It\^{o}--F\"{o}llmer Calculus in Banach Spaces I: The It\^{o} Formula

We prove F\"{o}llmer's pathwise It\^{o} formula for a Banach space-valued c\`{a}dl\`{a}g path. We also relax the assumption on the sequence of partitions along which we treat the quadratic variation of a path.


Introduction
In his seminal paper [22], Föllmer presented a new perspective on Itô's stochastic calculus. The main theorem of Föllmer [22] states that a deterministic càdlàg path satisfies the Itô formula provided it has quadratic variation along a given sequence of partitions. This theorem enables us to construct the Itô integral ∫ 0 ( − )d for a sufficiently nice function and a path with a quadratic variation. This suggests the possibility of developing an analogue of the Itô calculus in completely analytic, probability-free situations.
We call this framework Föllmer's pathwise Itô calculus or, more simply, the Itô-Föllmer calculus. It can be regarded as a deterministic counterpart of the classical Itô calculus.
Recently, the Itô-Föllmer calculus has been receiving increasing attention from the viewpoint of its financial applications. It is regarded as a useful tool to study financial theory under probability-free settings and has been used to construct financial strategies in a strictly pathwise manner (see, e.g., Föllmer and Schied [24], Schied [68], Davis, Obłój and Raval [13], and Schied, Speiser, and Voloshchenko [70]). We expect that the Itô-Föllmer calculus will have a growing presence in financial applications.
The Itô-Föllmer calculus can be applied to a stochastic process having quadratic variation. A standard example of such a process is a semimartingale. However, it is known that the class of processes possessing quadratic variation is strictly larger than that of semimartingales (see, e.g., Föllmer [22,23]). In this sense, the Itô-Föllmer calculus enables us to extend stochastic integration theory beyond semimartingales.
Among the various pathwise methods, we consider Föllmer's approach to be the simplest and the most intuitively clear. It needs only elementary arguments to establish calculation rules such as Itô's formula within this framework. Moreover, the Itô-Föllmer calculus requires only a minimal assumption that the integrator has quadratic variation. We believe that these are advantages of Föllmer's theory, and also that careful observation of this theory helps us to understand the path properties of processes better when we consider semimartingales and their stochastic integration.
To our knowledge, however, the Itô-Föllmer calculus in an infinite dimensional setting has not yet been sufficiently studied. Stochastic integration in infinite dimensions naturally appears when we treat stochastic partial differential equations (see, e.g. Da Prato and Zabczyk [12]). These have played an important role in modelling term structures of interest rates or forward variances in mathematical finance, and also in models of statistical mechanics and quantum field theories. Then we aim to extend Föllmer's theory to Banach space-valued paths. In this paper, we prove the Itô formula for a path in a Banach space with a suitably defined quadratic variation. We will study relations between various quadratic variations and prove some transformation formulae for quadratic variations in our second paper in this series [37]. We not only generalize the state space of paths but also relax the assumption on the sequence of partitions along which we consider the quadratic variation. In the context of the Itô-Föllmer calculus, two types of assumptions about a sequence of partitions are frequently used. One is | | → 0, as used in Föllmer [22], and the other is the condition − ( ; ) → 0, which is used in many papers handling continuous paths and some dealing with discontinuous paths such as Vovk [76]. In this paper, we introduce new conditions to a sequence of partitions and a càdlàg path (Definition 3.4), which gives a unified approach.
Our method can be interpreted as a deterministic counterpart of these stochastic integration theories in Banach spaces. Some of the works listed above, such as Metivier and Pellaumail [47] and Dinculeanu [20], give proofs of Itô's formula in a similar manner to Föllmer's calculus. One of the advantages of our approach appears in the statement of the Itô formula. For a function to satisfy the Itô formula, we require to be just 2 class, while a stochastic approach needs some additional assumptions about the boundedness of and its derivatives.
Before explaining our contribution, we begin by summarizing the main result of Föllmer [22]. Let Π = ( ) ∈ℕ be a sequence of partitions of ℝ ≥0 such that | | sup ] , ] ∈ | − | tends to 0 as → ∞. We say that a càdlàg path : ℝ ≥0 → ℝ has quadratic variation along Π if there exists a càdlàg increasing function [ , ] such that for all ∈ ℝ ≥0 An ℝ -valued càdlàg path = ( 1 , . . . , ) has quadratic variation along Π if the real-valued path + has quadratic variation along the same sequence for each and . Föllmer [22] proved that if has quadratic variation, then for any ∈ 2 (ℝ ) the path ↦ → ( ) satisfies Itô's formula. That is, holds for all ∈ ℝ ≥0 . The first term on the right-hand side of (1.1) is defined as the limit where , denotes the usual Euclidean inner product. We call this limit the Itô-Föllmer integral along Π. Föllmer's theorem claims that if has quadratic variation along Π, then the Itô-Föllmer integral above exists and it satisfies equation (1.1).
As stated above, we aim to extend Föllmer's pathwise Itô formula to Banach space-valued paths. Let us describe a simplified version of our main result. The precise statement will be given as Theorem 3.6 and Corollary 3.7. Let be a Banach space and ⊗ be the tensor product Banach space with respect to a reasonable crossnorm . We say that an -valued càdlàg path has strong/weak -tensor quadratic variation along Π = ( ) if there is a càdlàg path Let : ℝ ≥0 → be a càdlàg path that has strong/weak -tensor quadratic variation and finite 2-variation along ( ) ∈ℕ , and let : ℝ ≥0 → be a càdlàg path of finite variation in a Banach space. Suppose that ( ) satisfies Condition (C) for ( , ) and the left-side discretization of ( , ) along ( ) approximates ( − , − ) pointwise (see Definition 3.1 for the exact definition). If : × → is a 1,2 function such that the second derivative induces a continuous map 2 : × → L( ⊗ , ), then the composite function ( , ) satisfies The second integral on the right-hand side is defined respectively as the strong/weak limit of left-side Riemannian sums along ( ).
To conclude this section, we give an outline of the remainder of this paper. Section 2 is a preliminary part of this article. We introduce basic notation and terminology in the first subsection. The next subsection is devoted to a review of càdlàg paths and Stieltjes integrals in Banach spaces. In Section 3, we set up basic notions in Itô-Föllmer calculus in Banach spaces and state the main results of the paper (Theorem 3.6 and Corollary 3.7). In Section 4, we study conditions on the sequence of partitions and the relation between them and càdlàg paths. Fundamental properties of quadratic variations are studied in Section 5. The purpose of Section 6 is to show Lemma 3.10, which is essentially used in the proof of the main theorem. In the last section of the main part, Section 7, we prove the Itô formula for a Banach space-valued path having quadratic variation. In appendices, we present some auxiliary results related to differential calculus and integration in Banach spaces.

Notations and terminologies
In this section, we introduce basic notation and terminology used throughout this article.
If and are two real Banach spaces, L( , ) denotes the space of bounded linear maps from to . In addition, given another Banach space , we define L (2) ( , ; ) as the space of bounded bilinear maps from × to . Recall that L( , ) and L (2) ( , ; ) are Banach spaces with norms respectively. Now we introduce another topology on the space L( , ). Let K be the family of all compact subsets of . For each ∈ K , define a seminorm by the formula for each ∈ L( , ). Then the family ( ) ∈K induces a locally convex Hausdorff topology on L( , ).
We use the symbol L c ( , ) for this topological vector space.
and let be a Hausdorff topological vector space. A càdlàg path in is a function : ℝ ≥0 → that is right continuous at every ≥ 0 and has a left limit at every > 0. The terms RCLL and right-regular are also used to stand for the same property. Similarly, a càglàd (also called LCRL or left-regular) path in is a function : ℝ ≥0 → that is left continuous on ]0, ∞[ and has right limits on [0, ∞[. The symbols ( [0, ∞[, ) and (ℝ ≥0 , ) denote the set of all càdlàg paths in . If is an element of (ℝ ≥0 , ), we define We also use , − , and Δ to indicate the values ( ), ( −), and Δ ( ), respectively. Next, set We simply write , , and if there is no ambiguity. Given a discrete set ⊂ [0, ∞[ and a càdlàg path , we define Recall that a subset is discrete if it is a discrete topological subspace of [0, ∞[; i.e. every element of is an isolated point with respect to the subspace topology. By assumption, the set ∩ [0, ] is finite for all , and therefore the summation above is well-defined. Then ( ) is a càdlàg path of finite variation. For abbreviation, we often write ( ) instead of ( ( ); ).

Remarks on càdlàg paths and Stieltjes integration
In this subsection, we review some basic properties of càdlàg paths that will be referred to later. Let be a Banach space. -valued right continuous and left continuous step functions are functions of the form ∑︁ respectively, where 0 = 0 < 1 < · · · < < · · · → ∞ and , ∈ for all ∈ ℕ. Right continuous step functions are càdlàg and left continuous step functions are càglàd. Every right continuous step function A càdlàg path in a Banach space satisfies the following properties.     (v) The function d defined by is again a càdlàg path of finite variation.
Note that the summation in (v) of Lemma 2.2 is defined in the following manner. Let be the set of all finite subsets of ]0, ]. We regard as a directed set with the order defined by inclusion. Then the net ( ∈ Δ ( )) ∈ converges in by Condition (iv) of Lemma 2.2, and hence we can define Because there is a measure associated with , we can consider the Stieltjes integral with respect to . Let : × → be a continuous bilinear map between Banach spaces. Set 1 loc ( ; ) = 1 loc (| |; ), where | | denotes the variation measure of introduced in Appendix B.1. Then for each ∈ 1 loc ( ; ) and a bounded interval = ] , ], the Stieltjes integral is defined by the formula .
is a càglàd path, we can also define the Stieltjes integral as where the integral on the right-hand side is constructed in Appendix B.2. Finally, note that the decomposition = c + d gives a decomposition of the integral

Settings and the main result
In this section, we introduce the main results of this paper, namely, the Itô formula within the framework of the Itô-Föllmer calculus in Banach spaces. The statement is to be found in Theorem 3.6 and Corollary 3.7. First, we introduce some notations about difference operators. Given a function : ℝ ≥0 → and ≥ 0, define functions and on I into by the formulae By using this notation, the left-side Riemannian sum along a partition is expressed as We also define in the norm topology of ; (b) for all ∈ ℝ ≥0 , the jump of ( , ) is given by Then the path ( , ) is called the -quadratic covariation of and . If = and = , we call ( , ) the -quadratic variation of .
(ii) If the convergence of (i)-(a) holds in the weak topology of , we say that ( , ) has the weak -quadratic covariation ( , ).
We often call a -quadratic covariation a strong -quadratic covariation if we stress that the convergence holds in the norm topology. The quadratic covariation ( , ) depends on the sequence of partitions Π. Given a partition , we often write Because and are càdlàg, the map ↦ → ( , ) is also càdlàg. It is not, however, of finite variation unless and are of finite variation. With this notation, we can say that the strong/weak -quadratic covariation ( , ) is the pointwise limit of ( ( , )) ∈ℕ respectively in the norm/weak topology satisfying the jump condition (i)-(b) of Definition 3.1.
An important class of a bounded bilinear map is the canonical bilinear map into a tensor product of Banach spaces. Let be a norm on the algebraic tensor product ⊗ of two Banach spaces. The norm is called a reasonable crossnorm if it satisfies the following two conditions: holds for all ∈ and ∈ ; (ii) the inequality * ⊗ * ( ⊗ , ) * ≤ * * holds for all * ∈ * and * ∈ * , where ( ⊗ , ) * denotes the usual operator norm on the normed space ( ⊗ , ).
The completion of the normed space ( ⊗ , ), which is generally incomplete, is denoted by ⊗ . See Diestel and Uhl [18] and Ryan [66] for basic facts about tensor products of Banach spaces. The quadratic covariation of ( , ) with respect to the canonical bilinear map ⊗ : × → ⊗ is denoted by [ , ], and it is called the -tensor quadratic covariation. We also write [ , ] = ⊗ ( , ) and call it the discrete tensor quadratic covariation of ( , ) along . If = , we can consider -tensor quadratic covariations There are various important reasonable crossnorms in Banach space theory. The greatest crossnorm , also called the projective tensor norm 1 , is defined by the following formula is finite, ∈ and ∈ for all ∈ , and = ∑︁ ∈ ⊗ .
We simply write ⊗ = ⊗ and call it the projective tensor product of and . -tensor quadratic variations are also called projective tensor quadratic variations. An advantage of the projective tensor product is that there is a canonical isometric isomorphism L( , L( , )) L( ⊗ , ) L (2) ( , ; ). We use this identification without mention. If and are Hilbert spaces, Hilbert-Schmidt crossnorm is also a natural object. For ( 1 ) and ( 2 ) defined on some measure spaces, there is a crossnorm such that . See Defant and Floret [16,Section 7].
A càdlàg path : ℝ ≥0 → ℝ has tensor quadratic variation if and only if it has quadratic variation in the sense of Definition 2.3 of Hirai [35].

Remark 3.2.
(i) Our definition of strong tensor quadratic variations can be regarded as a pathwise analogue of tensor quadratic variations in classical stochastic integration theory in infinite dimensions such as Metivier and Pellaumail [47] and Metivier [46]. See also Dinculeanu [20]. Although one can define tensor quadratic variations in general Banach spaces, classical existence results only deal with Hilbert spaces.
(ii) Another important approach is developing recently in the context of the martingale theory in UMD Banach spaces. Yaroslavtsev [80] shows that a local martingale in a UMD Banach space has the covariation bilinear form [[ ]]. In our terminology, the covariation bilinear form corresponds to the cylindrical quadratic variation in the sense of Corollary 3.5 in Hirai [37]. By this corollary, we see that a càdlàg path with weak tensor quadratic variation has cylindrical quadratic variation.
Now we introduce a different type of quadratic variation, namely, scalar quadratic variation. Again we assume that Π = ( ) is a sequence of partitions of ℝ ≥0 . Definition 3.3. Let be a Banach space and be an -valued càdlàg path.
(ii) A càdlàg path : ℝ ≥0 → has scalar quadratic variation along Π if there exists a real-valued càdlàg increasing path ( ) such that We call the increasing path ( ) the scalar quadratic variation of along ( ).
If has scalar quadratic variation along Π, then it has finite 2-variation along Π.
The scalar quadratic variation of a Hilbert space valued path coincides with the bilinear quadratic variation , ( , ), where , is the inner product of the state space. If a càdlàg path = ( 1 , . . . , ) : ℝ ≥0 → ℝ has tensor quadratic variation along ( ), then it has scalar quadratic variation given by This trace representation is still valid for Hilbert space-valued càdlàg paths. This result is proved in Hirai [37]. Next, we introduce some conditions on a sequence of partitions and a càdlàg path. Let ∈ Par( [0, ∞[) and ∈ ]0, ∞[. The symbol ( ) denotes the element of that contains . By definition, there exists only one such interval. In addition, set ( ) = sup ( ) and ( ) = inf ( ). Then we have ( ) = ] ( ), ( )].
Let : → be a function into a Banach space and be a subset of . Define the oscillation of on by Using this notation, we introduce two kinds of oscillation of a path ∈ (ℝ ≥0 , ) along a partition ∈ Par(ℝ ≥0 ) as follows.
The second equality in the definition of − ( ; ) is valid because is supposed to be right continuous. These oscillations satisfy the relation − ( ; ) ≤ + ( ; ) for all ≥ 0. If is continuous, these two quantities coincide.

Remark 3.5.
(i) Let ( ) be a sequence of partitions along which we consider a quadratic variation of a path. In this paper, we often require that ( ) satisfies Condition (C) defined above. Therefore, we can say that (C) is a condition for integrators of Itô-Föllmer integration.
(ii) In contrast to (i), Condition (ii) of Definition 3.4 needs to be satisfied by integrands of Itô-Föllmer or Stieltjes integrals. This will be mainly used in Theorem 3.6 and Lemma 3.10.
Under these assumptions, we have the following 1,2 type Itô formula for Banach space-valued paths. and approximates ( , ) from the left. Suppose that has weak -quadratic variation and finite 2-variation along ( ) and suppose also that has finite variation. Moreover, let : × → be a function satisfying the following conditions: Then the Itô-Föllmer integral ∫ 0 ( − , − )d exists in the weak topology and it satisfies Moreover, if the quadratic variation exits in the strong sense, the convergence of the Itô-Föllmer integral holds in the norm topology of .
with suitable topology (see Definition 7.1.) As a direct consequence of Theorem 3.6, we can derive the Itô formula related to tensor quadratic variations.
Corollary 3.7. Let and be Banach spaces and let be a reasonable cross norm on ⊗ . Suppose that ∈ (ℝ ≥0 , ) has strong or weak -tensor quadratic variation and finite 2-variation along ( ) and suppose also that ∈ (ℝ ≥0 , ). Moreover, let : × → be a function satisfying the following conditions: (iii) the second derivative of induces a continuous function 2 : × → L c ( ⊗ , ).
Then ( , ) admits the following Itô formula: The convergence of the Itô-Föllmer integral holds in the norm or weak topology, respectively.
Then every element of C satisfies the assumptions in Corollary 3.7. Remark 3.9. A possible direction of development is a functional extension of Theorem 3.6 and Corollary 3.7, that is, infinite-dimensional functional Itô calculus. For such an extension, we should introduce Dupire's derivative of a functional of infinite dimensional paths. This seems an important future work. See, as referred to in Section 1, Dupire [21], Cont and Fournié [9,8,10], and Ananova and Cont [1], for functional Itô calculus in finite-dimensional state space.
The following lemma is essentially used to prove Theorem 3.6.
holds in the weak topology of . If, in addition, ( , ) is the strong -quadratic variation, then (3.2) holds in the norm topology.

Auxiliary results regarding sequences of partitions
In this section, we investigate conditions on a sequence of partitions along which we deal with quadratic variations and Itô-Föllmer integrals. Recall that basic notions were defined in Definition 3.4.    Then, by assumption, we can choose an ∈ ℕ such that − ( , ) < /2 holds for all ≥ . We will show that = ( ) holds for all ≥ . Take an from the interval ] ( ), [ such that This shows that ∉ ] ( ), ( ) [, and therefore = ( ). Hence ( ) exhausts the jumps of . This contradicts the condition that − ( , ) < .
Step 2: (ii) =⇒ (i). Suppose that ( ) satisfies the conditions (ii)-(a) and (b). Fix an > 0 and a > 0 arbitrarily. Because is càdlàg, we can take a sequence 0 = 0 < 1 < · · · < = such that ( ; ] , +1 [) < /2 for all (see Lemma 2.1). By assumption, we can choose an ∈ ℕ satisfying the following conditions:  As we mentioned in Section 1, two types of assumptions about a sequence of partitions are frequently used in the context of the Itô-Föllmer calculus. One is that | | → 0 and the other is that ( ) controls the oscillation of . In the next proposition, we show that both conditions imply that ( ) satisfies Condition (C) for . (ii) Assume that ( ) controls the oscillation of . Then we see that ( ) approximates from the left by the following estimate. In the last part of this section, we give an additional lemma about a sequence of partitions. Here, note that the second inequality holds by Condition (a) above. Thus, we get for arbitrary ≤ 1 . This implies (C3) for .

Properties of quadratic variations
This section is devoted to studying some basic properties of quadratic variations introduced in Section 3. Throughout this section, suppose that we are given a sequence Π = ( ) ∈ℕ of partitions of ℝ ≥0 . First, we give some examples of quadratic variations.
Example 5.1. Let be a path of finite variation in a Banach space . If ( ) satisfies | | → 0, then has projective tensor and scalar quadratic variations given by This result will be proved later in this section. Given arbitrary 1 -function : ℝ → into a Banach space , let us set ( ) = ( ( )). If has quadratic variation along ( ), then has projective tensor quadratic variation given by This is one of the main results of the second article in this series [37]. For the construction of a real continuous path of nontrivial quadratic variation, refer to Schied [69], Mishura and Schied [52], and Cont and Das [7].
The next examples are from the theory of stochastic processes.

Example 5.3.
(i) Let (Ω, F, (F ) ≥0 , ) be a filtered probability space satisfying the usual conditions. Consider a semimartingale = ( ) ≥0 in a separable Hilbert space . Moreover, let = ( ) ∈ℕ be an increasing sequence of bounded stopping times such that → ∞ as → ∞ and sup ( +1 − ) → 0 as → ∞ almost surely. Then the process [ , ] converges to the quadratic variation process [ , ] uniformly on compacts in probability (ucp). By passing to an appropriate subsequence, we see that almost all paths have quadratic variation along the subsequence. See Gravereaux and Pellaumail [28] or Metivier and Pellaumail [47] for details.
(ii) In addition to the assumptions of (i), let : → be a 1 function into a Banach space . Then, along a suitable subsequence of ( ), almost all paths of ( ) have quadratic variation. If, moreover, is of 2 class, the Itô-Föllmer integral ∫ · 0 ( − )d exists and its paths have quadratic variation along the same subsequence. Now, since the Banach space is chosen arbitrarily, it may fail to satisfy some useful properties required by the martingale theory, e.g. UMD property or martingale type 2 property. The path ( ), however, behaves well enough from the viewpoint of the Itô-Föllmer calculus.  Using the matrix notation, we can also express equation (5.1) as .
Proof. By the definition of B, we can easily check that hold for all ≥ 0. These immediately prove the assertion.
Applying Proposition 5.4 to the canonical bilinear map ⊗ : × → ⊗ , we obtain the following corollary.
for every reasonable crossnorms and on ⊗ and ⊗ , respectively.
Using Corollaries 5.7 and 5.9, we obtain the following. for every reasonable crossnorm on ⊗ .
By a discussion similar to the proof of Proposition 5.8, we see that a path of finite variation has scalar quadratic variation.

Proposition 5.11. Let be a càdlàg path of finite variation in a Banach space . If ( ) satisfies Condition (C) for , then has the scalar quadratic covariation given by
In the preceding part of this paper, we have used the summation to define the quadratic covariation. We can also consider a different summation which is a slight modification of that used in the original paper by Föllmer [22]. Let us investigate the relation between these two summations. Proof. We show the assertion about strong convergence. For each > 0 Hence, by assumption, the convergences of these two sequences are equivalent and their limits coincide.
In the weak case, we have a similar estimate for the pairing * , , which shows the assertion about the weak quadratic covariation.
According to Proposition 5.12, we see that the two definitions of quadratic covariation are equivalent provided that ( ) satisfies the assumption in the proposition. The first definition, which is given in Definition 3.1, is more intuitive. The second one has some technical advantages because the path ↦ →

Proof of Lemma 3.10
In this section, we prove Lemma 3.10, which is essentially used to show the main theorems of this paper. To prove it, we prepare some additional lemmas. Though Lemma 3.10 includes both weak and strong convergence results, we mainly focus on the proof of the weak case 3 .
Throughout this section, let the symbols , 1 , and denote Banach spaces and : × → 1 denote a bounded bilinear map.
holds for all ∈ ℝ ≥0 in the weak topology. If has strong -quadratic variation, then the convergence of (6.1) holds in the norm topology.
Proof. We show the case of weak convergence. By considering the decomposition we can assume that = 0 without loss of generality.
for all * ∈ * 1 . Next, assume < . Then, by Lemma 6.1, ( ) → as → ∞. Hence, Note that the second equality follows from the same argument as the proof of Proposition 5.12.
In both cases, we have the desired convergence. If has strong -quadratic variation, we can directly show the norm convergence of the sequence without taking the pairing * , . Lemma 6.3. Let be a càdlàg path in with weak -quadratic variation along a sequence ( ) satisfying Condition (C) for . Suppose that ∈ (ℝ ≥0 , L( 1 , )) has the representation where 0 = 0 < 1 < · · · < < +1 < · · · → ∞ and each is an element of L( 1 , ). If ( ) approximates from the left, then the Stieltjes integral of − with respect to ( , ) is approximated as in the weak topology. If has strong -quadratic variation, then (6.3) holds in the norm topology of .
Proof. We show the weak convergence case. First, note that the Stieltjes integral on the right-hand side of (6.3) has the representation On the other hand, the summation on the left-hand side of (6.3) is calculated as Therefore, it suffices to show that for all * ∈ * and ≥ 1. This follows directly from Lemma 6.2. If ( , ) is the strong -quadratic variation, the sequence of discrete sums converges in the norm topology by the strong version of Lemma 6.2.

Lemma 6.4. Let be a locally convex Hausdorff topological vector space of which topology is generated by the family of seminorms ( ) ∈ .
(i) Let : ℝ ≥0 → be a càdlàg path. Then for every ∈ and > 0, there is a right continuous step function ℎ such that ( ( ) − ℎ( )) ≤ for all ≥ 0.
(ii) If a sequence of partitions ( ) approximates from the left, then the step function ℎ in (i) can be chosen so that ( ) still approximates ℎ from the left.
(i) Let us construct a partition of ℝ ≥0 recursively. First, let 0 = 0. Next, assume that there is a sequence and then define otherwise.  For ∈ [ , +1 [, we see that Hence is a right continuous step function satisfying the desired condition. (ii) We shall show that the function defined above is approximated from left by the left-approximation sequence ( ) for . Note that, by the definition of , holds for all > 0. Now fix > 0 arbitrarily and choose a unique 0 ∈ ℕ such that ∈ ] 0 , 0 +1 ]. If is discontinuous at 0 , then we see that ( 0 ) → 0 by the same argument as the proof of Lemma 6.1. In this case, 0 ≤ ( 0 ) ≤ ( ) < for large enough and therefore we have Next, assume that is continuous at 0 and is not constant on any interval of the form [ 0 , [ for > 0 . Take such that 0 < < and ( −) ≠ ( 0 ) = ( 0 −). If ( ) ≤ 0 for infinitely many , then we can take a subsequence such that We will observe the behaviour of each part of the right-hand side. We can deduce from Lemma 6.3 that the second term converges to 0 as → ∞. By the choice of ℎ, we find that Therefore, On the other hand, we have Because is chosen arbitrarily, we get the desired conclusion. If ( , ) is the strong quadratic variation, we replace (6.5) with a similar norm inequality. In this case, we see that the corresponding second term converges to 0 by the strong version of Lemma 6.3. The remaining terms are estimated in almost the same way as above.

The Itô formula
This section is devoted to showing the Itô formula within our framework of the Itô-Föllmer calculus in Banach spaces. Let us begin by defining Itô-Föllmer integrals. Definition 7.1. Let be a locally convex space, and let and be Banach spaces. Consider càdlàg paths and in and , respectively, and a continuous bilinear map : × → . Suppose that a sequence of partitions ( ) approximates from the left. We call the limit (ii) Suppose that the following two Itô-Föllmer integrals exist: Then, for every , ∈ ℝ, the Itô-Föllmer integral of with respect to + exists and satisfies First, we consider the case where the integrator is a path of finite variation. From the dominated convergence theorem, we can easily deduce the following proposition. Here, the integral of the left-hand side is the Itô-Föllmer integral by Definition 7.1, and that of the right-hand side is the usual Stieltjes integral.
Now we start to prove our main theorem.
Proof of Theorem 3.6. We show the weak convergence of the Itô-Föllmer integral.
First, fix > 0 arbitrarily and choose compact convex sets 1 ⊂ , 2 ⊂ , and 3 ⊂ such that Step 1: Convergence of the summation in the formula (3.3). In this step, we confirm that the summation of jump terms converges absolutely. This is proved by the following estimate, which follows from Taylor's formula (Proposition A. Note that the uniform boundedness principle combined with the strong continuity shows that sup ( , ) 2 ( , ) and sup ( , ) ( , ) are finite. Moreover, we see that ∑︁ This shows that equation (3.1) is equivalent to the following equation: We will therefore prove (7.1) instead of (3.1).
Step 2: The Taylor expansion. Let = ] , ] ∈ and consider the first-order Taylor expansion with respect to the variable on ∩ [0, ] = ] ∧ , ∧ ]. Then we have Here, recall that the notation ' ( , ) ' was introduced in the second paragraph of Section 3. Next, we consider the second-order Taylor expansion with ( ) given by By the definition of 2 , we can rewrite (7.3) as Moreover, by summing up this equality along , we see that Step 3: Behaviour of ( ) 1 ( ), . . . , ( ) 8 ( ) of (7.6). Since ( ) satisfies Condition (C) for ( , ), we can easily verify that Finally, by letting → 0, we see that the right-hand side of (7.6) converges weakly to that of (7.1). This completes the proof for the weak case. If ( , ) is the strong -quadratic variation, then ( ) 5 converges in the norm topology by the strong version of Lemma 3.10. In this case, we obtain the norm convergence of the Riemannian sums by replacing (7.7) with a similar norm estimate.
Combining Corollaries 3.7 and 5.9, we obtain the integration by parts formula. Note that the existence of the Itô-Föllmer integral

A Differential calculus in Banach spaces
In this section, we review some basic results about differential calculus in Banach spaces. (ii) A function : → is Fréchet differentiable at ∈ if there exits an ∈ L( , ) such that The function is Fréchet differentiable if it is Fréchet differentiable at all points in .
The bounded operators in (i) and (ii) are called Gâteaux and Fréchet derivatives, respectively, and denoted by the same symbol ( ). If is Fréchet differentiable, then it is Gâteaux differentiable and both derivatives coincide. Hence this notation is consistent.
Higher-order Gâteaux and Fréchet derivatives are defined inductively by the formula +1 = . The -th Gâteaux derivative ( ) at ∈ is an element of L( ⊗ , ) L ( ) ( ; ), where L ( ) ( ; ) denotes the set of bounded -linear maps. Moreover, if is -times Fréchet differentiable, the multilinear map ( ) : → is symmetric. Before introducing Taylor's theorem, we define a mild continuity condition for a function defined on a subset of a Banach space. One can easily see that a Gâteaux differentiable function : → is continuous along line segments. The following proposition gives a version of Taylor's formula that we use in this article. It plays an essential role to prove the Itô formula in Section 7. Although that theorem seems classical, we give a proof for the reader's convenience.
for all ∈ and ∈ with + ∈ .
Proof. The proof is by induction on . Suppose that is Gâteaux differentiable and is strongly continuous along line segments. Then the map ↦ → ( + ) has the continuous derivative d d Hence, by the fundamental theorem of calculus, we get Next, assume that the assertion holds for . Let : → be an + 1-times Gâteaux differentiable function whose derivative +1 is strongly continuous along line segments. By assumption, satisfies Computation by the Leibniz rule shows d d Therefore, Combining (A.1) and (A.2), we get the equation This completes the proof.
The following proposition is also used in the proof of the main theorem. Proof. Fix an ( 1 , 2 ) ∈ 1 × 2 and take an arbitrary directional vector (ℎ 1 , ℎ 2 ) ∈ 1 × 2 . Applying Taylor's formula to the first variable, we get for all ≠ 0. Since 1 is strongly continuous, we see that Note that the inequality above follows from the uniform boundedness principle. Therefore, by the dominated convergence theorem, This shows lim →0, ≠0 which means that ( 1 , 2 ) is the Gâteaux derivative of at ( 1 , 2 ). The strong continuity of = ( 1 , 2 ) is obvious. The last claim follows from Lemma A.5 below.
Lemma A.5. Let be a compact topological space and let and be Banach spaces. Assume that a map : → L( , ) is strongly continuous. Then, the map : × → induced by is continuous.
Because is strongly continuous, the first term on the right-hand side converges to 0. The strong continuity and the uniform boundedness principle imply that sup ∈ ( ) < ∞. Hence the second term also converges to 0. As a consequence, we find that ( , ) ( ) converges to ( , ) ( ) .

B Vector integration B.1 A brief review of vector integration in Banach spaces
In this subsection, we will review an integration theory for vector functions and vector measures of finite variation. Let , , and be three Banach spaces and : × → be a bounded bilinear map. We aim to introduce integrals of the form for all such and . This estimate guarantees that the dominated convergence theorem remains valid in this situation. Finally, we introduce a decomposition of a vector measure on ℝ ≥0 into atomic and nonatomic parts. As before, let : D → be a -additive measure of finite variation. Set Because has finite variation, we see that is countable. Besides, since each singleton { } belongs to D, for every ∈ 1 (| |; ).

B.2 Extension of vector integration
In Section B.1, integrands are assumed to be strongly measurable with respect to the norm topology of ; i.e. integrands can necessarily be approximated pointwise by simple functions in the norm topology. A function : ℝ ≥0 → that is càdlàg in a weaker topology may not satisfy this condition. We need, however, to integrate such functions for our Itô-Föllmer formula (Theorem 3.6). For this purpose, we extend the vector integration introduced in a previous section to a suitable setting.
As in Appendix B.1, let and be Banach spaces and let : D → be a -additive vector measure of finite variation. Here, recall that a seminorm is defined by (2.1) in Section 2.1.
Proof. Let = ∈Λ 1 be an I-simple function with Λ being disjoint. Then, by the definition of the integral and the variation of , we see that
The general case is proved by approximation. By applying the dominated convergence theorem to the integral on the right-hand side, we obtain the desired convergence. for all * ∈ * and ∈ ℝ ≥0 .
Proof. First, assume that has the form = ∈Λ 1 with Λ ⊂ I disjoint. By a direct calculation, we see that * , ∫ Hence the formula holds for simple functions. It can be extended to general integrands by the density argument.