Rough Functional Itô Formula
Franziska Bielert

§ ABSTRACT 
We prove a rough Itô formula for path-dependent functionals of α-Hölder continuous paths for α∈(0,1). Our approach combines the sewing lemma and a Taylor approximation in terms of path-dependent derivatives.

: Leveraging Composability and Diversity to Design Fault and Intrusion Resilient Chips
    Ahmad T. Sheikh, Ali Shoker, Suhaib A. Fahmy and Paulo Esteves-Verissimo
CEMSE Division, King Abdullah University of Science and Technology (KAUST)

Thuwal 23955-6900, Kingdom of Saudi Arabia

{ahmad.sheikh, ali.shoker, suhaib.fahmy, paulo.verissimo{@kaust.edu.sa}}
    September 9, 2024
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§ INTRODUCTION

In dupireFunctionalItoCalculus2009 Dupire developed an Itô calculus for causal functionals F, i.e., functionals that depend at time t∈[0,T] on a path X [0,T] → up to time t. He introduced suitable notions of directional derivatives. A `time' derivative DF by a perturbation of time t+h with stopped paths X_t and a `space' derivative ∇ F by fixing the time and perturbing of the end point of the stopped path X_t + h_[t,T]. Similar results were established in a number of papers by R. Cont and D.A. Fournié using purely analytical arguments for paths X that have finite quadratic variation in a pathwise sense. A thorough treatment can be found in BallyVlad2016Sibp. In particular they proved a pathwise functional Itô formula using Föllmer type integrals introduced in Föllmer1981.

Afterwards the pathwise Itô formula for path-dependent functionals (as well as for standard functions) was extended to paths X with arbitrary regularity by R. Cont and N. Perkowski in contPathwiseIntegrationChange2019. For non path-dependent functions they also investigated the relation to rough path theory. By identifying a natural candidate for the reduced rough path  induced by a multidimensional path X, it was shown that the Föllmer integral in the pathwise Itô formula coincides with a rough integral. 

The main result of this paper is Theorem <ref>. It constructs a rough integral ∫∇ F(t, X) (t) for multidimensional α-Hölder continuous paths for α∈(0,1) and provides a rough functional Itô formula under suitable regularity assumptions on the causal functional F.  

Similar results for less regular functionals F and cadlag paths X with finite p-variation have been obtained independently by Christa Cuchiero, Xin Guo and Francesca Primavera and are to be published in cuchieroFunctionalItoformulaTaylor. Their proof relies on a density argument, passing from linear functions of the
signature of the path to general path functionals.

This work was intended to give a simple proof that follows the standard approach in rough path theory. So we will allow for strong regularity assumptions on F and apply the sewing lemma. 
Namely Corollary <ref> gives an error bound for higher order Taylor approximations of F(t, X) in terms of the causal derivatives. This is a generalization of Lemma 2.2 from A. Ananova and R. Cont in ananovaPathwiseIntegrationRespect2017. The higher order Taylor approximation allows to adapt the techniques in contPathwiseIntegrationChange2019 to the path-dependent setting.



 §.§ Notation

Let T>0 and D denote the set of cadlag paths X [0, T] →^d equipped with the uniform norm |·|_∞. For such paths and t∈ [0,T] we denote by X(t) the value of the path at time t and by X_t the stopped path X_t = X(·∧ t). Let further X_t- denote the path X stopped right before t, namely for u∈[0,T], X_t-(u) = X(u)_[0,t)(u) + lim_r↑ tX(r) _[t, T](u) .






Let Δ_T:= { (s,t)∈ [0,T]× [0,T] 0≤ s≤ t≤ T}.
We call = { [t_k-1, t_k] k=1,…,n} with t_k∈[0,T] for all k=0,…,n, partition of [0,T] if 0=t_0< t_1<… < t_n = T. The mesh of a partition  is defined as || = max_[s,t]∈| t-s|. interval@||

For α∈(0,1), a two-parameter path ΞΔ_T→^d is α-Hölder continuous if 

    |Ξ|_α := sup_(s,t)∈Δ_T|Ξ(s,t)|/| t-s|^α < ∞,

here |·| denotes the euclidean norm.
Then a path X [0,T]→^d is α-Hölder continuous if its increments (δ X)(s,t):= X(t) - X(s) are.


For two terms x,y we abbreviate the existence of some constant C>0 such that x≤ C y to x≲ y and by ≲_p we indicate a dependency C= C(p) on some parameter p.≲





 §.§ Causal Derivatives

Following dupireFunctionalItoCalculus2009 and oberhauserExtensionFunctionalIto2012 (from where we took the present definitions), we consider for causal functionals F [0,T]× D →, i.e., 
F(t, X) = F(t, X_t),  the following notions of differentiability:

[Causal Space Derivative]
	If for all (t,X)∈ [0,T]× D the map
	
    ^d∋ h↦ F(t, X_t + h _[t,T])

	is continuously differentiable at h = 0 we say that F has a causal space derivate. We denote it by ∇ F(t,X) = (∂_1 F(t,X), …, ∂_n F(t,X)). Recursively, we define for n∈ the nth causal space derivative and denote it by ∇^n F.
 
[Causal Time Derivative]
	If for all (t,X)∈ [0,T]× D the map 
	
    [0,∞)∋ h ↦ F(t + h, X_t)

is continuous and right-differentiable at h=0 we denote this derivative by D F(t, X). If additionally t↦ D F(t, X) is Riemann integrable, then we say that F has a causal time derivative.


For n∈ we write F∈ℂ^1,n_b, if F has a causal time derivative and n causal space derivatives such that F, DF and for k=1, …, n, ∇^k F are continuous in [0,T]× D and bounded in the sense that sup_(t, X)∈[0,T]× D| F(t, X)| <  ∞. 
We refer to [Definition 19]oberhauserExtensionFunctionalIto2012 for weaker regularity notions. Since the purpose of this paper is to give a simple proof of a rough functional Itô formula, we follow the rough path tradition and keep the assumptions simple. 



§ TAYLOR APPROXIMATION FOR CAUSAL FUNCTIONALS


To derive a Taylor formula for t ↦ F(t, X) we use the signature X of a paths that have bounded variation. 
We briefly specify the (for us necessary) theory.



 §.§ Symmetric Part of the Signature of a Path

Set T_0(^d) := 1 and for k∈ , T_k(^d) := (^d)^⊗ k the space of k-tensors and T(^d) = ⊕_k=0^∞ T_k(^d) the tensor algebra. 
A word w in the alphabet := { 1, … d} of length k is a tuple (w_1, …, w_k) such that for j=1, …, k, w_j ∈. 
Denote for i=1, …, d by e_i := (0, …, 0, 1, 0, …, 0) the ith unit vector and e_w := e_w_1⊗…, ⊗ e_w_k. Then the set { e_w w  word in  of length k} is a basis of T_k(^d). 
We write ⟨·, ·⟩ for the natural inner product in T_k(^d). 
Abusing the notation a bit, we also write for T∈ T_k(^d), S∈ T_m(^d) with m<k, T, S∈ T_m-k, where for h∈ T_k-m(^d), T, S (h) :=  T, S ⊗ h. Finally note that we can choose compatible norms |·| on T_k(^d), i.e. for v_1, …, v_k∈^d, 

    | v_1 ⊗…⊗ v_k|≤∏_j=1^k | v_j|.

Let _k denote the projection from T(^d) onto T_k(^d).


Let further X [0,T] →^d be continuous and of bounded variation, i.e. there exists finite signed measures μ^i([0,T]) →, such that for all t∈[0,T], μ^i([0,t]) = X^i(t).
Then the signature is a two-parameter path XΔ_T → T(^d), where for every (s,t)∈Δ_T, k∈,

    _k Xs,t = ∑_w = (w_1, …, w_k),
    
    			w_j∈⟨Xs,t, e_w⟩ e_w,

with

    ⟨Xs,t, e_w⟩ := ∫_s^t ∫_s^s_k…∫_s^s_2 X^w_1(s_1) … X^w_k(s_k).

The symmetric part T of a k-tensor T is given via 

    ⟨T, e_w⟩ = 1/k!∑_σ∈𝔖_k⟨ T, e_(w_σ1, …, w_σ k)⟩,
  
where 𝔖_k denote the permutation group of degree k. 
We next define a commutative product on tensors indexed by words. Let  m, k_1,…, k_m∈ and k= ∑_j=1^m k_j. The shuffles sh(k_1,…, k_m) of words of length k_1,…, k_m are those permutations σ∈𝔖_k such that σ 1 < … < σ k_1, σ (k_1+1) < … <  σ (k_1+k_2) and so on. Then for a word w of length k, we define

    e_(w_1, …, w_k_1)… e_(w_k - k_m + 1, …, w_k) = ∑_σ∈sh(k_1, …, k_m) e_(w_σ 1, …, w_σ k).



Note that for letters w_1, …, w_k∈, this reduces to

    e_w_1… e_w_k = ∑_σ∈𝔖_k e_(w_σ 1,…, w_σ k).
 
It is easy to check that the signature has the remarkable property that for two words w and u it holds that

    ⟨Xs,t, e_w e_u ⟩ = ⟨Xs,t, e_w⟩⟨Xs,t, e_u⟩,

compare [Exercise 2.2]RoughBook. We deduce that the symmetric part of the kth level signature is 

    _k Xs,t = 1/k! (X(t) - X(s))^⊗ k,

since it follows from (<ref>), (<ref>), (<ref>) and (<ref>) for a word w of length k that

    ⟨_k Xs,t, e_w ⟩   =1/k!∑_σ∈𝔖_k⟨Xs,t, e_(w_σ1, …, w_σ k)⟩
    	= 1/k!⟨Xs,t, e_w_1… e_w_k⟩
       = 1/k!∏_j=1^k ⟨Xs,t, e_w_j⟩ = 1/k!∏_j=1^k (X^w_j(t) - X^w_j(s)).

Finally we point out that [Proposition 3.5]frizUnifiedSignatureCumulants2021 shows that the symmetric part of the signature satisfies 

    _k Xs,t = ∫_s^t _k-1Xs,r⊗ X(r).

Proof basically just X = X⊗ X and so

    ∫_s^t _k-1Xs,r⊗ X(r) = ∫_s^t _k-1Xs,r⊗ X(r) = _kXs,t.




 §.§ Taylor Formula for Causal Functionals


The first result establishes a Taylor formula in terms of path-dependent derivatives for paths of bounded variation. It is based on the Taylor expansion of one-dimensional and piecewise constant paths X that is used in BallyVlad2016Sibp, contPathwiseIntegrationChange2019 to prove the functional Itô formula with Föllmer integrals. It will prove very useful to have an explicit representation of the remainder.

[Taylor Formula for Functionals of Bounded Variation Paths]
Let n, d∈ and F∈ℂ^1,n_b, such that for k=1, …, n-1, ∇^k F∈ℂ^1,1_b. Then it holds for every path X [0,T]→^d that is continuous and of bounded variation and every (s,t)∈Δ_T, that 

    F(t, X) - F(s, X)   = ∑_k=0^n-11/k!∫_s^t  D∇^k F(u, X), (X(t) - X(u))^⊗ k u + ∑_k=1^n-11/k!∇^k F(s, X),(X(t) - X(s))^⊗ k
          + 1/(n-1)!∫_s^t ⟨∇^n F(u, X),(X(t) - X(u))^⊗ n-1⊗ X(u)⟩ .




The proof is by induction on n.

Note that in the case that d=1, the result follows for  n=1 from [Theorem 1.10]contPathwiseIntegrationChange2019 applied with p=2 and X^2 = 0 with higher regularity assumption on F. For consistency we give the start of the induction with minor changes due to d≥ 1 and X more regular.

For n=1 let () be a sequence of partitions of [s,t] with ||→ 0. We consider the piecewise constant approximation of X on [s,t]:

    X^(u) = X(u)_[0,s)(u) + ∑_[t_j, t_j+1]∈ X(t_j+1)_[t_j, t_j+1) + X(t)_[t, T].

Since X^→ X uniformly and X^_t- = X^_t, X^_s- = X_s, it holds 

    F(t, X) - F(s,X) = lim_||→ 0∑_[t_j, t_j+1]∈ F(t_j+1, X^_t_j+1-) - F(t_j,X^_t_j-).
 
Noting that X^_t_j = X^_t_j+1- on [0, t_j+1] and X^_t_j = X^_t_j- + (X(t_j+1) - X(t_j))_[t_j, T] we decompose the difference into the time and space perturbation,

    F(t_j+1, X^_t_j+1-) - F(t_j,X^_t_j-)
       = F(t_j+1, X^_t_j) - F(t_j, X^_t_j)  + F(t_j, X^_t_j- + (X(t_j+1) - X(t_j))_[t_j, T]) - F(t_j,X^_t_j-).

By construction of the causal time derivative [t_j, t_j+1)∋ u ↦ F(u, X^_t_j) is right-differentiable with Riemann integrable derivatives, thus by the fundamental theorem of calculus, cf. botskoStrongerVersionsFundamental1986, it holds 

    ∑_[t_j, t_j+1]∈ F(t_j+1, X^_t_j) - F(t_j, X^_t_j)
    = ∑_[t_j, t_j+1]∈∫_t_j^t_j+1 D F(u, X^_t_j)  u.

Since for every u∈[s,t], ∑_[t_j, t_j+1]∈ F(u, X^_t_j) _[t_j, t_j+1)(u) = F(u, X^_u) → F(u, X_u) as ||→ 0 and DF bounded, the last expression converges to ∫_s^t  DF(u, X)  u. Similarly it follows for the space perturbation in (<ref>) that

    F(t_j, X^_t_j- + (X(t_j+1) - X(t_j))_[t_j, T]) - F(t_j,X^_t_j-)
       = ∫_0^1 ⟨∇ F(t_j, X^_t_j- + λ(X(t_j+1) - X(t_j)_[t_j, T]), (X(t_j+1) - X(t_j))⟩λ
       =: ⟨∇ F(t_j,X^_t_j-), (X(t_j+1) - X(t_j))⟩ + R_j.

For i=1,…, d let now μ^i denote measures of bounded variation related to component X^i. Since ∑_[t_j, t_j+1]∈∂_i F(t_j,X^_t_j-)_[t_j, t_j+1)(u) →∂_i F(u, X) as ||→ 0 and ∂_i F bounded, it follows that

    ∑_[t_j, t_j+1]∈∂_i F(t_j,X^_t_j-) (X^i(t_j+1) - X^i(t_j)) 
       = ∫_s^t ∑_[t_j, t_j+1]∈∂_i F(t_j,X^_t_j-)_[t_j, t_j+1)(u) μ^i(u)
       →∫_s^t ∂_i F(u, X) μ^i(u)
    = ∫_s^t ∂_i F(u, X)  X^i(u).

Moreover using that the images of (𝕀,X^) are compact in [0,T]× D, we may assume that ∇ F is compactly supported and therefore uniformly continuous. 


For every n∈, K_n = {(t_j, X^_t_j-+ λ(X(t_j+1) - X(t_j))_[t_j, T]) j = 0, … ,n-1, λ∈[0,1]} is compact: Any sequence in K_n corresponds to a sequence t_m in the finite set {t_j j=0,…, n-1} and (λ_m)⊂[0,1]. Then there are note relabeled subsequences with t_m → t_j^* and λ_m →λ*∈[0,1]. Clearly for large m, t_m = t_j^*. Then trivially 
    (t_m, X^_t_m-+ λ_m(X(t_m+1) - X(t_m)_[t_m, T]) = (t_j^*, X^_t_j^*-+ λ_m(X(t_j+1^*) - X(t_j^*)_[t_j^*, T]) → (t_j^*, X^_t_j^*-+ λ^*(X(t_j+1^*) - X(t_j^*))_[t_j^*, T])∈ K_n.

Hence the remainders 

    R_j = ∫_0^1 ∇ F(t_j, X^_t_j- + λ(X(t_j+1) - X(t_j))_[t_j, T]) - ∇ F(t_j,X^_t_j-) λ· (X(t_j+1) - X(t_j))

satisfy 

    ∑_[t_j, t_j+1]∈| R_j|≤ C(|∇ F|_∞, ) ∑_[t_j, t_j+1]∈| X(t_j+1) - X(t_j)|≤ C(|∇ F|_∞, )|μ|([s,t])

where |μ| denotes the total variation of μ and C(|∇ F|_∞, )→ 0 as ||→∞.


For n → n+1 we apply the previous result componentwise to get 

    ∇^n F(u, X) - ∇^n F(s, X) = ∫_s^u D∇^n F(r, X)  r + ∫_s^u ⟨∇^n+1 F(r, X),  X(r) ⟩ (·).





Plugging that into the remainder (<ref>) and using Fubini, it follows

    ∫_s^t  ⟨∇^n F(u,  X), ( X(t) -  X(u))^⊗ n-1⊗ X(u)⟩
       = ∫_s^t  D∇^n F(r,X),  ∫_r^t (X(t) -  X(u))^⊗ n-1⊗ X(u) r 
           + ∫_s^t ∇^n+1 F(r,X),∫_r^t (X(t) -  X(u))^⊗ n-1⊗ X(u) ⊗ X(r)
          + ∇^n F(s,  X),∫_s^t( X(t) -  X(u))^⊗ n-1⊗ X(u)

For every r∈[s,t] the function g(h):= F(r, X_r + h_[r,T]) is (n+1)-times continuously differentiable in zero by assumption. Thus Schwarz' lemma shows that the causal space derivative ∇^n+1 F(r, X) = ∇^n+1 g(0) is a symmetric tensor (i.e., ∇^n+1 F(r, X) = ∇^n+1 F(r, X)). 

Consequently,

    ∇^n F(s,  X),∫_s^t( X(t) -  X(u))^⊗ n-1⊗ X(u)= ∇^n F(s,  X), ∫_s^t( X(t) -  X(u))^⊗ n-1⊗ X(u).

It holds that

    ∫_s^t( X(t) -  X(u))^⊗ n-1⊗ X(u)
       = ∑_l=0^n-1 (-1)^n-l-1n-1l(X(t)-X(s))^⊗ l⊗∫_s^t (X(u) - X(s))^⊗ n-l-1⊗ X(u).
 
Recalling (<ref>) and (<ref>) we deduce

    1/(n-l-1)!∫_s^t (X(u) - X(s))^⊗ n-l-1⊗ X(u)
    
    = 1/(n-l)! (X(t)- X(s))^⊗ n-l.


And plugging that into (<ref>) yields

    ∫_s^t( X(t) -  X(u))^⊗ n-1⊗ X(u)   = (X(t)-X(s))^⊗ n∑_l=0^n-1 (-1)^n-l-1n-1l(n-l-1)!/(n-l)!
       =(n-1)!/n! (X(t)-X(s))^⊗ n,

since ∑_l=0^n-1 (-1)^n-l-1nl = 1. Using similar arguments for the inner product with D∇^n F and ∇^n+1 F in (<ref>) we conclude that


    n!/(n-1)!∫_s^t  ⟨∇^n F(u,  X), ( X(t) -  X(u))^⊗ n-1⊗ X(u)⟩
       =∫_s^t  D∇^n F(r,X), ( X(t) -  X(r))^⊗ n r +∫_s^t ∇^n+1 F(r,  X),(X(t) -  X(r))^⊗ n⊗ X(r)
           + ∇^n F(s,X),(X(t)-X(s))^⊗ n.



 Note that it is sufficient that the functional F and its causal derivatives are continuous. As seen in the proof the images (𝕀, X^) lie in a compact subset of [0,T]× D if X is continuous. Then any continuous functional restricted to this compact metric space is uniformly continuous and bounded.  


The next corollary is a generalization of [Lemma  2.2]ananovaPathwiseIntegrationRespect2017. 
It estimates the error of a lower order Taylor approximation of F composed with an α-Hölder continuous path X [0,T]→^m. The reader may notice that the previously mentioned result [Theorem 1.10]contPathwiseIntegrationChange2019 can be applied to less regular paths using Föllmer integrals. But since we want to estimate the remainder (<ref>) for the next result, we prefer to use integrals against paths of bounded variation.


Let X [0,T] →^d be α-Hölder continuous for some α∈(0,1) and F as in Theorem <ref>. Assume additionally that F and D F are Lipschitz continuous for fixed times with bounded Lipschitz constants. Then it holds for every (s,t)∈Δ_T with | t-s|≤ 1 and 0≤ l≤ n-1 that

    | F(t, X) - ∫_s^t DF(r,X) r- ∑_k=0^l 1/k!∇^k F(s, X),(X(t) - X(s))^⊗ k|
        ≲| t-s|^α + (n-1)α^2 + | t-s|^1+α + | t-s|^(l+1)α,

with a constant depending on n, l, | F|_∞, | D∇^k F|_∞ and |∇^k+1F|_∞ for k=l+1,…, n-1 as well as sup_r∈[s,t]{ Lip(F(r, ·), DF(r, ·))} and | X|_α.



We point out the differences of (<ref>) to a typical Taylor approximation. As usual the exponent (l+1)α connected to the l space derivatives used in the approximation. The appearance of (1+α) is due to the path-dependent time derivatives. And finally α + (n-1)α^2 due to an approximation of X by piecewise constant paths.



Let  be a partition of [s,t] whose subintervals are all of length ||.
Consider a piecewise linear approximation X^ of X on [s,t] such that X^_s = X_s, and for every [u,v]∈ it holds X^(u)= X(u) and X^(v) = X(v) and in between X^ is linearly interpolated. Then X^ is continuous and on [s,t] of bounded variation. Hence the previous theorem shows that

    F(t, X^) - ∫_s^t DF(r,X^) r- ∑_k=0^l 1/k!∇^k F(s, X^),(X^(t) - X^(s))^⊗ k
       = ∑_k=1^n-11/k!∫_s^t  D∇^k F(r, X^), (X^(t) - X^(r))^⊗ k r+ ∑_k=l+1^n-11/k!∇^k F(s, X^),(X^(t) - X^(s))^⊗ k
           + 1/(n-1)!∫_s^t ⟨∇^n F(r, X^),(X^(t) - X^(r))^⊗ n-1⊗ X^(r)⟩.

Note that X^ is also α-Hölder continuous  with | X^|_α≲| X|_α
 u≤ v∈ [t_j, t_j+1] it holds X^(v) - X^(u)= v-u/t_j+1 - t_j (X(t_j+1) - X(t_j)), so 
    | X^(v) - X^(u)|/| v-u|^α≲_| X|_αv-u/t_j+1 - t_j^1-α≤ 1.

And for u∈[t_j, t_t_j+1], v∈[t_i, t_i+1] with j<i it follows that 
    | X^(v) - X^(u)|/| v-u|^α≲_| X|_α(t_j+1-u/v-u)^α + (t_i - t_j+1/v-u)^α +  (v - t_i/v-u)^α≤ 3.
  
and that for every [u,v]∈ it holds on (u,v),

    | X^/ r| = |X(v) - X(u)/v- u|≤| v-u |^α - 1 = ||^α - 1.

In the case l≤ n-2, it follows that (<ref>) is bounded by 

    ∑_k=1^n-11/k!| D∇^k F|_∞| X^|_α^k| t - s|^kα| t-s|  + ∑_k=l+1^n-11/k!|∇^k F|_∞| X|_α^k| t-s|^kα
          + 1/(n-1)!|∇^n F|_∞| X^|_α^(n-1)| t-s|^(n-1)α||^α - 1| t-s|
       ≲_n,l, | D∇^k F|_∞, | X|_α, |∇^k F|_∞| t-s|^1+α + | t-s|^(l+1)α + | t-s|^1+ (n-1)α||^α - 1.

Since F(s,X^) = F(s,X) and X^(t) -X^(s) = X(t) - X(s) it follows that

    F(t, X) - ∫_s^t DF(r,X) r- ∑_k=0^l 1/k!∇^k F(s, X),(X(t) - X(s))^⊗ k
       = F(t, X)-F(t, X^) - ∫_s^t DF(r, X) - DF(r, X^)  r
          +  F(t, X^) - ∫_s^t DF(r,X^) r- ∑_k=0^l 1/k!∇^k F(s, X^),(X^(t) - X^(s))^⊗ k.

Since F and DF are Lipschitz continuous for fixed times it holds that 

    | F(t, X)-F(t, X^) |≲_Lip(F(t,·))| X - X^|_∞≲_| X|_α||^α

and similar

    |∫_s^t DF(r, X) - DF(r, X^)  r |≲_sup_r∈[s,t] Lip(DF(r,·))| X - X^|_∞| t-s|≲_| X|_α| t-s|||^α.

Together with estimate (<ref>), we deduced that (<ref>) is bounded by a constant depending on n, l, Lip(F),  sup_r∈[s,t] Lip(DF(r, ·), | X|_α, | D∇^k F|_∞, |∇^k F|_∞ times

    ||^α + | t-s|||^α  + | t-s|^1+α +| t-s|^(l+1)α + | t-s|^1+ (n-1)α||^α - 1.

Optimizing the choice of ||, by balancing ||^α≈| t-s|^1+ (n-1)α||^α - 1, i.e. ||≈| t-s|^1+(n-1)α we obtain the assertion for 0≤ l≤ n-2. Finally note that for l=n-1, the second sum in (<ref>) is empty, so the RHS is simply | t-s|^1+α+| t-s|^1+ (n-1)α||^α - 1. Nevertheless there is nothing wrong in writing | t-s|^nα in the assertion (<ref>), since α + (n-1)α^2 < nα. 
||≈| t-s|^1+(n-1)α, in the sense that the number of subintervals m=| t-s|/||∈ satisfies 
    | t-s|^-(n-1)α≤ m ≤ 2| t-s|^-(n-1)α,
 possible because | t-s|^-(n-1)α≥ 1. Then 
    ||^α≤| t-s|^α + (n-1)α^2,   ||^α-1≤ 2^1-α| t-s|^-(1+(n-1)α) + α+(n-1)α^2.





§ ROUGH FUNCTIONAL ITÔ FORMULA


Throughout this section we denote the point evaluation of two-parameter paths Ξ by Ξ_s,t = Ξ(s,t). 

We now define for an α-Hölder continuous path X and (s,t)∈Δ_T, ^0_s,t:=1 and for k≥1,

    ^k_s,t := 1/k!(X(t) - X(s))^⊗ k.

Further we write for their collection := (^0, ^1, …). It was shown in [Definition 4.6, Lemma 4.7]contPathwiseIntegrationChange2019 that for every k≥ 1, ^kΔ_T → T_k(^d) is a kα-Hölder continuous two-parameter path and a reduced Chen relation holds: For every (s,u), (u,t)∈Δ_T,

    _s,t = _s,u⊗_u,t.



[Rough Functional Itô Formula]
	Let X be α-Hölder continuous for α∈ (0,1) and n be the smallest natural number such that 2α + (n-1)α^2>1. Let further F be as in Corollary <ref> to parameter n+1. Assume additionally that for each k=1, …, n, ∇^k F and D∇^k F are also Lipschitz continuous for fixed times with bounded Lipschitz constants.  
	Then 
	
    ∫_0^T ∇ F(u, X) (u) := lim_||→ 0∑_[s,t]∈∑_k=1^n⟨∇^k F(s, X),^k_s,t⟩,

	is a well defined limit.
Moreover if F satisfies Corollary <ref> to parameter ñ such that α + (ñ-1)α^2>1, then 

    F(T, X) = F(0, X) + ∫_0^T DF(u, X)  u + ∫_0^T ∇ F(u, X) (u).




	We show existence of the rough integral by adapting the proof of [Proposition 4.10] contPathwiseIntegrationChange2019 to our path-dependent  setting. 
Set for (s,t)∈Δ_T, k=1,…, n,

    Ξ_s,t^X := ∑_k=1^n ⟨∇^k F(s, X), _s,t^k ⟩,     
    	R_s,t^X, k := ∇^k F(t, X) - ∑_l=k^n ⟨∇^l F(s, X), _s,t^l-k⟩.
	
As usual in rough path theory (<ref>) follows from the sewing lemma once we show that for every (s,u), (u,t)∈Δ_T (w.l.o.g | t -s|≤ 1), 
	
    |Ξ_s,t-Ξ_s,u-Ξ_u,t|≲| t-s|^θ

	for some θ >1.
	
Recalling that ∇^k F(s, X) is symmetric, the reduced Chen relation (<ref>) implies that
	
    ∇^k F(s,X), _s,t^k 
    	= ∇^k F(s,X), _k_s,u⊗_u,t
    	= ∑_l=0^k ∇^k F(s,X), _s,u^k-l⊗_u,t^l.

Plugging that into Ξ_s,t and interchanging the summation order, it follows that

    Ξ_s,t - Ξ_s,u = ∑_k=1^n ∑_l=k^n ∇^l F(s,X), _s,u^l-k⊗_u,t^k .

Therefore 

    Ξ_s,t-Ξ_s,u-Ξ_u,t = - ∑_k=1^n  R_s,u^X, k, _u,t^k.


	For k=1, …, n, it holds by Corollary <ref> applied for ∇^k F∈ℂ_b^1,(n+1-k) with l= n-k, that
	
    | R_s,u^X, k|≲| u-s|^α + (n-k)α^2.

The other terms in (<ref>) don't appear since α∈(0,1) and the minimality of n imply α + (n-k)α^2 < (n-k+1) α and 1> α + (n-k)α^2.
	1≥α + (n-k)α^2, if not for some k, then 1 <α +(n-1)α^2. But then 2α + (n-2)α^2 > α + (n-1)α^2 > 1, which is a contradiction to n smallest natural number satisfying 2α + (n-1) α^2 > 1.
	Recalling that ^k is kα-Hölder continuous, we deduced that
	
    | R_s,u^X, k, _u,t^k|≲| u-s|^(k+1)α + (n-k)α^2.

Since
	
    (k+1)α + (n-k)α^2 = nα^2 + α + k(α - α^2) > 2α + (n-1)α^2,

and by assumption 2α+(n-1)α^2>1, (<ref>) now follows from (<ref>).

It is left to show the functional Itô formula. Applying  Corollary <ref> to F∈ℂ_b^1,ñ with l=n, shows that 

    | F(t, X) - F(s, X) - ∫_s^t DF(u, X)  u - ∑_k=1^n⟨∇^k F(s, X),^k_s,t⟩|≲| t-s|^1 + α + | t-s|^(n+1)α + | t-s|^α+(ñ-1)α^2.

By assumption α+(ñ-1)α^2 and (n+1)α are both greater one.
Together with the estimate from the sewing lemma it follows that t↦ F(t, X) - F(0,X) - ∫_0^t DF(u, X)  u - ∫_0^t ∇ F(u,X)(u) is θ̃-Hölder continuous for some θ̃>1. Consequently the map is constant zero.


 Clearly ñ≥ n+1.
For Brownian sample paths the theorem can be applied with n=2 and ñ = 4. (Indeed 2α + α^2 <1 ⇔α > √(2) -1 ≈ 0,41 and α + 3α^2 >1 ⇔α > (√(13) - 1)/6≈ 0,43).
So for the existence of the integral it is sufficient that the functional F has 3-causal space derivatives, but for the Itô formula we need 4-causal space derivatives. This additional regularity is comparable to the regularity change in the standard setting [Lemma 4.1, Proposition 5.8]RoughBook. But there are regimes of α where the the change in regularity exceeds one. For example for α∈(√(2)-1, (√(16)-1)/6], it holds n+1=3 and ñ=5. This gap increases for α→ 0.
It remains an open question if the loss of regularity from α to α^2 in Corollary <ref> can be circumvented.


 
We can easily change to  

    ^n_s,t := ^n_s,t - 1/n!μ((s,t]),

for a symmetric tensor-valued measure μ = ∑_wμ^w e_w (over words w of length n in the alphabet ), such that μ^w 
are finite signed measures with no atoms. Then it is immediate that

    F(T, X)    = F(0, X) + ∫_0^T DF(u, X)  u + ∫_0^T ∇ F(u, X) (u) - 1/n!∫_0^T ∇^nF(u, X), μ(u).

The measure μ could be for example a suitable notion of finite p-variation, cf. [Definition 4.1]contPathwiseIntegrationChange2019 or the stochastic quadratic variation if X is a sample path of a semimartingale. 




 §.§ Acknowledgement
 The author would like to thank Christa Cuchiero, Xin Guo and Francesca Primavera for making available the slides of Christa Cuchiero's talk on cuchieroFunctionalItoformulaTaylor at TU Berlin, 2023. Moreover, Nicolas Perkowski for his helpful comments. 
