 such that for all n ≥ 1, the number of points of A n in each interval of B n is bounded by M (A, B). So:

max t∈A n min s∈B n |t -s| ≤ M |B n | ≤ M C|π n | β and max s∈B n min t∈A n |t -s| ≤ M |A n | ≤ M C|π n | β .
Therefore for all n ≥ 1,

d(A n , B n ) ≤ M C|π n | β . Choosing L > M C leads to a contradiction.
Hence the result follows.

Lemma 3.6. Let π be a balanced sequence of partitions of [0, T ] and A, B ∈ C(β, π) be β-

coarsenings of π. Let β < γ < 1 2 . If for n ≥ 1, d(A n , B n ) ≤ c|π n | γ for some c < ∞ then ∀ω ∈ C 1 2 -([0, T ], R), [ω] A = [ω] B . Proof. Since d(A n , B n ) ≤ c|π n | γ for any t ∈ A n there exists at least one s ∈ B n such that |s -t| ≤ c|π n | γ . Let A n = (0 = a n 1 < a n 2 < • • • < a n N (A n ) = T ). For i ∈ {1, 2 • • • N (A n )} there exists m i ≥ 1 such that s a n i 1 < s a n i 2 < • • • < s a n i m n i ∈ B n such that sup j∈{1•••m a n i } |s a n i j -a n i | ≤ c|π n | γ
Now from Lemma 3.4 we can conclude that m n i < M < ∞ for all n ≥ 1 and i. Also notice that:

{s a n i j |i = 1, 2 • • • N (A n ) and j = 1, 2, • • • m n i } ⊇ B n as, d(A n , B n ) ≤ c|π n | γ . Let ω ∈ C 1/2-([0, T ], R). Then for any γ < α < 1/2: ω(a n i+1 ) -ω(a n i ) 2 - m n i j=2 ω(s a n i j+1 ) -ω(s a n i j ) 2 - m n i+1 j =2
ω(s

a n i j +1 ) -ω(s a n i j ) 2 = O(|π n | γα |A n | α ). So, |[ω] A n -[ω] B n | = O(N (A n )|π n | γα |A n | α ) = O(|π n | γα+βα-β ) = O(|π n | β( γ β α+α-1)
). Now from the construction we have

γ β > 1 which implies γ β + 1 > 2. Choosing β γ + β < α < 1 2 then leads to γ β α + α -1 > 0 so |[ω] A n -[ω] B n | = O(|π n | β( γ β α+α-1) ) n→∞ ----→ 0.
Remark 3.7. In fact one can obtain a more refined estimate for

A, B ∈ G(β, π, k). Let β < γ < 1 2 . Then d(A n , B n ) ≤ c|π n | γ implies ∀ω ∈ C 1 2 -([0, T ], R), ∀α < 1 2 , |[ω] A n -[ω] B n | ≤ cM 2 T k α ω α |π n | α(γ+β)-β ,
where ω α is the α-Hölder norm of ω. Furthermore, there exists a > 0 such that sup

A,B∈G(β,π,k) |[ω] A n -[ω] B n | = o(|π n | a 2 ).
Theorem 3.8 (Quadratic roughness of Brownian paths). Let W be a Wiener process on a probability space (Ω, F, P), T > 0 and (π n ) n≥1 a balanced sequence of partitions of [0, T ] with

(log n) ν |π n | n→∞ → 0 for some ν > 2. ( 11
)
Then the sample paths of W almost-surely satisfy the quadratic roughness property on [0, T ]:

∀β ∈ ( 2 ν , 1), P W ∈ R β π ([0, T ], R) = 1.
Proof. Let W be a Wiener process on a probability space (Ω, F, P), which we take to be the canonical Wiener space without loss of generality i.e Ω = C 0 ([0, T ], R), W (t, ω) = ω(t). Using Lévy's modulus of continuity result,

lim h→0 sup 0≤t≤T -h |W (t + h) -W (t)| 2h log(1/h) = 1
almost surely, so there exists δ, c > 0 and Ω 1 ⊂ Ω with P(Ω 1 ) = 1 such that for h < δ we have

∀ω ∈ Ω 1 , |ω(t + h) -ω(t)| ≤ c h log( 1 h ).
For any 0 < a < 1/2 we have

∀ω ∈ Ω 1 , sup 0<h≤δ |ω(t + h) -ω(t)| h a ≤ ch 1 2 -a log( 1 h ).
The right hand side is bounded and has a finite maximum C a on [0, δ] so

∀ω ∈ Ω 1 , sup 0<h<δ |ω(t + h) -ω(t)| h a ≤ C a . ( 12
)
Let

π n = (0 = t n 0 < t n 1 .. < t n N (π n ) = T
) be a balanced sequence of partitions of [0, T ] satisfying [START_REF] De | On almost sure convergence of quadratic Brownian variation[END_REF]. Then |π n | log n → 0 so the results of Dudley [START_REF] Dudley | Sample functions of the gaussian process[END_REF] imply that

P π n |W (t n i+1 ∧ t) -W (t n i ∧ t)| 2 n→∞ → t = 1. Furthermore P(C 1 2 -([0, T ], R)) = 1, so if we set Ω 0 = Ω ∩ Q π ([0, T ]) ∩ C 1 2 -([0, T ], R) ∩ Ω 1 then P (Ω 0 ) = 1 and any ω ∈ Ω 0 satisfies [ω] π (t) = t and (12) for any 0 < a < 1/2. Let β ∈ ( 2 ν , 1
) and C(β, π) be the set of all β-coarsenings of π i.e. sequences of subpartitions satisfying Definition 2.4, and

C n (β, π) = {A n , A ∈ C(β, π)}. Let A = (A n ) ∈ C(β, π) with A n = (0 = t n p(n,0) < t n p(n,1) < • • • < t n p(n,N (A n )) = T ). ( 13
)
Define

a ii = (t n i+1 -t n i )(t n i +1 -t n i ) if p(n, j-1) ≤ i = i < p(n, j) for some j ∈ {1, 2 • • • N (A n )} otherwise set a ii = 0. Let S(A n , W ) = N (A n ) j=1 p(n,j-1)≤i =i <p(n,j) W (t n i+1 ) -W (t n i ) T W (t n i +1 ) -W (t n i ) = N (π n ) i,i =1 a ii X n i X n i
where

X n i = W (t n i+1 ) -W (t n i ) t n i+1 -t n i ∼ N (0, 1) are IID variables for i = 0, • • • N (π n ) -1. Let Λ 2 = 1≤i,i ≤N (π n ) a 2 ii = N (A n ) j=1 p(n,j-1)≤i =i <p(n,j) a 2 ii ≤ N (A n ) j=1 p(n,j-1)≤i =i <p(n,j) ∆t n i ∆t n i ≤ N (A n ) j=1 |A n j | 2 ≤ |A n | N (A n ) j=1 |A n j | = |A n |T.
The Hanson-Wright inequality [START_REF] Hanson | A bound on tail probabilities for quadratic forms in independent random variables[END_REF] then implies that for any δ > 0, there exists constants C 1 and C 2 such that

∀n ≥ 1, P |S(A n , W )| > δ ≤ 2 exp(-min{C 1 δ |A n | , C 2 δ 2 |A n | }) n→∞ ≤ 2 exp -C 1 δ |A n | since |A n | |π n | β → 0. As |π n | β (log n) 2 → 0, if we denote ε n = |A n |(log n) then ε n → 0 and
we can rewrite this bound as

P |S(A n , W )| > δ ≤ 2 exp(-min{ C 1 δ log n ε n , C 2 δ 2 (log n) 2 ε 2 n }) ≤ 2C n C1δ/εn . ( 14
) Let β < γ < 1 2 and k ≥ 1. Denote R(n) = L(β, π, k)|π n | β-γ + 1, where L(β, π, k) is the constant defined in Lemma 3.5. For i ∈ {1, 2, • • • , R(n)}, we define F n i (β, π) = {B ∈ G(β, π, k), d(A n , B n ) ∈ [(i -1)|π n | γ , i|π n | γ )} ⊂ C(β, π).
Then by Lemma 3.5

G(β, π, k) = ∪ R(n) i=1 F n i (β, π). From Remark 3.7 there exists a N 0 ∈ N such that for all i ∈ {1, 2 • • • R(n)} and for all n ≥ N 0 : ∀ω ∈ Ω 0 , ∀ B, C ∈ F n i (β, π), |[ω] B n -[ω] C n | ≤ δ 2 so |S(B n , ω)| -|S(C n , ω)| ≤ δ 2 . So, if there exists C ∈ F n i (β, π) such that |S(C n , W )| > δ then for any B ∈ F n i (β, π) we have |S(B n , W )| > δ 2 , i.e. ∀B ∈ F n i (β, π), P (∃C ∈ F n i (β, π) : |S(C n , W )| > δ) ≤ P |S(B n , W )| > δ 2 .
Combining this with our previous Hanson-Wright estimate yields

P ∃C ∈ F n i (β, π), |S(C n , W )| > δ ≤ 2C n C1δ/εn .
So for all n ≥ N 0 ,

P ∃B ∈ G(β, π, k), |S(B n , W )| > δ ≤ R(n) i=1 P ∃C ∈ F n i (β, π) : |S(C n , W )| > δ ≤ 2C 2 R(n) n C1δ/εn ≤ 2C 2 ( L(β, π, k)|π n | β-γ + 1) n C1δ/εn ≤ 2C 2 (L(β, π, k)|π n | β-γ + 2) n C1δ/εn ≤ log n ε n 2(β-γ) β C 3 n C1δ/εn . Therefore P ∃B n ∈ C n (β, π), |S(B n , W )| > δ ≤ O   log n ε n 2(β-γ) β 1 n C1δ/εn   . The series n log n ε n 2(β-γ) β 1 n C1δ/εn
< ∞ is absolutely convergent. So we can apply the Borel-Cantelli lemma to obtain for each δ > 0 a set Ω δ with P(Ω δ ) = 1 and

N δ ∈ N such that ∀ω ∈ Ω δ , ∀A ∈ C(β, π), ∀n ≥ N δ , |S(A n , W )(ω)| ≤ δ.

Now if we set

Ω π = Ω 0 ∩ ∩ k≥1 Ω 1/k then P(Ω π ) = 1
and for paths in Ω π we have S(A n , ω) → 0 simultaneously for all β-coarsenings A ∈ G(β, π, k):

∀ω ∈ Ω π , ∀A ∈ G(β, π, k), S(A n , ω) n→∞ → 0, which is true for all k ∈ N. Noting that C(β, π) = ∪ k∈N G(β, π, k) allows to conclude ∀ω ∈ Ω π , ∀A ∈ ∪ k∈N G(β, π, k) = C(β, π), S(A n , ω) n→∞ → 0, therefore P W ∈ R β π ([0, T ], R) = 1.
Combining Theorem 3.8 with Remark 3.2 we obtain the almost-sure convergence of quadratic variation for Brownian motion simultaneously for all β-coarsenings of π: Corollary 3.9. Under the assumptions of Theorem 3.8,

P   W ∈ τ ∈C(β,π) Q τ ([0, T ], R d )   = 1.
Note that the set C(β, π) of β-coarsenings of π is uncountable, so the above result is far from obvious: metric arguments play a crucial role in Theorem 3.8.

Uniqueness of quadratic variation along balanced partitions 4.1 Main result

The following lemma shows that the quadratic roughness property is a necessary condition for the stability of quadratic variation with respect to the choice of the partition sequence:

Lemma 4.1. Let 0 < β < 1 and π = (π n ) n≥1 be a balanced partition sequence such that x ∈ Q π ([0, T ], R d ) with [x] π strictly increasing. If x has the same quadratic variation along all β-coarsenings of π then x has the quadratic roughness property along π:

(∀τ ∈ C(β, π), [x] τ = [x] π ) ⇒ x ∈ R β π ([0, T ], R d ).
Proof. Let τ ∈ C(β, π) be a β-coarsening of π. Using the same notation as in Def. 3.1,

τ n = (0 = t n p(n,0) < t n p(n,1) < • • • < t n p(n,N (π n )) = T ).
From the assumption we have

x ∈ Q τ ([0, T ], R d ) and [x] τ = [x] π . So for all t ∈ [0, T ]: lim n→∞ [x] π n (t) -[x] τ n (t) = 0 = lim n→∞ N (τ n ) j=1 p(n,j-1)≤i =i <p(n,j) x(t n i+1 ∧ t) -x(t n i ∧ t) t x(t n i +1 ∧ t) -x(t n i ∧ t)
which is the quadratic roughness property, so the result follows.

We will now show that quadratic roughness is also a sufficient condition for the uniqueness of quadratic variation along balanced partition sequences. We consider partitions satisfying the following assumption:

Assumption 1. σ = (σ n ) n≥1 is a balanced sequence of partitions of [0, T ] with |σ n | → 0 and lim sup n |σ n | |σ n+1 | < ∞, or equivalently lim sup n |N (σ n+1 )| |N (σ n )| < ∞ (15)
The dyadic partition (and any geometric partition) obviously satisfies this assumption.

Our main result is that quadratic roughness along such a sequence implies uniqueness of pathwise quadratic variation along all balanced partition sequences: Theorem 4.2. Let σ be a sequence of partitions satisfying Assumption 1 and

x ∈ C α ([0, T ], R d )∩ R β σ ([0, T ], R d ) for some 0 < β ≤ α. Then for any balanced sequence of partitions τ = (τ n ) n≥1 , if x ∈ Q τ ([0, T ], R d ) then [x] τ = [x] σ .
Proof. We first prove the result in the case where lim sup

n |τ n | |σ n | < 1.
k(n) = inf{k ≥ n, |σ k | ≤ |τ n |} < ∞ since |τ k | k→∞ → 0. ( 16
)
then the subsequence (σ k(n) ) n≥1 of σ satisfies lim sup

n |τ n | |σ k(n) | ≥ 1 and lim sup n |τ n | |σ k(n)-1 | < 1. For i = 1..N (τ n ), j(i, n) = inf{j ≥ 1, s k(n) j ∈ (t n i , t n i+1 ]}. Then we have s k(n) j(i,n)-1 ≤ t n k < s k(n) j(i,n) < • • • < s k(n) j(i+1,n)-1 ≤ t n i+1 < s k(n) j(i+1,n) .
We now show that the size j(i+1, n)-j(i, n) of these clusters is uniformly bounded. Assume that sup

i=1..N (τ n ) j(i + 1, n) -j(i, n) → ∞ as n → ∞.
Then, from the above definition of k(n) and using the balanced property of τ n and σ k(n) we have lim sup

n |τ n | |σ k(n) | → ∞. Since lim sup n |τ n | |σ k(n)-1 | is bounded from (16) we thus have lim sup n |τ n | |σ k(n) | - |τ n | |σ k(n)-1 | = lim sup n |τ n | |σ k(n)-1 | |σ k(n)-1 | |σ k(n) | -1 = ∞
which contradicts the assumption that lim sup

j |σ j-1 | |σ j | < ∞. Hence the sequence sup i=1..N (τ n ) j(i + 1, n) -j(i, n) is bounded as n → ∞: ∃M > 0, ∀i, n ≥ 1, j(i + 1, n) -j(i, n) ≤ M < ∞.
Therefore (σ k(n) ) n≥1 and (τ n ) n≥1 are (asymptotically) comparable i.e. the sequences

N (σ k(n) ) N (τ n ) and |σ k(n) | |τ n | (17)
are bounded. Define now

l n = inf{l ≥ n : N (σ l ) ≥ N (σ k(n) ) 1/β } Then N (σ ln-1 ) < N (σ k(n) ) 1/β ≤ N (σ ln ).
Since the subsequence (σ ln ) is also balanced, there exists constants c 1 and c 2 such that c

1 |σ ln-1 | > |σ k(n) | 1/β ≥ c 2 |σ ln |.
The points of the partition τ n are interspersed among those of σ ln . Define for

k = 1 • • • N (τ n ): p(n, k) = inf{m ≥ 1 : s ln m ∈ (t n k , t n k+1 ]}.
Then we have

s ln p(n,k)-1 ≤ t n k < s ln p(n,k) < • • • < s ln p(n,k+1)-1 ≤ t n k+1 < s ln p(n,k+1) (18)
where p(n, N (τ n )) -1 = N (σ ln ). We will show that by grouping the points of σ ln according to the intervals defined by τ n we obtain a β-coarsening of σ ln and use the quadratic roughness property of x to conclude that for all t ∈ [0, T ]:

[x] τ n (t) = N (τ n )-1 k=0 x(t n k+1 ∧ t) -x(t n k ∧ t) t x(t n k+1 ∧ t) -x(t n k ∧ t) and [x] σ ln (t) = N (σ ln )-1 k=0 x(s ln k+1 ∧ t) -x(s ln k ∧ t) t x(s ln k+1 ∧ t) -x(s ln k ∧ t)
have the same limits. We shall give the proof for t = T ; for t < T we have an additional boundary term which goes to zero. Decomposing ∆ n k = x(t n k+1 ) -x(t n k ) along the partition points of σ ln , we obtain,

x(t n k+1 ) -x(t n k ) ∆ n k = (x(t n k+1 ) -x(s ln p(n,k+1)-1 ) D k -(x(t n k ) -x(s ln p(n,k)-1 )) B k + p(n,k+1)-1 i=p(n,k) (x(s ln i ) -x(s ln i-1 )) C k
Grouping together the terms in [x] σ ln according to (18) yields

[x] τ n (T ) -[x] σ ln (T ) = N (τ n ) k=1   ∆ n k t ∆ n k - p(n,k+1)-1 i=p(n,k) x(s ln i ) -x(s ln i-1 ) t x(s ln i ) -x(s ln i-1 )   = N (τ n ) k=1 ∆ n k t ∆ n k -C t k C k + N (τ n ) k=1   C t k C k - p(n,k+1)-1 i=p(n,k) x(s ln i ) -x(s ln i-1 ) t x(s ln i ) -x(s ln i-1 )   .
Since

N (τ n ) k=1 C t k C k = N (τ n ) k=1 (∆ n k -D k + B k ) t (∆ n k -D k + B k ) = N (τ n ) k=1 ∆ n k t ∆ n k + N (τ n ) k=1 (D k -B k ) t (D k -B k ) + N (τ n ) k=1 ∆ n k t B k - N (τ n ) k=1 ∆ n k t D k
we finally obtain

N (τ n ) k=1 [∆ n k t ∆ n k -C t k C k ] ≤ N (τ n ) k=1 (D k -B k ) t (D k -B k ) + N (τ n ) k=1 ∆ n k t B k + N (τ n ) k=1 ∆ n k t D k .
Now we will show that as n → ∞,

N (τ n ) k=1 (∆ n k t ∆ n k -C t k C k ) → 0. Since x ∈ C α ([0, T ], R d ) we have : ∀t ∈ [0, T -h], ∀h > 0, x(t + h) -x(t) ≤ x α h α . Now, N (τ n ) k=1 D t k D k ≤ N (τ n ) i=1 ||D k || 2 ≤ N (τ n ) k=1 x 2 α |σ ln | 2α ≤ x 2 α N (τ n )|σ ln | 2α ≤ cN (σ k(n) )|σ k(n) | 2α/β n→∞ → 0 since α ≥ β. Similarly we have N (τ n ) k=1 B t k B k → 0. Therefore, N (τ n ) k=1 |(D k -B k ) t (D k -B k )| ≤ 2 N (τ n ) k=1 D t k D k + 2 N (τ n ) k=1 B t k B k → 0.
Using the Cauchy-Schwarz inequality,

N (τ n ) k=1 D t k ∆ n k ≤ N (τ n ) k=1 D k 2 1 2 N (τ n ) k=1 ∆ n k 2 1 2 .
Since the quadratic variation of x along the sequence of partitions τ is finite the sequence

N (τ n ) k=1
∆ n k 2 is bounded. Combining this wit the estimate above we obtain

N (τ n ) k=1 D t k ∆ n k → 0.
Similarly, we have,

N (τ n ) k=1 B t k ∆ n k → 0. Therefore, N (τ n ) k=1 [∆ n k t ∆ n k -C t k C k ] → 0. Hence lim n→∞ [x] τ n (T )-[x] σ ln (T ) = lim n→∞ N (τ n ) k=1   C t k C k - p(n,k+1)-1 i=p(n,k) x(s ln i ) -x(s ln i-1 ) t x(s ln i ) -x(s ln i-1 )   .
The partition points of σ ln and τ n are ordered according to

s ln p(n,k)-1 ≤ t n k < s ln p(n,k) < s ln p(n,k)+1 < • • • < s ln p(n,k+1)-1 ≤ t n k+1 < s ln p(n,k+1) . (19)
Expanding the term C t k C k we obtain: Proposition 3.3(iii) x also has the quadratic roughness property along the subsequence σ l = (σ ln ) n≥1 with the same coarsening index β. We will now show that

N (τ n ) k=1   C t k C k - p(n,k+1)-1 i=p(n,k) x(s ln i ) -x(s ln i-1 ) t x(s ln i ) -x(s ln i-1 )   = N (τ n ) k=1 x(s ln p(n,k+1)-1 ) -x(s ln p(n,k)-1 ) t x(s ln p(n,k+1)-1 ) -x(s ln p(n,k)-1 ) - p(n,k+1)-1 i=p(n,k) x(s ln i ) -x(s ln i-1 ) t x(s ln i ) -x(s ln i-1 ) = N (τ n ) k=1 p(n,k+1)-1 i,j=p(n,k) i =j x(s ln i ) -x(s ln i-1 ) t x(s ln j ) -x(s ln j-1 ) . Since x ∈ R β σ ([0, T ], R d ), by
A n = (0 = s ln 0 < s ln p(n,1)-1 < s ln p(n,2)-2 • • • < s ln p(n,N (τ n ))-1 = T ) is a β-coarsening of (σ ln ) n≥1 . For large enough n, 1 2 τ n ≤ A n ≤ |A n | ≤ 2|τ n | so (A n ) is a balanced sequence of partitions. Also, lim sup n |τ n | |σ ln | β ≤ c 3 lim sup n |τ n | |σ k(n)+1 | ≤ c 4 lim sup n |τ n | |σ k(n) | × |σ k(n) | |σ k(n)+1 | < ∞.
On the other hand lim inf

n |τ n | |σ ln | β ≥ c 5 lim inf n |τ n | |σ k(n) | > 0.
Therefore (A n , n ≥ 1) is a β-coarsening of (σ ln ) n≥1 . The quadratic roughness property of x along (σ ln ) n≥1 then implies

N (τ n ) k=1 p(n,k+1)-1 i,j=p(n,k) i =j x(s ln i ) -x(s ln i-1 ) t x(s ln j ) -x(s ln j-1 ) n→∞ ----→ 0. Therefore [x] σ -[x] τ = lim n→∞ [x] τ n (T ) -[x] σ ln (T ) = 0.
Let us now examine the case where lim sup

n |τ n | |σ n | ≥ 1. Using Lemma 2.3 (i) there exists a subsequence (π n ) n≥1 = (τ k(n) ) n≥1 of τ such that: lim sup n |σ n | |π n | ≥ 1.

So using the above result

[x] π = [x] σ . Since π is a subsequence of τ we have [x] π = [x] σ = [x] τ .
As an application, we show that the roughness property and the quadratic variation of a path are invariant under a reparameterization of the path: Proposition 4.3 (Stability of quadratic variation under reparameterization). Let α > 0 and g ∈ C 1 ([0, T ], R + ) be an increasing function with inf g > 0 and π a sequence of partitions satisfying Assumption 1. Then for

x ∈ C α ([0, T ], R) ∩ R β π ([0, T ], R), we have x • g ∈ Q π ([0, T ], R) and ∀t ∈ [0, T ], [x] π (g(t)) = [x • g] π (t). Proof. Let π n = (0 = t n 1 < t n 2 < • • • < t n N (π n ) = T ) be a balanced partition. Then g(π n ) = (g(t n k ), k = 0..N (π n )) defines a partition of [g(0), g(T )]. From Proposition 2.2(iii) the sequence of partitions g(π) = (g(π n )) n≥1 is balanced. From the mean value theorem there exists U n k ∈ [t n k , t n k+1 ] such that g(t n k+1 ) -g(t n k ) = g (U n k )(t n k+1 -t n k ). Therefore lim sup n→∞ sup π n (g(t n k+1 ) -g(t n k )) sup π n+1 (g(t n+1 k+1 ) -g(t n+1 k )) = lim sup n→∞ sup π n g (U n k )(t n k+1 -t n k ) sup π n+1 g (U n+1 k )(t n+1 k+1 -t n+1 k ) ≤ max g min g lim sup n→∞ |π n | |π n+1 | < ∞.
The last inequality follows from Assumption 1 on π. The assumptions on g then imply that g(π) π. The sequence of partitions g(π n ) then satisfies Assumption 1. Therefore we can apply theorem 4.2 to conclude that

∀t ∈ [0, T ], [x] π (g(t)) = [x • g] π (t).

Intrinsic definition of quadratic variation

For γ ≥ 0, define P γ ([0, T ]) as the set of balanced partition sequences (σ n ) n≥1 satisfying Assumption 1 such that

|σ n | = o(| log n| -(γ+ )
) for some > 0:

P γ ([0, T ]) = {σ ∈ B([0, T ]), lim sup n→∞ |σ n | |σ n+1 | < ∞, ∃ > 0, (log n) γ+ |σ n | n→∞ → 0}. ( 20
)
Let Q([0, T ], R d ) be the set of paths which are α-Hölder continuous for α < 1/2 and satisfy the quadratic roughness property along some partition sequence σ ∈ P 4 ([0, T ]):

Q([0, T ], R d ) = C 1 2 -([0, T ], R d ) ∩   σ∈P4([0,T ]) R 1 2 σ ([0, T ], R d )   . (21)
Lemma 4.4. The class Q([0, T ], R d ) is non-empty and contains all 'typical' Brownian paths.

Proof. Let W be a Wiener process on a probability space (Ω, F, P), which we take to be the canonical Wiener space without loss of generality. For any σ ∈ P 4 ([0, T ]), Theorem 3.8 implies that

P W ∈ R 1 2 σ ([0, T ], R d ) = 1.
Brownian paths are almost-surely α-Hölder for α < 1 2 [START_REF] Lévy | Théorie de l'addition des variables aléatoires[END_REF], so

P W ∈ R 1 2 σ ([0, T ], R d ) ∩ C 1 2 -([0, T ], R d ) = 1
and the result follows.

Based on the results above we can now give an 'intrinsic' definition of pathwise quadratic variation for paths in Q([0, T ], R d ) which does not rely on a particular partition sequence: Proposition 4.5 (Quadratic variation map). There exists a unique map

[ . ] : Q([0, T ], R d ) → C 0 ([0, T ], S + d ) such that ∀π ∈ B([0, T ]), ∀x ∈ Q π ([0, T ], R d ) ∩ Q([0, T ], R d ), ∀t ∈ [0, T ], [x] π (t) = [x](t).
We call [x] the quadratic variation of x.

Proof. Let π ∈ B([0, T ]). For any

x ∈ Q π ([0, T ], R d ) ∩ Q([0, T ], R d ) there exists σ ∈ P 4 ([0, T ]) such that x ∈ R 1 2 σ ([0, T ], R d ) ∩ C 1 2 -([0, T ], R d ).
Then Theorem 4.2 implies that for any balanced partition sequence π ∈ B([0, T ]) we have:

∀t ∈ [0, T ], [x] π (t) = [x] σ (t).
By the same argument the quadratic variation does not depend on the choice of σ ∈ P 4 ([0, T ])

such that x ∈ R 1 2
σ ([0, T ], R d ), so the result follows.

Remark 4.6. If X is a continuous P-semimartingale then its image [X] under the map defined in Proposition 4.5 coincides almost-surely with the usual definition of quadratic variation. Building on [START_REF] Karandikar | On the quadratic variation process of a continuous martingale[END_REF], Karandikar and Rao [START_REF] Karandikar | On quadratic variation of martingales[END_REF] construct a (different) quadratic variation map which shares this property. In contrast to [START_REF] Karandikar | On quadratic variation of martingales[END_REF], our construction does not rely on specific path-dependent partitions, identifies explicitly the domain of definition of the map (rather than implicitly in terms of the support of a probability measure) and does not use any probabilistic tools.

I : C 2 (R d ) × Q([0, T ], R d ) → Q([0, T ], R) (f, x) → I(f, x) = . 0 (∇f • x).dx such that: ∀π ∈ B([0, T ]), ∀x ∈ Q π ([0, T ], R d ) ∩ Q([0, T ], R d ), ∀t ∈ [0, T ], I(f, x)(t) = t 0 (∇f • x).d π x = lim n→∞ π n ∇f (x(t n i )).(x(t n i+1 ∧ t) -x(t n i ∧ t)).
We denote I(f, x)

= . 0 (∇f • x)dx. Furthermore ∀f ∈ C 2 (R d ), ∀π ∈ B([0, T ]), ∀x ∈ Q π ([0, T ], R d ) ∩ Q([0, T ], R d ), f (x(t)) -f (x(0)) = t 0 (∇f • x).dx + 1 2 t 0 < ∇ 2 f (x), d[x] > (22)
and . 0

(∇f • x) dx) π (t) = t 0 < (∇f • x) t (∇f • x), d[x] > . (23)
Proof. For any balanced partition sequence

π ∈ B([0, T ]) if x ∈ Q π ([0, T ], R d )∩Q([0, T ], R d ) then there exists σ ∈ P 4 ([0, T ]) such that x ∈ R 1 2 σ ([0, T ], R d ) ∩ C 1 2 -([0, T ], R d ).
Then the pathwise Ito formula [START_REF] Föllmer | Calcul d'Itô sans probabilités[END_REF] .

implies t 0 (∇f • x).d π x = f (x(t)) -f (x(0)) - 1 2 t 0 < ∇ 2 f (x), d[x] π > and t 0 (∇f • x).d σ x = f (x(t)) -f (x(0)) - 1 2 t 0 < ∇ 2 f (x), d[x] σ > .
(∇f • x).d π x = . 0 (∇f • x).d σ x ∈ Q π ([0, T ], R) ∩ Q σ ([0, T ], R) and . 0 ∇f • x dx π (t) = t 0 < (∇f • x) t (∇f • x), d[x] π > .
0 ∇f • x dx σ (t) = . 0 ∇f • x dx π (t) = t 0 < (∇f • x) t (∇f • x), d[x] > .
Since [x] is strictly increasing by assumption, the right hand side is a strictly increasing function as soon as ∇f • x = 0 (otherwise the result trivially holds). Since we can choose any π ∈ C( 1 2 , σ)

in the above, we can apply Lemma 4.1 to conclude that I(f, x) has the quadratic roughness property along σ: I(f, x) ∈ R 1/2 σ ([0, T ], R). So finally I(f, x) ∈ Q π ([0, T ], R).

Local time

Pathwise analogues of (semimartingale) local time have been considered in [START_REF] Bertoin | Temps locaux et intégration stochastique pour les processus de Dirichlet[END_REF][START_REF] Cont | Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity[END_REF][START_REF] Davis | Arbitrage bounds for prices of weighted variance swaps[END_REF][START_REF] Kim | Local time for continuous paths with arbitrary regularity[END_REF][START_REF] Perkowski | Local times for typical price paths and pathwise Tanaka formulas[END_REF][START_REF] Wuermli | Lokalzeiten für martingale[END_REF] in the context of extension of Föllmer's pathwise Ito formula to convex functions or functions with Sobolev regularity. In the aforementioned studies, local time of a path is constructed as a limit of a sequence of discrete approximations along a sequence of time partitions. Given a partition sequence σ = (σ n ) n≥1 and a path

x ∈ C 0 ([0, T ], R) ∩ Q σ ([0, T ], R), one defines the function L σ n t : R → R by L σ n t (u) := 2 t n j ∈σ n ∩[0,t] 1 [[x(t n j ),x(t n j+1 ))] (u) |x(t n j+1 ∧ t) -u|.
where

[[u, v)] := [u, v) if u ≤ v and [[u, v)] := [v, u) if u > v. L σ n t is bounded and zero outside [min x, max x].
Following [START_REF] Wuermli | Lokalzeiten für martingale[END_REF][START_REF] Bertoin | Temps locaux et intégration stochastique pour les processus de Dirichlet[END_REF][START_REF] Davis | Arbitrage bounds for prices of weighted variance swaps[END_REF][START_REF] Perkowski | Local times for typical price paths and pathwise Tanaka formulas[END_REF] we say that x has (L 2 -)local time on [0, T ] along σ if the sequence (L σ n t , n ≥ 1) converges weakly in L 2 (R) to a limit L σ t for all t ∈ [0, T ]:

∀t ∈ [0, T ], ∀h ∈ L 2 (R), L σ n t (u)h(u)du n→∞ → L σ t (u)h(u)du.
The local time along π satisfies the occupation time formula [START_REF] Wuermli | Lokalzeiten für martingale[END_REF][START_REF] Bertoin | Temps locaux et intégration stochastique pour les processus de Dirichlet[END_REF][START_REF] Perkowski | Local times for typical price paths and pathwise Tanaka formulas[END_REF]: for every Borel set

A ∈ B(R), A L π t (u)du = 1 2 t 0 1 A (x)d[x] π
and the following extension of the pathwise Ito formula [START_REF] Karandikar | On the quadratic variation process of a continuous martingale[END_REF] to functions in the Sobolev space W 2,2 (R) (see e.g.[10, Thm 3.1]):

∀f ∈ W 2,2 (R), f (x(t)) -f (x(0)) = t 0 (f • x).d π x + 1 2 R L π t (u)f (u)du, (24)
where the first integral is a limit of left Riemann sums along π:

t 0 (f • x).d π x = lim n→∞ π n f (x(t n i )).(x(t n i+1 ) -x(t n i )).
Unlike the intrinsic definition of local time for real functions (see e.g. [START_REF] Geman | Local times for real and random functions[END_REF]), the above construction depends on the choice of the partition sequence π and a natural question is therefore to clarify the dependence of this local time on choice of the partition sequence. Note that, differently from [START_REF]Occupation densities[END_REF], L π t is the density of a weighted occupation measure, weighted by quadratic variation [x] π so a necessary condition for the uniqueness of L π t is the uniqueness of [x] π . We now show that the quadratic roughness property implies an invariance property of the local time with respect to the sequence of partitions:

∈ C α ([0, T ], R) ∩ R β σ ([0, T ], R) with 0 < β ≤ α ≤ 1 2 . Assume x has local time L σ t on [0, t] along σ. Then if x has local time L π t on [0, t] along some balanced partition sequence π ∈ B(0, T ]) then L π t (u) = L σ t (u) du -a.e.
This defines a unique element L t ∈ L 2 (R) which we call the local time of x on [0, t].

This result shows that for paths satisfying the quadratic roughness property, the (L 2 -)local time is an intrinsic object associated with the path x, independent of the (balanced) sequence of partitions used in the construction.

Proof. From [START_REF] Wuermli | Lokalzeiten für martingale[END_REF]Satz 9] for any Borel set A ∈ B(R) we have the occupation density formula:

A L σ t (u)du = 1 2 t 0 1 A (x)d[x] σ .
If π is a balanced sequence of partitions and the local time along π exists, we also have

∀A ∈ B(R), A L π t (u)du = 1 2 t 0 1 A (x)d[x] π .
Since π is balanced and lim sup

n |σ n | |σ n+1 | < ∞, Theorem 4.2 implies that [x] π = [x] σ . Hence ∀A ∈ B(R), A L π t (u)du = A L σ t (u)du,
which implies L π t = L σ t almost everywhere.

A Proofs of lemmas

A.1 Proof of Proposition 2.2

(i) For any sequence of partitions π of [0, T ] and for any n ≥ 1:

N (π n )π n ≤ T ≤ N (π n )|π n |.
For proof of (⇒): Using the balanced property, lim inf

n→∞ N (π n )π n = lim inf n→∞ N (π n )|π n | π n |π n | ≥ lim inf n→∞ 1 c N (π n )|π n | ≥ T c > 0. Similarly, lim inf n→∞ N (π n )|π n | = lim inf n→∞ N (π n )π n |π n | π n ≤ lim inf n→∞ cN (π n )π n ≤ cT < ∞.
For proof of (⇐): lim sup

n→∞ |π n | π n = lim sup n→∞ N (π n )|π n | N (π n )π n = lim sup n→∞ N (π n )|π n | lim sup n→∞ N (π n )π n < ∞. (ii)
For any sequence of partitions π, and any fixed h > 0 there exists a N 0 such that for all n ≥ N 0 , |π n | < h. So for all n ≥ N 0 and for all t ∈ [0, T -h], N (π n , t, t + h) ≥ 1. Hence:

π n ≤ h N (π n , t, t + h) ≤ |π n |. So lim sup n→∞ sup t∈[0,T -h] N (π n , t, t + h) inf t∈[0,T -h] N (π n , t, t + h) ≤ lim sup n→∞ |π n | h × h π n < ∞.
(iii) For any balanced sequence of partitions π of [0, T ] and for any n ≥ 1:

c 1 N (π n )|π n | ≤ N (π n )π n ≤ T ≤ N (π n )|π n | ≤ c 2 N (π n )π n .
c 1 and c 2 are constants > 0. So the equivalence follows.

(iv) Let π = (π n ) n≥1 be any balanced sequence of partitions of [0, T ]:

π n = (0 = t n 1 < t n 2 < • • • < t n N (π n ) = T ).
Now, define the new partition g(π) = (g(π n )) n≥1 as follows: A.2 Proof of Lemma 2.3

g(π n ) = g(0) = g(t n 1 ) < g(t n 2 ) < • • • < g(t n N (π n ) = g(T )) . Now,
Denote the partition points of τ n and σ n respectively by (t n k , k = 0..N (τ n )) and (s n l , l = 0..N (σ n )).

Proof of (i): From the assumption we have, lim sup

k(n) = inf{k ≥ n, |τ k | ≤ |σ n |} < ∞ since |τ k | k→∞ → 0. ( 26
)
We now consider the subsequence (τ k(n) ) n≥1 of τ . From the definition of k(n):

lim sup n |σ n | |τ k(n) | ≥ 1.
Proof of (ii): Define k(n) as in [START_REF] Lévy | Théorie de l'addition des variables aléatoires[END_REF] for i = 1..N (σ n ),

j(i, n) = inf{j ≥ 1, t k(n) j
∈ (s n i , s n i+1 ]}. Then we have

t k(n) j(i,n)-1 ≤ s n k < t k(n) j(i,n) < • • • < t k(n) j(i+1,n)-1 ≤ s n i+1 < t k(n) j(i+1,n) .
If for some i, j(i+1,n)-j(i,n)→ ∞ as n → ∞ then, from the above construction of k(n)

and using the well balanced property of σ n and τ k(n) we have: lim sup

n |σ n | |τ k(n) | → ∞ and lim sup n |σ n | |τ k(n)-1 | < 1. Hence, lim sup n |σ n | |τ k(n)-1 | |τ k(n)-1 | |τ k(n) | -1 → ∞ which is a contradiction
because of our assumption. Hence the size j(i + 1, n) -j(i, n) of clusters is uniformly bounded:

∀k, n ≥ 1, j(i + 1, n) -j(i, n) ≤ M < ∞.
So there exists a constant c 0 such that

1 ≤ lim sup n |σ n | |τ k(n) | ≤ c 0 < ∞. (27)
Therefore (τ k(n) ) n≥1 and (σ n ) n≥1 are (asymptotically) comparable. A.3 Proof of Proposition 3.3

The proof of 1 -3 are direct consequences of Definition 3.1. The proof of 4 is as follows.

LetA n = (0 = t n p(n,0) < t n p(n,1) < From the quadratic roughness property of x, the first term in the sum goes to zero as n increases.

For the last sum we have This completes the proof.

Lemma 2 . 3 (

 23 Adjusting the mesh of a balanced sequence). Let τ = (τ n ) n≥1 and σ = (σ n ) n≥1 be two balanced partition sequences of [0, T ] with lim sup n |σ n | |τ n | < 1 and |τ n | n→∞ ----→ 0.

5

  Pathwise Itô calculus 5.1 Pathwise integration and the pathwise Itô formula

Theorem 4 .

 4 2 and Proposition 4.5 allow to give an intrinsic formulation of Föllmer's pathwise integration and pathwise Itô calculus, without relying on a specific sequence of partitions.

Theorem 5 . 1 (

 51 Uniqueness of the Föllmer integral). There exists a unique map

From Theorem 4 . 2 0 ( 0 ( 0 ( 1 2

 420001 we have [x] σ = [x] π . So: ∀t ∈ [0, T ], t ∇f • x).d π x = t ∇f • x).d σ xi.e. the pathwise integral t ∇f • x).d π x along a balanced sequence of partitions π does not depend on choice of π. By the same argument, it does not depend on the choice of σ ∈ P 4 ([0, T ]).To show I(f, x) ∈ Q([0, T ], R) we first note that by[START_REF] Ananova | Pathwise integration and functional calculus for paths with finite quadratic variation[END_REF] Lemma 4.11] we haveI(f, x) ∈ C -([0, T ], R).Applying the pathwise isometry formula [2, Theorem 2.1], to the integral . 0 (∇f • x).d π x we obtain that . 0

From

  Theorem 4.2 we have [x] σ = [x] π , so . 0 ∇f • x dx π (t) does not depend on choice of balanced partition π. As a consequence:

Theorem 5 . 2 (

 52 Uniqueness of local time for rough functions). Let σ be a sequence of partitions satisfying Assumption 1 and x

n |τ n | |σ n | > 1 .

 1 Then there exists N 0 ∈ N such that for n ≥ N 0 , |τ n | |σ n | ≥ 1. Since we are only concerned about the limiting behaviour when n → ∞ we will only consider n > N 0 throughout the rest of the proof. If lim sup n |τ n | |σ n | < ∞ we set k(n) = n; otherwise if lim sup n |τ n | |σ n | = +∞ we define:

  Proof of (iii):If lim sup n |σ n | |τ n | < 1 then the set {n ≥ 1, |σ n | |τ n | ≥ 1} is finite and the set, A = {n ≥ 1, |σ n | |τ n | < 1}is infinite. Now define r : N → N as follows: we set r(n) = n for n / ∈ A andr(n) = inf{k ≥ 1, |σ k | > |τ n |} < ∞ for n ∈ A.

  • • • < t n p(n,N (A n )) = T ) be any β-coarsening of π. Then |A n | |π n | β . Let I n j = (t p(n,j-1) , t p(n,j) ] ∩ [0, t].Then for all t ∈ (0, T ] :

	N (A n )
	(x + y)(t n i+1 ) -(x + y)(t n i )	t (x + y)(t n i +1 ) -(x + y)(t n i )
	j=1 t n i =t n i ∈I n j
	N (A n )
	=	x(t n i+1 ) -x(t n i )	t x(t n i +1 ) -x(t n i )
	j=1 t n i =t n i ∈I n j
	N (A n )
	+	y(t n i+1 ) -y(t n i )	t y(t n i +1 ) -y(t n i )
	j=1 t n i =t n i ∈I n j
	N (A n )
	+	x(t n i+1 ) -x(t n i )
	j=1 t n i =t n i ∈I n j

t y(t n i +1 ) -y(t n i ) .

  Since A n is a balanced sequence and N (A n ) → ∞, we have |A n | → 0 so by continuity of x, the first term goes to zero as n → ∞. The second term is bounded as y has bounded variation. So we have

									
		≤	 sup j	t n i ∈I n j	x(t n i+1 ) -x(t n i ) + sup k	(x(t n k+1 ) -x(t n k )		π n	y(t n i +1 ) -y(t n i )
					N (A n )
							x(t n i+1 ) -x(t n i )	t y(t n i +1 ) -y(t n i )	n→∞ ----→ 0.
						j=1 t n i =t n i ∈I n j
	Similarly,
					N (A n )
							y(t n i+1 ) -y(t n i )	t y(t n i +1 ) -y(t n i )
						j=1 t n i =t n i ∈I n j
						N (A n )
								x(t n i+1 ) -x(t n i )	t y(t n i +1 ) -y(t n i )
							j=1 t n i =t n i ∈I n j
						N (A n )
					=		y(t n i +1 ) -y(t n i )	t	x(t n i+1 ) -x(t n i )
						j=1 t n i ∈I n j		t n i ∈I n j ,t n i =t n i
						N (A n )
					≤		y(t n i +1 ) -y(t n i )	t	x(t n i+1 ) -x(t n i )
						j=1 t n i ∈I n j		t n i ∈I n j ,t n i =t n i
		N (A n )						
		≤				y(t n i +1 ) -y(t n i )	t		x(t n i+1 ) -x(t n i ) + (x(t n i +1 ) -x(t n i )	
			j=1 t n i ∈I n j			t n i ∈I n j
										N (A n )
	≤	 sup j	t n i ∈I n j	x(t n i+1 ) -x(t n i ) + sup k	(x(t n k+1 ) -x(t n k )		j=1 t n i ∈I n j	y(t n i +1 ) -y(t n i )

n→∞

----→ 0.

Acknowledgements. We thank Anna Ananova, Henry Chiu and David Prömel for helpful remarks and suggestions.

An important consequence of this result is the uniqueness of limits of left Riemann sums for integrands in the Sobolev space W 1,2 (R) and a robust version of the pathwise Tanaka formula: Corollary 5.3 (Uniqueness of Föllmer integral on W 1,2 (R) and pathwise Tanaka formula). Under the assumptions of theorem 5.2 we have:

Designating this common value by

where the pathwise integral and the local time may be computed with respect to any balanced partition sequence along which x has local time.