Fractional Ito calculus

We derive It\^o-type change of variable formulas for smooth functionals of irregular paths with non-zero $p-$th variation along a sequence of partitions where $p \geq 1$ is arbitrary, in terms of fractional derivative operators, extending the results of the F\"ollmer-Ito calculus to the general case of paths with 'fractional' regularity. In the case where $p$ is not an integer, we show that the change of variable formula may sometimes contain a non-zero a 'fractional' It\^o remainder term and provide a representation for this remainder term. These results are then extended to paths with non-zero $\phi-$variation and multi-dimensional paths. Finally, we derive an isometry property for the pathwise F\"ollmer integral in terms of $\phi$ variation.

Hans Föllmer derived in [13] a pathwise formulation of the Ito formula and laid the grounds for the development of a pathwise approach to Ito calculus, which has been developed in different directions in [1,2,5,6,7,10,8,11,19,24]. Föllmer's original approach focuses on functions of paths with finite quadratic variation along a sequence of partitions.In a recent work [10], Cont and Perkowski extended the Föllmer-Ito formula of [13] to function(al)s of paths with variation of order p ∈ 2N along a sequence of partitions and obtained functional change of variable formulas, applicable to functionals of Fractional Brownian motion and other fractional processes with arbitrarily low regularity (i.e.any Hurst exponent H > 0).These results involve pathwise integrals defined as limits of compensated left Riemann sums, which are in turn related to rough integrals associated with a reduced order-p rough path [10].As the notion of p-the order variation may be defined for any p > 0, an interesting question is to investigate how the results in [10] extend to 'fractional' case p / ∈ N. In particular one may ask whether the change of variable formula contains a fractional remainder term in this case and whether the definition of the compensated integral needs to be adjusted.We investigate these questions using the tools of fractional calculus [23].Given that fractional derivative operators are (non-local) integral operators, one challenge is to obtain non-anticipative, 'local' formulas which have similar properties to those obtained in the integer case [10].We are able to do so using a 'local' notion of fractional derivative and exhibit conditions under which these change of variable formulas contain (or not) a 'fractional Itô remainder term'.It is shown that in most cases there is no remainder term; we also discuss some cases where a non-zero remainder term appears and give a representation for this term.These results are first derived for smooth functions then extended to functionals, using the concept of vertical derivative [9].We extend these results to the case of paths with finite φ-variation [17] for a class of functions φ and we obtain an isometry formula for the pathwise integral in terms of φ-variation, extending the results of [1,10] to the fractional case.Finally, we extend these results to the multi-dimensional case.Our change of variable formulas are purely analytical and pathwise in nature, but applicable to functionals of fractional Brownian motions and other fractional processes with arbitrary Hurst exponent, leading in this case to non-anticipative 'Itô' formulas for functionals of such processes.However, as probabilistic assumptions play no role in the derivation of our results, we have limited to a minimum the discussion of such examples.
Outline Section 1 recalls some results on pathwise calculus for functions of irregular paths (Sec.1.1) and fractional derivative operators and associated fractional Taylor expansions (Sec.1.2).Section 2 contains our main results on change of variable formulas for function(al)s of paths with fractional regularity.We first give a change of variable formula without remainder term for time-independent functions (Theorem 2.7), followed by a discussion of an example where a remainder term may appear (Example 2.9).We then provide a formula for computing this fractional remainder term using an auxiliary space(Theorem 2.12).Sections 2.2 extends these results to the time-dependent case and Section 2.3 extends them to the case of path-dependent functionals using the Dupire derivative.In Section 3 we show that these results may be extended to the case where the p−th variation is replaced by the more general concept of φ−variation [17].In Section 4 we derive a pathwise isometry formula extending a result of [1] to the case of φ−variation.Finally, in Section 5 we discuss extensions to the multidimensional case.These extensions are not immediate, as the space V p (π) is not a vector space.
Definition 1.1 (p-th variation along a sequence of partitions).Let p > 0. A continuous path S ∈ C([0, T ], R) is said to have a p-th variation along a sequence of partitions π = (π n ) n≥1 if osc(S, π n ) → 0 and the sequence of measures converges weakly to a measure µ without atoms.In that case we write S ∈ V p (π) and [S] p (t) := µ([0, t]) for t ∈ [0, T ], and we call [S] p the p-th variation of S.
1. Functions in V p (π) do not necessarily have finite p-variation in the usual sense.Recall that the p-variation of a function f ∈ C([0, T ], R) is defined as [12] f p-var := sup where the supremum is taken over the set Π([0, T ]) of all partitions π of [0, T ].A typical example is the Brownian motion B, which has quadratic variation [B] 2 (t) = t along any refining sequence of partitions almost surely while at the same time having infinite 2-variation almost surely [12,25]: 2. If S ∈ V p (π) and q > p, then S ∈ V q (π n ) with [S] q ≡ 0.
Cont and Perkowski [10] obtained the following change of variable formula for p ∈ N, S ∈ V p (π) and f ∈ C p (R, R): holds, where the integral is defined as a (pointwise) limit of compensated Riemann sums.
Remark 1.5 (Relation with Young integration and rough integration).Note however that, given the assumptions on S, the pathwise integral appearing in the formula cannot be defined as a Young integral.This relates to the observation in Remark 1.2 that p-variation can be infinite for S ∈ V p (π).
When p = 2 it reduces to an ordinary (left) Riemann sum.For p > 2 such compensated Riemann sums appear in the construction of 'rough path integrals' [14,16].Let X ∈ C α ([0, T ], R) be α-Hölder continuous for some α ∈ (0, 1), and write q = ⌊α −1 ⌋.We can enhance X uniquely into a (weakly) geometric rough path (X 1 s,t , X 2 s,t , . . ., X q s,t ) 0≤s≤t≤T , where and therefore the controlled rough path integral where |π| denotes the mesh size of the partition π, and which is the type of compensated Riemann sum used to define the integral above, plus an additional term (Ito term) absorbed into the sum).

Fractional derivatives and fractional Taylor expansions
We recall some definitions on fractional derivatives and their properties.Several different notions of fractional derivatives exist in the literature and it is not clear which ones are the right tools for a given context.Our goal here is to shed some light on the advantages of different notions of fractional derivative.Much of this material may be found in the literature [23].We have provided some detailed proofs for some useful properties whose proof we have not been able to find in the literature.
Definition 1.6 (Riemann-Liouville fractional integration operator).Suppose f is a real function and α > 0, the left Riemann-Liouville fractional integration operator of order α is defined by if integration exists for x > a ∈ R. Similarly, the right Riemann-Liouville fractional integration operator if given by This may be used to define a (non-local) fractional derivative of a real function f associated with some base point, as follows: Definition 1.7 (Riemann-Liouville fractional derivative).Suppose f is a real function and n ≤ α < n + 1 for some integer n ∈ N. Then left Riemann-Liouville fractional derivative of order α with base point a at x > a is defined by if it exists.Similarly, the right Riemann-Liouville fractional derivative of order α with base point b and b > x is given by Some basic properties of the Riemann-Liouville fractional derivative are described below.
Proposition 1.8.Suppose f is a real function, α, β are two real numbers.We will also use the convention that I α a+ = D −α a+ .Then we have It is continuous in uniform topology for α > 0 and strongly continuous for α ≥ 0.
) is valid in the following cases: , then we have and similarly, Here D κ a + f (a) = lim y→a,y>a D κ a + f (y) for any κ ∈ R and similar for the right Riemann-Liouville fractional derivative.
The Riemann-Liouville derivative has several shortcomings.One of them is that the fractional derivative of a constant is not zero.To overcome this, we introduce a modification of Riemann-Liouville fractional derivative which is called the Caputo derivative.Definition 1.9 (Caputo derivative).Suppose f is a real function and n + 1 ≥ α > n.We define the left and right Caputo fractional derivatives of order α at x ∈ (a, b) by here f (k) denotes the k-th derivative of f .
We enumerate below some useful properties of the Caputo derivative: Proposition 1.10.Suppose f is a real function and α, β > 0.
for all n ∈ AE.
Example 1.11.We give some simple examples of Caputo fractional derivative here.
1. Consider f (x) = |x| α and 0 < β ≤ α < 1.Then we have So we can see directly that f (β+) (a) = 0 for any a = 0 or β < α but f (α+) (0) = Γ(α + 1).It can be seen further that for all a ≥ 0, C α a + f is continuous on [a, ∞] but for a < 0, C α a + f has singularity at point 0. In particular, for β = α, we have As we were not able to find a proof of this expansion in the literature, we provide a detailed proof below.Let us start with a simple but useful lemma, which corresponds to Theorem 3.3 in [18] for the case ρ = 1: ), then for any 0 < α < n and α / ∈ N, we have actually From proposition 1.8, we have Now we have by linearity of Riemann-Liouville fractional derivative operator and definition of Caputo fractional derivative that by definition.Hence we have Now since f ∈ C n and (• − a) k ∈ C ∞ , we see that actually D α−j a + g(a) = 0 for every j = 1, • • • , n by lemma 1.13.For the case j = n + 1, we actually have that , hence the result.The derivation of the other formula is similar.
The above derivative operators are non-local operators.We now introduce the concept of local fractional derivative: Definition 1.14 (Local fractional derivative).Suppose f is left(resp.right) fractional differentiable of order α on [a, a+ δ]([a− δ, a]) for some positive δ, then the left(resp.right) local fractional derivative of order α of function f at point a is given by Remark 1.15.Using the integration by parts formula, we can see that we actually have Hence we will have furthermore that . This means the existence of one side will imply the existence of the another side.And if we take the limit we can see that the existences of f (α+) (x) and (f (n) ) (α+) (x) are equivalent and they are equal.This is very important in the following proofs.
Proof.We only prove for the left Caputo derivative case.
Thus we proved the result.
A similar mean value theorem holds for the non-local fractional derivative.The following result, which we state for completeness, is a direct consequence of proposition 1.12: A similar formula holds for the Caputo right derivative.A rather complex proof of this proposition is given in [4, Corollary 3].We here give a simple proof of this property using only properties of monotone functions.

A characterization of the local fractional derivative
Proof.We first suppose 0 < p < 1.And by Taylor expansion theorem for fractional derivative, we actually have Hence for any sequence y n → x, y n > x, we have Now let Here L denotes Lebesgue measure on the real line.Let's consider the open set ∪ x∈K (x, x + δ x ) in R, where δ x is the largest number so that f (y) > f (x), ∀y ∈ (x, x + δ x ).We can then write ∪ x∈K (x, x + δ x ) = ∪ ∞ k=1 I k for some open interval I k .By the additive property of measure, ∃I r such that L(K ∩ Īr ) > 0. For any x < ȳ ∈ K ∩ Īr , there exists x 0 ∈ K ∩ [x, ȳ] such that , which is a contradiction.Hence x 0 = ȳ and we have f (ȳ) > f (x), ∀x < ȳ ∈ K ∩ Īr .Now define f : Īr → R such that f = f on K ∩ Īr and f is linear outside of K ∩ Īr .In fact, let x ∈ Īr , if x / ∈ K, then x ∈ (z 0 , z 1 ) with z 0 , z 1 ∈ K.By above reason, f (z 1 ) ≥ f (z 0 ), and we perform linear interpolation to define the value f (x) here.It is then easy to conclude that f is an increasing function.Hence f is differentiable almost everywhere on Īr .Now we go back to the function f on set K. It could be concluded that almost surely on K, f is differentiable and we can simply suppose here f is differentiable and f = f on K whose Lebesgue measure is positive.Now choose a point x 1 ∈ K, either x 1 is isolated in K or it is an accumulated point in K.For the latter case, we would have Here {y n } is a sequence in K approximating x 1 .Since the set of all isolated points in K is of zero measure, we could get g = 0 almost surely on K, which is a contradiction.Hence , we could conclude again that f (p+) (x) = 0 almost surely on R.
Remark 1.19.Here actually we only need a weaker condition on the function, namely that exists for x ∈ R with m = ⌊p⌋ and p = m + α, which could be thought as classical definition of α order derivative.We call left 'classical' fractional derivative of order α if it exists and similar for right derivatives.However, when this limits exists, it is not guaranteed that local fractional derivative of f exists.
We can actually give a stronger result.Let E = {x ∈ R : lim y→x,y>x Proposition 1.20.The Hausdorff dimension of the set E is at most α.
Proof.For each ǫ, δ, we define the set E δ ǫ+ to be the subset of E such that for all x ∈ E δ ǫ+ , we have Then by the countably additive of measure, we only need to show the Hausdorff dimension of the set E δ ǫ+ is zero.Furthermore, we can restrict the set E δ ǫ+ on the interval [0, δ] due to the countably additive of measure.Hence we will work on [0, δ] with E δ ǫ+ .First of all, suppose the Hausdorff dimension of set E δ ǫ+ is larger than α.In this case, we considers partitions we choose the leftmost and rightmost points of the set E δ ǫ+ and denote this interval as ǫ+ and by the definition of Hausdorff measure, we know that The following result is a consequence of the fractional Taylor expansion ( 2)-( 3): ), then for any β < α and β / ∈ N, we have f (β+) (x) = 0 if f (β+) (x) exists.And for β > α with same integer part, f (β+) (x) will either be ±∞ or not exist.

Fractional Itô Calculus
In this section we derive Itô-type change of variable formulas for smooth functions of paths with p−th variation along a sequence of partitions {π n } n∈N , first for functions (Sections 2.1 and 2.2) then for path-dependent functionals (Section 2.3).In each case the focus is the existence or not of a 'fractional' Itô remainder term: we will see that the existence of a non-zero Itô term depends on the fine structure of the function and its fractional derivative.

A change of variable formula for time-independent functions
We now derive an Itô-type change of variable formula for smooth functions of paths with p−th variation along a sequence of partitions {π n } n∈N .Let S ∈ V p (π) be a path which admits p−th order variation along some sequence of partitions {π n } n∈N .We make the following assumptions: and admits a left local fractional derivative of order p everywhere.
Assumption 2.3.The set is locally finite, i.e. for any compact set K ∈ R, the set Γ c f ∩ K has only finite number of points.Assumption 2.1 will be automatically satisfied if the occupation measure of S is atom-less.In fact, We first give a simple lemma regarding the set Γ f : Lemma 2.4.∀x ∈ Γ f , the left local fractional derivative of order p of f exists and equals to zero, i.e.
Assumption 2.1 will be satisfied in particular if S admits an occupation density [15], then .However, as the following example shows, Assumption 2.1 may fail to be satisfied even if the path has a non-zero p−th order variation.

Remark 2.5 (A counterexample)
. We now give an example of path failing to satisfy Assumption 2.1, in the spirit of [11,Example 3.6.]. 1et p > 2 and define the intervals , which is the Cantor ternary set [3].Let c : [0, 1] → R + be the associated Cantor function, which is defined by We can see it is a non-decreasing function increasing only on Cantor set C. Consider the function We are going to construct a sequence of partitions such that the p−th variation of S along this sequence will be the Cantor function c.In this case, we will see which shows Assumption 2.1 is not satisfied.We begin with the partition {π i j,n }, which denotes the n−th partitions in the interval I i j .Define t i,0 j,n = inf I i j and k n is an integer to be determined.We then do the calculation Then the sum for the n−th partition will be n i=1 we see that [S] p π will not change on the interval I i j and hence due to symmetry, we finally can show that [S] p (t) = c(t), the Cantor function.
Remark 2.6.Assumption 2.3 is necessary for Theorem 2.7.There are continuous function with f (p+) = 0 everywhere but no joint continuity for the Caputo derivative.We note here that the joint continuity can lead to the uniform convergence of the equation The following theorem extends the result of [10] to the case of functions with fractional regularity: Theorem 2.7.Let S ∈ V p (π) satisfying assumptions 2.1.If f is a continuous function satisfying assumptions 2.2 and 2.3 for m = ⌊p⌋, then where the last term is a limit of compensated Riemann sums of order m = ⌊p⌋: Proof of Theorem 2.7.We first suppose that for any t i ∈ π n other than 0 and T , f (p+) (S ti ) = 0.By Taylor's formula with integral remainder, we have It is easy to notice that there exists a constant M > 0 and ) is away from Γ c f with at least ǫ 2 distance, which means we have the estimation for some modulus of continuity Here we also take ǫ small enough such that [k − 2ǫ, k + 2ǫ] are disjoint and ||g ǫ || ∞ ≤ 1.Then by definition of the p−th order variation, we see By assumption 2.1 we then see that Next for the set C n 1,ǫ and α = p − m, we have Then by inequality ( 5) and the fact that and the lemma 2.4), we have Thus we obtain Now for the set C n 2,ǫ , we have Again by the inequality (5), we will have Hence, we can obtain the estimation: Thus we could obtain that And finally for the case t i = 0 or t i+1 = T , the continuity of f (m) and S will show that Thus, by summing all these together and let ǫ tends to 0, we have Hence we see lim n→∞ L n exists and we denote it as t 0 (f ′ •S).dS.And this leads to the Ito formula Remark 2.8.We can not expect the Itô formula in the fractional case to have the same form as in the integer case, i.e. there might be no Itô term in the fractional case even if Assumption 2.1 does not hold.For the example in Remark 2.5, we can show that for 2 < p < 3 and f (x) = |x| p , the Itô term still vanishes even though we have , since {S(t) = 0} = C is the support of the function c.In this case, we will show that In fact, we will need to calculate which is in fact the 'rough integral' of f ′ along the reduced order-p Ito rough path associated with S [10].Splitting the terms across each I i j we have Now we consider the term involving the second derivative.On I i j , we have Summing the second order terms over π n we obtain: Adding together the first order term − np 2kn , we need to calculate Since it is obvious to see that Use the inequality We then obtain first that Notice that lim Thus in this case we see the remainder term is zero, which is different from the integer case.Hence we see that the pathwise change of variable formula in Theorem 2.7 may hold even if Assumption 2.1 does not hold.
We now give an example of path with the same p−th variation as above but leading to a non-zero remainder term in the change of variable formula.
Example 2.9.Define the intervals I i j as in (4).Then we define S| I i j by induction on i.Let g(t) = 2min{t, 1 − t} on [0, 1] and 0 otherwise.For a < b, we define g(t, [a, b]) = g t−a b−a .First for i = 1, we divide the interval I i j into r i smaller intervals ri and two of each are non-intersecting.On each interval I i j,k , we define In other words, S is defined as the limit Let π n = (τ n l ) be the dyadic Lebesgue partition associated with S: We can choose Then similar to what was discussed in the above remark, we have [S] p (t) = c(t), the Cantor function.Let f (x) = |x| p , 2 < p < 3 2 and T = 1, we are going to calculate the Itô remainder term for f (S(1)) − f (S(0)).We calculate the limit Here We have Hence n i=1 Now we let n → ∞, we have ) − f (S(0)) = 0, which means there will be a non-zero remainder.And in this case, we also have so Assumption 2.1 is not satisfied.
In fact, we can provide a formula for the Itô remainder term for this path and function f = |x| p .Take T = 1, m = ⌊p⌋ and α = p − m.Let for a = b and take the limit value when exists for a = b.For the function f (x) = |x| p , we can see G p f is defined on R 2 /{(0, 0)} and for k > 0, which indicate us to consider G p f as a function on the unit circle S 1 .We define the projection map P : R 2 /{(0, 0)} → S 1 by p(x) = x ||x|| and define a sequences of measures by where N n is the number of intervals in the partition π n .Furthermore, we define Ĝp for θ ∈ [0, 2π).Since the distance between two successive points in partition π n is 2 −n , the remainder term can be rewritten as and we have Hence the remainder term is By definition, we have lim n→∞ 2 n−np n i=1 r i = 1.And if we have the weak convergence of νn , then we can obtain the limit expression.Now let's calculate the limit of the measures νn .Since S(t i ) = k2 −n for some positive number k and t i ∈ π n , the result tan (P (S(t i ), S(t i+1 ))) will equal to either k k+1 or k k−1 , which means the final measure ν will be supported on these points.And By symmetry, we have Hence we have the Itô formula for the path S defined in the example 2.9 and f (x) = |x| p : The above calculations show that the remainder term may be non-standard when assumption 2.1 is not satisfied.Furthermore, through the calculation of exact expression of Itô's remainder term for the path in example 2.9, we can give a general method to calculate Itô's remainder term for paths satisfying some assumptions.For a function f and a positive number p = m + α, where m = ⌊p⌋, we define the map Denote by N t n the number of intervals in π n on [0, t].In order to compute the remainder term in the fractional Ito formula, we 'stratify' the increments across the partition by the values of G p f , to build an auxiliary quotient space X which is required to have certain properties: Assumption 2.10 (Auxiliary quotient space).There exists a space X and a map converges weakly to a measure ν on [0, T ] × X.
Remark 2.11.The measure ν in the example 2.9 is actually the measure ν([0, 1], dx) defined in assumption 2.10 on the space X.In fact, due to the fact that partitions in the example 2.9 are of Lebesgue type, we have Hence assumption 2.10 is a reasonable condition in order to calculate Ito's remainder term such as example 2.9.Furthermore, since the first marginal measure of ν n is we see S ∈ V p (π n ) is equivalent to the weak convergence of ν n (dt, X) on the interval [0, T ].In this sense, we see assumption 2.10 actually requires other properties in addition to the existence of p−th variation along the sequence {π n }.
The choice of X is crucial in making the calculation of the remainder term tractable.One can always choose X = R but this choice may not lead to easy calculations (see below).
Theorem 2.12.Let {π n } be a sequence of partitions and S ∈ V p (π) such that there exists (X, P p f ) satisfying Assumption 2.10.We define the map Ĝp where Proof.Using a Taylor expansion, we have actually by the definition of measures ν n , we have Using the weak convergence of ν n in the assumption 2.10, so taking limits yields the desired result.
Remark 2.13.We can perturb the function f without changing the choice of the space X, by working in a neighborhood of the singularity of G p f .Take S 1 as an example.Suppose f ∼ |x| p near zero, which means that G p f has a unique singularity at (0, 0).If furthermore, on each ray emanating from the origin, the function G p f converges as point tends to origin 'uniformly'.Here uniform means that for each ǫ > 0, there exists δ > 0 such that on the band {(a, b) : d((a, b), y = x) < δ}, we have ) and (a, b), (c, d) are on the band.Above assumption describe the behaviour of G p f around the singular point (0, 0) and can be used to calculate the Itô remainder term.For such function f and path S, we can derive the formula Here g(x) is defined as the radial limit at the origin of G p f .Furthermore if f admits a bounded p − th order Caputo derivative, then the uniform convergence can be checked by uniform convergence of the Caputo derivative C p a f (b).
We can see from the theorem that when Itô's remainder term is non-zero, the formula is not in the classical form.Hence we will need more information to describe the Itô remainder term in the fractional case.
Let us go back to zero remainder term case.Consider the set (E p ) ′ of all functions satisfying assumptions 2.2 and 2.3.Then (E p ) ′ is a subset of C p (R).Let E p be the closure of (E p ) ′ in C p (R) equipped with the semi-norms here p = m + α with m < p < m + 1.Then we have the following theorem Theorem 2.14.Let S ∈ V p (π) satisfying Assumption 2.1 and f ∈ E p .Then for any t ∈ [0, T ] where the integral is a limit of compensated Riemann sums of order m = ⌊p⌋: Let K be a compact set containing the path of S([0, T ]).Since f k → f in C p , we have for every ǫ > 0, there exists N > 0 and for all k > N , sup Hence we have lim sup Finally let ǫ tends to 0, we have Hence L n converges when n tends to infinity and we have Remark 2.15.It can be seen in the proof of the theorem 2.7 that we only need for k ∈ Γ c f .And we can denote P f as the set of paths which satisfies equation ( 9) on Γ c f .On the other hand, if we fix the path S, then by Fubini's theorem, we actually have that which means equation ( 9) is satisfied by almost surely all k ∈ R. Hence we can consider the set (E p S ) ′ with all functions f such that Γ f contains all points that break the equation ( 9).And then we can obtain the closure E p S for fixed path S. We can also ignore p in the subscript because p is included in the information of S. So we can just write E S .And it is obvious that t dk, we can see that the set of k which violates equation ( 9) is actually countable.Hence we could define set (E p x ) ′ to be all functions f such that Γ f contains point x and E p x to be the completion under Hölder norm.If we furthermore define R S = k ∈ R :  3. The linear combinations of |x − k| p .For example Remark 2.17.Smooth (C ∞ ) functions belong to E p .Denote by C p+ (R) ⊂ E p the completion of the smooth function under Hölder norm i.e. the set of functions f ∈ C p such that on every compact set K: Then we can easily see that for any q > p, we will have Here q is a non-integer.
Example 2.18 (Examples for P f with f (x) = |x| p ).
1. Fractional Brownian Motion: Let p = 1/H.We have which means that almost surely B H ∈ P p f .
Remark 2.19.We give here another remark on Assumption 2.3.In fact, as we see in the remark 2.6, Assumption 2.3 means uniform convergence of the equation on every compact set K. Hence, according to the proof of the theorem 2.7, we can change 3 assumptions to the following one Assumption 2.20.There exists a sequence of open set U i such that Ūi+1 ⊂ U i and If we replace the 3 assumptions in the theorem 2.7 by the single Assumption 2.20, then Theorem 2.7 still holds.And in this case we do not need to calculate the Caputo derivative of the function f .The only thing that we need to check is the uniform Hölder continuity.This means we have Theorem 2.21.Suppose S ∈ V p (π) and f satisfies assumption 2.20 for m = ⌊p⌋.Then where the last term is a Föllmer integral, defined a limit of compensated Riemann sums of order m = ⌊p⌋: This is a special case of Theorem 2.30 on the path-dependent Itô formula, whose proof will be given in Sec.2.3.

Change of variable formula for time-dependent functions
We now extend the change of variable formula to time dependent functions such as Here U s means the intersection of U and t = s and P t (U ) is the projection of U to t−axis.For the change of variable formula for time-dependent functions, we need the following assumption: Under Assumption 2.22, we have where m = ⌊p⌋ and Proof.For simplicity, we only prove the case t = T here.As before we write And for the first sum we have Then above sum can be written as Now let's deal with the second sum By assumption 2.22.Hence and on the set C n ǫ,2 , we have ) is continuous in a neighborhood of S(t i ), hence they are identically zero.Then by uniform continuous property, we can have a modulus of continuity ω ǫ such that f .Hence we have for j = 1, 2 for some constant M .And let n → ∞, we see that Hence we finally obtain that and hence the result.
We can do as the time-independent case to extend the space of functions for fixed path S using approximation in the C 1,p space and denote the result space as E S .
Remark 2.24.As in the time-independent case, we can use a uniform convergence assumption to replace Assumption 2.22: Assumption 2.25.There exists a sequence of open set Under this assumption, Theorem 2.23 still hold.
where m = ⌊p⌋ and

Extension to path-dependent functionals
In this subsection S t will designate the path stopped at t, i.e. S t (s) = S(s ∧ t).
We consider the space D([0, T ], R) of càdlàg paths from [0, T ] to R. Let be the space of stopped paths.This is a complete metric space equipped with We will also need to stop paths 'right before' a given time, and set for t > 0 ω t− (s) := ω(s), s < t lim r↑t , s ≥ t while ω 0− = ω 0 .We recall here some concepts from the non-anticipative functional calculus [9,7].
Definition 2.27.A non-anticipative functional is a map F : Λ T → R. Let F be a non-anticipative functional.
• We write F ∈ C 0,0 l if for all t ∈ [0, T ] the map F (t, •) : D([0, T ], R) → R is continuous and if for all (t, ω) ∈ Λ T and all ǫ > 0, there exists δ > 0 such that for all  • F is vertically differentiable at (t, ω) ∈ Λ T if its vertical derivative exists.If it exists for all (t, ω) ∈ Λ T , then ∇ ω F is a non-anticipative functional.In particular, we define recursively ∇ k+1 ω F := ∇∇ k ω F whenever this is well defined.
• For p ∈ N 0 , we say that F ∈ C 1,p b (Λ T ) if F is horizontally differentiable and p times vertically differentiable in every (t, ω) ∈ Λ T , and if In order to extend our result to the functional case, we need to give the definition of fractional Caputo derivative of a non-anticipative functional.But here we choose another approach using the uniform convergence of Hölder semi-norm as stated in Remark 2.6.Definition 2.28 (Vertical Hölder continuity).Let 0 < α < 1.
• F is vertically α−Hölder continuous at (t, ω) ∈ Λ T if and we call L(t, ω) the Hölder coefficient of and we call L the Hölder coefficient of F over E and we denote F ∈ C 0,α (E).
• For m < p < m + 1, we say We now give assumptions on the functional F ∈ C 1,m b ∩ C 0,p loc , where m = ⌊p⌋.Given a path S which admits p−th order variation along a sequence of partitions π.Assumption 2.29.There exists a sequence of continuous functions 2. There exists a bounded function g on Λ T such that g k → g pointwise and ∩ C 0,p loc satisfies Assumption 2.29 then the limit exists and we have Proof.We write We only need to consider j with t j+1 ≤ t since the remainder is another o(1) as n → ∞ by left continuity of functional F .Now we split the difference into two parts: Now since S n tj = S n tj+1− on [0, t j+1 ], we have From the definition of C 0,p , we see that there exists M > 0 such that and by the definition of vertically Hölder continuity, we have lim ǫ→0 sup ω∈E/E k L(ω, ǫ) = 0 for each k with E a bounded subset of Λ T containing all stopped paths occurred in this proof.
Then one may choose smooth functions y k with support g k and g then satisfy 2) in Assumption 2.29.

Extension to φ-variation
The concept of φ−variation φ−variation along a sequence of partitions [17] extends the notion of p-th order variation to more general functions: The function φ we considered in this section satisfies some assumption given here.It is obvious that in this case φ(0) = 0. Furthermore, we let φ map R + to R + and φ is either odd or even function.
We say a function f is φ−continuous if and only if for each compact set K, there exists a constant We denote the collection of these functions by C φ .
Remark 3.3.As in the fractional case, if we consider the Taylor expansion of f up to 'φ', we should write it as it is obvious that if g is continuous, then we would have the remainder term in the change of variable formula.However, for the function φ satisfying assumption 3.2, the function g in the Taylor expansion above would be almost surely zero, the same as stated in proposition 1.18 for |x| p .What we need is just to change (y − x) p by φ(y − x) in the proof of proposition 1.18 and notice that if f is differential at some point x, then for the case m = 0, Follow the steps in the proof of Proposition 1.18, we can show that g = 0 almost everywhere.Hence a regular remainder should not appear in this case.
Proof.We have the same with γ F (s, t) := ∇ ω F (s.S s )(S(t) − S(s)), we have from lemma 4.2 that , we know Then by the definition of p(φ), we see that where ω is continuous on R + and ω(0) = 0. Combine all above, we get and we also have In particular, if we set φ(x) = |x| p for p > 0, we have Remark 4.5.There exists a non empty set of p and α such that assumption is valid.In fact, what we need to verify is that should not be identical zero for all α > . Due to the definition of p(φ), φ(S(t i+1 )−S(t i )) should behave like C(π n )|S(t i+1 )−S(t i )| p(φ) when the oscillation of π n is small enough.Here C(π n ) are uniformly bounded.Hence it remains to check can not be zero for all S ∈ C α .By the definition of Hölder property, it remains to verify p(φ) which is equivalent to  5 Multi-dimensional extensions The extension of the above results to the multidimensional case is not entirely straightforward, as the space V p (π) is not a vector space [24].
In the case of integer p, some definitions may be extended to the vector case by considering symmetric tensor-valued measures as in [10] but this is not convenient for the fractional case.We use a definition which is equivalent to the definition in the paper [10] when p is an integer.Remark 5.2.Although we can define the situation when S admits a p-th variation, we can not give an object directly related to the p-th variation of the path S in this definition.However, it can be seen that multi-dimension fractional Brownian motion will definitely admit p−th variation along sequences of partitions with fine enough mes.
We now state the assumptions on the function f that will appear in the Itô formula.By taking sum over the partition π n and taking the limit of n we then obtain desired result.
Proof of theorem 5.5.The technique for the proof is the same as the previous theorems.We only consider the situation t = T , the case t < T is similar with an o(1) additional term.

αProposition 1 . 12 .
Associated with the Caputo derivative is a fractional Taylor expansion: Let f ∈ C n ([a, b]) and n + 1 ≥ α > n.If the Caputo fractional derivative of order α of f exists on [a, b] and C α a + f ∈ L 1 ([a, b]), then we have the Taylor expansion:

Corollary 1 . 16 .
Let f ∈ C n ([a, b]) and n + 1 ≥ α > n.Suppose left(right) local fractional derivative of order α of f exists at a.This leads to the following fractional Taylor formula with remainder term for x ∈ [a, b]

T 0 1 1 .
{S(t)=k} d[S] p (t) = 0 , then we have actually E S = ∩ x∈RS E p x .Now let's give some examples of functions belonging to E p .Example 2.16 (Examples of functions belonging to E p .).All functions f ∈ C m+1 (R).

2 .
The function f (x) = |x − k| p for some k ∈ R. It can be seen in the example 1.11 that C p a + f (x) is continuous on {(a, x) ∈ U × U : a ≤ x} for any compact set U contained in positive real line or negative real line, which means Γ f = R/{0} and hence is a function in E ′ p ⊂ E p .

Assumption 3 . 2 .
There exists m ∈ N such that φ ∈ C m and

2
which is obviously true.

Definition 5 . 1 .
Let p ≥ 1 and S = (S 1 , • • • , S d ) ∈ C([0, T ], R d ) be a continuous path.Let {π n } be a sequence of partition of [0, T ].We say that S has a p−the order variation along π = {π n } if osc(S, π n ) → 0 and for any linear combination S a := d i=1 a i S i , we have S a ∈ V p (π).Furthermore, we denote µ Sa to be the weak limit of measuresµ n Sa := [tj ,t j+1] ∈πn δ(• − t j )|S a (t j+1 ) − S a (t j )| p
m for each t j when n is large enough.This shows that Remark 2.31.One may derive theorems 2.23 and 2.26 from above theorem.Let Γ f defined as Assumption 2.20, E k n M |S(t j+1 ) − S(t j )| p ≤ lim n→∞ [tj ,tj+1]∈πn M g k ((t j , S n tj − )|S(t j+1 ) − S(t j )| p = t o M g k ((t, S t ))d[S] tby assumption 2.29.Let k tends to infinity, we get the result.