aa r X i v : . [ m a t h . P R ] D ec Functional calculus and pathwise integration
Henry CHIU ∗ Rama CONT † December 30, 2019
Abstract
We construct a bespoke topology on path spaces and introduce Func-tional calculus on generic domains, without any assumption on the vari-ation index of a path. We obtain change of variable formulas and extendthe notion of a pathwise integral [8], which may be understood as theanalytic analogue of a martingale. For paths that possess quadratic vari-ation, we obtain the analytic analogue of Itô ’s isometry, in the spirit ofthe pathwise (BDG) inequality [9]. We illustrate that pathwise integralsare solutions to the path-dependent heat equation.
Contents1 Introduction 2 ∗ Dept of Mathematics, Imperial College London. [email protected] † Laboratoire e Probabilités, Statistiques et Modélisation, CNRS-Sorbonne Université[email protected] Introduction1.1 Motivation
Let π := ( π n ) n ≥ be a sequence of partition of [0 , ∞ ) and denote QV π the setof càdlàg paths with finite quadratic variation along π in the sense of Föllmer’s [1], then for any f ∈ C ( R ) , Itô ’s formula holds pathwise for every x ∈ QV π .Moreover, the pathwise integral (a.k.a Föllmer integral) can be computed as alimit of Riemann sum Z T f ′ ( x ( t − )) dx ( t ) := lim n X π n ∋ t i ≤ T f ′ ( x ( t i ))( x ( t i +1 ) − x ( t i )) , (1)without the use of any probabilistic machinery. Based on the key observation Proposition 1.1.
For any semi-martingale X , there exists a sequence of par-tition π such that a.s. sample paths are contained in QV π , Föllmer showed in [1] that for any integrand of the form f ′ ( X ( t )) , where f ∈ C ( R ) and X is a semi-martingale, the pathwise integral (1) coincides with Itô’s integral with probability one, hence it provides a path-by-path (deterministic)interpretation of a stochastic integral. In an earlier work [8, Thm. 4], Föllmer’s pathwise formula is extended to functionals driven by x ∈ QV π under thecondition [8, Rem.7] (which is not required in Föllmer ’s [1]) that J ( x ) ⊂ lim inf n π n , (2)where J ( x ) is the set of discontinuity points of x . We come to the understandingthat condition (2) may be less than ideal: Denote QV π := { x ∈ QV π | J ( x ) ⊂ lim inf n π n } , the pathwise formula of [8, Thm.4] then hold upto QV π and we may encounter Proposition 1.2.
There exists a semi-martingale X such that a.s. sample pathsof X are not in QV π for all π .Proof. Appendix § 7.In light of Prop. 1.2, the pathwise interpretation (1) of a stochastic integral,may become lost. Moreover, we may yet encounter
Proposition 1.3.
Let F be a C , functional in the sense of [8], I ( t, x ) := R t ∇ x F d π x , then x I ( t, x ) may be neither continuous nor differentiable (inthe sense of [8]). roof. The map x I ( t, x ) is in general, discontinuous in the uniform topologyon QV π and is undefined outside of QV π , a region where at every t / ∈ lim inf n π n ,the operator ∇ x sends x (even when x is continuous) to.In other words, condition (2) contributes to the non-differentiability of path-wise integral I . Why is it important to have a calculus that works on I ? Sincethe differentials of a functional is independent of the variation index of a path,if I were to be smooth and if one were able to characterise I in terms of itsdifferentials, then one may be able to extend the notion of a pathwise integralto generic domains, by using only these differentials. Most importantly, I couldthen become solution to otherwise unsolvable problems, for example as solutionto path-dependent PDEs. The key idea [7] behind functional Itô calculus, as elaborated in [8] & [10,s4.1,s5.2], can be summarised as follows: First, construct a calculus for contin-uous functional F on piecewise constant path x n . Second, extend the calculusto all càdlàg paths x , using a density argument and the continuity of F i.e. F ( x n ) → F ( x ) . Clearly, the second step is where topology becomes important.However, functional Itô calculus was built on top of the uniform topology [10,s5.1]. As is well known, piecewise constant approximation of a càdlàg path underthe uniform topology requires exact knowledge of all points of discontinuity. Itis then clear as to why condition (2) would be required in [8, Thm.4]. To remove(2), the uniform topology must be replaced accordingly. Unfortunately, the off-the-shelf topologies on the Skorokhod space D , as referenced in [6, s5], are unfit.Consider the identity map I d ( u ) := u on R , the simple functional F ( x ) := I d ( x ( t )) is not J continuous on D [5, VI. 2.3] and the same applies to all weaker topolo-gies. Since I d ∈ C ( R ) , it may be a lost cause to obtain a functional calculusbuilt on top of weak topologies on D . In this article, we introduce functional calculus based on a new topology. Weextend the notion of pathwise integral to generic domains, give existence condi-tion of pathwise integral and obtain a general change of variable formula withoutany assumption on the variation index of paths. We show that pathwise inte-gral belongs to class M , a subclass of continuous and infinitely differentiablefunctionals. Using class M as primitives and an analogue of the Fundamen-tal theorem of calculus (FTC), we show how to compute pathwise integral andrelate class M to the notion of martingale .3e then apply our functional calculus to treat paths with finite quadraticvariation. We obtain Itô ’s formula and an analytic analogue of Itô ’s isometryin the spirit of the pathwise BDG inequality of [9], a formula of which, one mayresort in order to extend the pathwise integral under (possibly a collection of)positive linear functional(s). In addition, we show that class M are canonicalsolutions to the path-dependent heat equations.The layout of this article is as follows: We introduce our notations in Pre-liminaries. In section 2, we prove a new limit theorem which allows us to treatfunctionals that involve quadratic variation. In section 3, we introduce a newtopology on path spaces, relate it with other well-known topologies and give ex-amples of continuous functionals. In section 4, we introduce functional calculusand smooth functionals, the class C , , X and M . In section 5, we extend thenotion of pathwise integral and prove change of variable formulas. In the finalsection 6, we obtain an analytic analogue of Itô ’s isometries and discuss therelation of class M to path-dependent PDEs. Denote D m to be the Skorokhod space of R m -valued càdlàg functions t x ( t ) := ( x ( t ) , . . . , x m ( t )) ′ on R + := [0 , ∞ ) . Denote S m (resp. BV m ) the subset of step functions (resp.locally bounded variation functions) in D m . For m = 1 , we will omit thesubscript m . By convention, x (0 − ) := x (0) and ∆ x ( t ) := x ( t ) − x ( t − ) . Thepath x ∈ D m stopped at ( t, x ( t )) (resp. ( t, x ( t − )) ) s x ( s ∧ t ) shall be denoted by x t ∈ D m (resp. x t − := x t − ∆ x ( t )1 I [ t, ∞ ) ∈ D m ). We write ( D m , d J ) when D m is equipped with a complete metric d J which induces theSkorokhod (a.k.a. J ) topology.Let π := ( π n ) n ≥ be a fixed sequence of partitions π n = ( t n , ..., t nk n ) of [0 , ∞ ) into intervals t n < ... < t nk n < ∞ ; t nk n ↑ ∞ with vanishing mesh | π n | ↓ on compacts. By convention, max( ∅ ∩ π n ) := 0 , min( ∅ ∩ π n ) := t nk n . Since π isfixed, we will avoid superscripting π .We denote QV m ⊂ D m the subset of càdlàg paths with finite quadraticvariation along π . We denote t ′ n := max { t i < t | t i ∈ π n } , the following piecewise constant approximations of x along π by x n := X t i ∈ π n x ( t i +1 )1 I [ t i ,t i +1 ) , x along π by x ( n ) .We set lim n a n := ∞ whenever a real sequence ( a n ) does not converge. Forreal-valued matrices of equal dimension, we write h· , ·i to denote the Frobeniusinner product and | · | to denote the Frobenius norm. If f (resp. g ) are R m × m -valued functions on [0 , ∞ ) , we write Z t f dg := X i,j Z t f i,j ( s − ) dg i,j ( s ) whenever the RHS makes sense. If x ∈ QV m and f ∈ C ( R m ) , we write Z t ∇ f ◦ xdx := Z t ∇ f ( x ( s − )) dx ( s ) to denote the Föllmer integral [1]. For a function F defined on a domain Λ , theco-domain of F will always be R k,l for some k, l . In this section, we shall extend [8, Lem.12] (atomless measure) to Lem. 2.6(measure with atoms). The goal is to obtain Thm. 2.7, a general limit theoremto treat functional of quadratic variation.
Definition 2.1.
We say that x ∈ D m has finite quadratic variation along π ifthe following sequence of step functions: q n ( t ) := X π n ∋ t i ≤ t ( x ( t i +1 ) − x ( t i ))( x ( t i +1 ) − x ( t i )) ′ converges in the Skorokhod topology. The limit [ x ] := ([ x i , x j ]) ≤ i,j ≤ m ∈ D m × m is called the quadratic variation of x . Proposition 2.2.
Let x ∈ D m , then x ∈ QV m if and only if x i , x i + x j ∈ QV .If x ∈ QV m , then we have the polarisation identity [ x i , x j ]( t ) = 12 ([ x i + x j ] − [ x i ] − [ x j ]) ( t ) ∈ BV = [ x i , x j ] c ( t ) + X s ≤ t ∆ x i ( s )∆ x j ( s ) (3) Proof.
The proof can be found in [11, Thm. 3.6].
Lemma 2.3.
Let v n , v be non-negative Radon measures on R + and J be theset of atoms of v , then v n → v vaguely on R + if and only if v n → v weakly on [0 , T ] for every T / ∈ J Proof.
See for example [11, Lem. 2.2]. 5 emma 2.4.
Let x ∈ QV , d [ x ] be the Radon measure associated with [ x ] . Forevery [0 , T ] , T n := max { t i < T | t i ∈ π n } , T n +1 := min { t i ≥ T | t i ∈ π n } anddefine a sequence of non-negative Radon measures on R + by: µ n ([0 , T ]) := X t i ∈ π n ( x ( t i +1 ) − x ( t i )) δ t i +1 ([0 , T ))+ ( x ( T n +1 ) − x ( T n )) , then it holds(i) ξ n := P t i ∈ π n ( x ( t i +1 ) − x ( t i )) δ t i −→ d [ x ] vaguely on R + ,(ii) µ n −→ d [ x ] vaguely on R + .Proof. (i) follows from [11, Thm. 2.7]. By Lem. 2.3, we may assume T to bea continuity point of d [ x ] . Let f be a continuous function on [0 , T ] . If T = 0 ,then µ n ( { } ) ≡ d [ x ]( { } ) = 0 . If T > , observe that ξ n ([0 , T )) −→ d [ x ]([0 , T )) (by (i)), f is uniform continuous on [0 , T ] and that x is right continuous, put T ′ n +1 := min { t i > T | t i ∈ π n } , it follows that for sufficiently large n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T f dξ n − Z T f dµ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X π n ∋ t i Let v n , v be non-negative Radon measures on R + ; v n −→ v vaguely on R + and J be the set of atoms of v . If for every T ∈ J , thereexists a sequence ( T n ) in R + , T n ↑ T ; v n ( { T n } ) −→ v ( { T } ) , (4) then v n −→ v weakly on [0 , T ] for all T ≥ .Proof. For every T ≥ , ˜ v n ([0 , T ]) := v n ([0 , T ]) − v n ( { T n } ) and ˜ v ([0 , T ]) := v ([0 , T ]) − v ( { T } ) . If T / ∈ J , the claim follows immediately from Lem. 2.3.Thus, we may assume T ∈ J . If T = 0 ∈ J , then T n ≡ . Let T > and f ∈ C ([0 , T ] , k · k ∞ ) . Since f = ( f ) + − ( f ) − , we may take f ≥ and forsufficiently small ǫ > , we define the following extensions: f ǫ ( t ) := f ( t )1 I [0 ,T ] ( t ) + f ( T ) (cid:18) T − tǫ (cid:19) I ( T,T + ǫ ] ( t ) f ǫ ( t ) := f ( t )1 I [0 ,T − ǫ ] ( t ) + f ( T ) (cid:18) T − tǫ (cid:19) I ( T − ǫ,T ] ( t ) , f ǫ , f ǫ ∈ C K ([0 , ∞ )) , ≤ f ǫ ≤ f I [0 ,T ] ≤ f ǫ ≤ k f k ∞ . and we have Z ∞ f ǫ d ˜ v n ≤ Z T f d ˜ v n ≤ Z ∞ f ǫ d ˜ v n . Since v n → v vaguely and (4) holds, we obtain ≤ lim sup n Z T f d ˜ v n − lim inf n Z T f d ˜ v n ≤ Z ∞ f ǫ − f ǫ d ˜ v ≤ f ( T ) ( v ([ T − ǫ, T + ǫ ]) − v ( { T } )) ǫ −→ , hence by monotone convergence lim n Z T f d ˜ v n = lim ǫ Z ∞ f ǫ d ˜ v = Z T f d ˜ v. By (4), it follows lim n R T f dv n = R T f dv . Lemma 2.6. Let v n , v be non-negative Radon measures on R + , f n , f be real-valued left continuous functions on R + and J be the set of atoms of v . Supposethere exists a sequence ( T n ) ∈ R + , T n ↑ T ;(i) v n −→ v vaguely; v n ( { T n } ) −→ v ( { T } ) for every T ∈ J ,(ii) ( f n ) is locally bounded and converges pointwise to f ,then Z T f n dv n −→ Z T f dv. for every T ≥ .Proof. By the Lebesgue decomposition theorem, we can decompose v into acontinuous part v c and a discrete measure v d . By (i) and Lem. 2.5, we immedi-ately see that ( v n − v d ) −→ v c weakly for every [0 , T ] . Since v c has no atoms,by an application of [8, Lemma 12] we have Z T f n d ( v n − v d ) −→ Z T f dv c . By (ii) and dominated convergence, the proof is complete. Theorem 2.7. Let x ∈ QV , f n , f be real-valued left continuous functions on R + ; ( f n ) is locally bounded and converges pointwise to f on R + , then ( i ) X π n ∋ t i ≤ T f n ( t i )( x ( t i +1 ) − x ( t i )) −→ Z T f d [ x ]( ii ) X π n ∋ t i ≤ T f n ( t i +1 ∧ T )( x ( t i +1 ) − x ( t i )) −→ Z T f d [ x ] for every T ≥ . In particular, the convergence(s) also hold under P π n ∋ t i Remark . t R t f d [ x ] is in BV and the continuous part is Z t f d [ x ] c = Z t f d [ x ] − X s ≤ t h f ( s − ) , ∆ x ( s )∆ x ( s ) ′ i . In this section, we shall construct a topology on suitable subsets of E := R + × D m , x as an integrator.(i.e. which makes these type of functionals continuous.) Since we are concernedin causal system , a domain should be { ( t, x t ) | t ∈ R + , x ∈ Ω } ⊂ E, for a suitable Ω ⊂ D m . An important question is: What sort of Ω ⊂ D m mayconstitute a domain in the context of functional calculus? An arbitrary Ω ⊂ D m may not necessarily be closed under vertical perturbation (Dupire direction) i.e. x ∈ Ω = ⇒ x t + e I [ t, ∞ ) ∈ Ω , hence F ( x t + e I [ t, ∞ ) ) may not make sense. Definition 3.1 (generic) . A non-empty subset Ω ⊂ D m is called generic if Ω satisfies the following two closure properties (under operations):i For every x ∈ Ω , T ≥ , ∃ N ∈ N ; x n ∈ Ω , ∀ n ≥ N .ii For every x ∈ Ω , t ≥ , ∃ convex neighbourhood U of containing − ∆ x ( t ) ; x t + e I [ t, ∞ ) ∈ Ω , ∀ e ∈ U . Remark . Def. 3.1(ii) implies that −U is a convex neighbourhood of con-taining ∆ x ( t ) ; x t − + e I [ t, ∞ ) ∈ Ω , ∀ e ∈ −U . We will expand N as N T ( x ) and U (resp. −U ) as U t ( x ) (resp. U t − ( x ) ) whendependency matters. Example 3.3. S m , BV m , QV m , QV + m (i.e. positive paths in QV m ) and D m areall generic. If Ω is generic, then Ω ba := { x ∈ Ω | a < | x | < b } for all constants a, b are all generic. All subsets of continuous paths are notgeneric. Example 3.4. Let Ω be generic, then Ω := Ω ∩ QV m (5)is generic. Proof. We observe S m ⊂ QV m and if x ∈ QV m , then x + S m ∈ QV m . Definition 3.5 (rich) . A generic subset Ω ⊂ QV m is called rich if:9 ∃ x ∈ Ω , N ∈ N ; [ x ] is continuous, non-vanishing and x ( n ) ∈ Ω , ∀ n ≥ N . Example 3.6. QV m and QV + m are rich. S m and BV m are not. Definition 3.7 (domain) . Let Ω be generic, Λ := { ( t, x t ) | t ∈ R + , x ∈ Ω } is called a domain .On E , there already exists two well known (product) topologies, the uniformU (resp. the Skorokhod J ) topology, generated by the standard topology on R + and local uniform (resp. the Skorokhod J ) topology on D m . On a domain Λ ⊂ E , the uniform U and J topologies are defined as follows Definition 3.8. The U (resp. J ) topology on Λ is the subspace topology withregard to the U (resp. J ) topology on E . Remark . Every J continuous functional is U continuous. (i.e. The localuniform topology is strictly finer than the J topology on D m [5, VI].)We shall now define a new topology on a domain Λ , relate it with the U andJ topologies on Λ and give examples of continuous functionals on Λ . Definition 3.10 (The Λ topology) . For every t ∈ R + , x ∈ Ω , we define t ′ n := max { t i < t | t i ∈ π n } and x n := X t i ∈ π n x ( t i +1 )1 I [ t i ,t i +1 ) (6)for all n sufficiently large. Denote X the set of functionals F : Λ R satisfying: . ( a ) lim s ↑ t ; s ≤ t F ( s, x s − ) = F ( t, x t − ) , ( b ) lim s ↑ t ; s Proposition 3.12. C (Λ) = X .Proof. By the definition of initial topology, we have X ⊂ C (Λ) and that z n Λ −→ z ⇐⇒ F ( z n ) → F ( z ) ∀ F ∈ X , (7)hence C (Λ) ⊂ X by (7) and the construction of X (i.e. Def. 3.10). Definition 3.13 (continuous functional) . A functional F on Λ is called continuous if it is continuous with regard to the Λ topology. F is called left- (resp. right-) continuous if F satisfies property3.10.1 (resp. property 3.10.2). Remark . A function F on Λ is continuous (resp. left/right) if all its com-ponents F i,j are continuous (resp. left/right). Remark . If F , G are functions on Λ ; F , G both satisfy any one of theconditions in Def. 3.10, then αF + βG and F G also satisfies the same conditionfor all α, β ∈ R . In particular, C (Λ) is an algebra. Definition 3.16 (Strictly causal) . Let F be a function on Λ and denote F − : ( t, x t ) F ( t, x t − ) . F is called strictly causal if F = F − . Remark . If F, G are strict causal, then αF + βG and F G are strict causal. Lemma 3.18 (pathwise regularity) . Let F be a function on Λ .(i) If F is left continuous, then t F − ( t, x t ) is càg; t F ( t, x t ) is làg.(ii) If F is right continuous, then t F ( t, x t ) is càd; t F − ( t, x t ) is làd.(iii) If F is continuous, then t F − ( t, x t ) (resp. t F ( t, x t ) ) is càglàd(resp. càdlàg ) and its jump at time t is equal to ∆ F ( t, x t ) .Proof. It follows from Def. 3.10.1(a)&(b) and Def. 3.10.2(a)&(b). Example 3.19. Let Ω ⊂ QV m , then the functionals(i) F ( t, x t ) := f ( x ( t )) ; f ∈ C ( R m ) ,(ii) F ( t, x t ) := f ([ x ]( t )) ; f ∈ C ( R m × m ) ,(iii) F ( t, x t ) := R t f ◦ xd [ x ] ; f ∈ C ( R m , R m × m ) ,11iv) F ( t, x t ) := R t ∇ f ◦ xdx ; f ∈ C ( R m ) ,are continuous. Proof. In light of Prop. 3.12, F is continuous if and only if F satisfies Def. 3.10.1& 2 for all ( t, x t ) ∈ Λ . Since conditions Def. 3.10.1(a),(b) & 2(a),(b) are easy toverify, we will put our focal on Def. 3.10.1(c),(d) & 2(c),(d). (i) is trivial. For(ii), we first remark from Def. 2.1 and (3) that q n J −→ [ x ];∆ q n ( t ′ n ) = ∆ x n ( t ′ n )∆ x n ( t ′ n ) ′ −→ ∆ x ( t )∆ x ( t ) ′ = ∆[ x ]( t ) . (8)Since [ x n ]( t ) = q n ( t ) , and by (8), if t n −→ t , the limits of q n ( t n ) and q n ( t n − ) are readily determinedaccording to the rules laid down in [11, s4.2] and (ii) immediately follows fromthe continuity of f .To show (iii) and (iv), it is suffice to assume t n −→ t ; t n ≥ t ′ n (i.e. the othercriteria follow similar lines of proof, see [11, s4.2]). By (8) and [11, s4.2] | q n ( t n ) − q n ( t ′ n ) | −→ . (9)Upon a closer look at (iii) and by Cor. 2.8, we observe F ( t n , x nt n )= Z t n f ◦ x n d [ x n ]= X π n ∋ t i Let T ≥ , x ∈ D m , then ( x T ) n J −→ x T .Proof. See for example [4, Lemma 12.3] and [5, VI]. Lemma 3.22. Let ( t, x ) ∈ E , t ′ n := max { t i < t | t i ∈ π n } , t n −→ t , then ( i ) t n ≤ t ′ n = ⇒ x nt n − J −→ x t − , ( ii ) t n < t ′ n = ⇒ x nt n J −→ x t − , ( iii ) t n ≥ t ′ n = ⇒ x nt n J −→ x t , ( iv ) t n > t ′ n = ⇒ x nt n − J −→ x t . Proof. Let t n ≤ t ′ n , by Lem. 3.21, we have ( x t − ) n J −→ ( x t − ) . Since x is càdlàgwe observe k x nt n − − ( x t − ) n k ∞ ≤ sup s ∈ [ t n ,t ′ n ] | x ( t n ) − x ( s ) | + | x ( t n ) − x ( t − ) | −→ , and (i) follows immediately from [5, VI.1.23]. (ii)-(iv) follow similar lines ofproof. Lemma 3.23. Let Ω be rich. Then the functionals(i) F ( t, x t ) := | [ x ]( t ) | ,(ii) F ( t, x t ) := R t xdx ,are not U continuous.Proof. If Ω is rich (Def. 3.5), there exists T > , continuous x, x ( n ) ∈ Ω ; | [ x ]( T ) | > . (10)Since x ( n ) T −→ x T in the local uniform topology on [0 , ∞ ) , it follows fromDef. 3.8 that ( T, x ( n ) T ) U −→ ( T, x T ) Λ . We observe that each x ( n ) T is a continuous function of bounded variationon [0 , ∞ ) , it follows that | [ x ( n ) ]( T ) | = 0 , ∀ n ≥ (11)and by (10) we have established that (i) is not U continuous. By (10), (11) andthat x, x ( n ) ∈ QV m , we obtain (by an application of Itô ’s formula [1] to | · | ): lim n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T xdx − Z T x ( n ) dx ( n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = lim n (cid:12)(cid:12)(cid:12) | x ( T ) | − | x (0) | − tr ([ x ]( T )) − (cid:16) | x ( n ) ( T ) | − | x ( n ) (0) | (cid:17)(cid:12)(cid:12)(cid:12) = tr ([ x ]( T )) > , hence (ii) is not U continuous on Λ Theorem 3.24. Let Ω be rich, then(i) Every J continuous functional is continuous.(ii) There exists a continuous functional which is not U continuous.(iii) There exists a U continuous functional which is not continuous.Proof. If F is J continuous, then F satisfies Def. 3.10.1(a),(b) & 2(a),(b) dueto Rem. 3.9 & 3.11. (i) now follows immediately from Lem. 3.22. (ii) is due toExample 3.19 and Lem. 3.23.It remains to show (iii). We first note that the U topology on Λ is metrisable,hence sequential continuity is equivalent to continuity. Let us fix a t > ; t / ∈ ∪ n π n , define F ( t, x t ) := | ∆ x t ( t ) | on Λ . Observe that if x n U −→ x in D m then it is well known that: ∆ x n ( s ) −→ ∆ x ( s ) (12)for all s ≥ . In particular, if t n −→ t ; x n ( · ∧ t n ) U −→ x t then (12) implies ∆ x n ( · ∧ t n )( s ) −→ ∆ x t ( s ) for all s ≥ , hence F is U continuous on Λ .On the other hand, we take an x ∈ Ω ; ∆ x ( t ) = 0 , it follows from ourchoice of t that F ( t , x nt ) = | ∆ x n ( t ) | ≡ , hence by Def. 3.10.2(c), F is not continuous on Λ and (iii) follows. Corollary 3.25. Let Ω be rich, then • The Λ topology is strictly finer than the J topology. • The Λ topology and the U topology are non-comparable.Proof. It is an immediate consequence of Thm.3.24 and Rem. 3.9.14 Smooth functional We recall that if f is a continuous function on a locally compact metricspace X , then one also obtains from the continuity of f , (local) uniform conti-nuity which implies (local) boundedness and the existence of a (local) modulusof continuity , two notions (esp. modulus of continuity ) that are essential toestablishing the pathwise Itô formula in [1].In this section, we shall introduce weaker notions of boundedness and modulusof continuity for function defined on a domain Λ (Def. 3.7) and define thecorresponding notion of a C , functional on Λ . We then introduce class X and M , which are important sub-classes of C , .For Ω ⊂ QV m , it is shown that certain functionals that involve quantitiessuch as quadratic variation and Föllmer integral are not only C , but alsobelong to class M , a sub-class of infinitely differentiable functionals.Following [7] & [8], we shall define the time/horizontal D F and space/vertical ∇ x F (Dupire) derivatives as follows: Definition 4.1 (differentiable in time/horizontally) . F : Λ R is called differentiable in time/horizontally if D F ( t, x t ) := lim h ↓ F ( t + h, x t ) − F ( t, x t ) h exists ∀ ( t, x t ) ∈ Λ . Definition 4.2 (differentiable in space/vertically) . F : Λ R is called dif-ferentiable in space/vertically if for every ( t, x t ) ∈ Λ , the map U t ( x ) R : e F (cid:0) t, x t + e I [ t, ∞ ) (cid:1) is differentiable at . We define ∇ x F ( t, x t ) := ( ∇ x F ( t, x t ) , . . . , ∇ x m F ( t, x t )) ′ ; ∇ x i F ( t, x t ) := lim ǫ → F (cid:0) t, x t + ǫ e i I [ t, ∞ ) (cid:1) − F ( t, x t ) ǫ . Remark . If F, G are differentiable in time (resp. space), then αF + βG and F G are differentiable in time (resp. space). Definition 4.4 (differentiable) . F : Λ R is called differentiable if F isdifferentiable in time and in space. Remark . We extend the above definitions to function F on Λ whose com-ponents F i,j satisfy the respective conditions. Proposition 4.6. A function on Λ is strictly causal if and only if it is differ-entiable in space with vanishing derivative. roof. The if part follows from the mean value theorem. The only if part: Let x ∈ Ω and put z := x t + e I [ t, ∞ ) then z t − = x t − and F ( t, x t + e I [ t, ∞ ) ) = F ( t, z t ) = F − ( t, z t ) = F − ( t, x t ) = F ( t, x t ) , by the strict causality of F (Def. 3.16). Definition 4.7 (locally bounded) . A function F on Λ is called locally bounded if for every x ∈ Ω and T ≥ , ∃ n ≥ N T ( x ) ; the family of maps t F ( t, x nt ) ,n ≥ n is locally bounded on [0 , T ] Remark . If F, G are locally bounded, then αF + βG and F G are locallybounded. Proposition 4.9. Every continuous function on Λ is locally bounded.Proof. Let F be continuous; if F is not locally bounded, there exists x ∈ Ω , T ≥ , and a sub-sequence ( n k ) ; | F ( t n k , x n k t nk ) | > k, ∀ k ≥ (13) ( t n k ) is bounded on [0 , T ] .For ease of notation, let t n k −→ t ∈ [0 , T ] without actually passing throughto the sub-sequence. Observe that one can always choose another sub-sequence,bounded (either above or below) by t ′ n k = max { t i < t | t i ∈ π n k } . Since F iscontinuous, if t n k < t ′ n k (resp. t n k ≥ t ′ n k ), then Def. 3.10.1(d) (resp. 2(c)) wouldcontradict (13) as k ↑ ∞ . Lemma 4.10. Let F be locally bounded;(i) F is left continuous then F − is locally bounded.(ii) F is left continuous then t F − ( t, x t ) is locally bounded.(iii) F is right continuous then t F ( t, x t ) is locally bounded.Proof. Since F is locally bounded, we first observe | F ( t, x nt ) | ≤ constfor all t ≤ T and all n sufficiently large. If F is left continuous, then Def. 3.10.1(b)implies const ≥ lim s ↑ t ; s Let f ∈ C ( R + × R m ) , then F ( t, x t ) := f ( t, x ( t )) admits a modulus. Proof. For a given x ∈ Ω and T ≥ , r > , put k x k T := sup t ≤ T | x ( t ) | , r := α k x k T + r ; α > , then f is uniform continuous on [0 , T ] × B r (0) and a modulusof continuity of f on [0 , T ] × B r (0) is given by ω ( δ ) := sup | t − s | + | u − v |≤ δ | f ( t, u ) − f ( s, v ) | which verifies (14). Remark . If F, G admit moduli, then αF + βG admits a modulus. If inaddition, F − , G − are locally bounded, then F G admits a modulus. Lemma 4.14. Let F be differentiable in space and ( ∇ x F ) − be locally bounded,if ∇ x F admits a modulus then F admits a modulus.Proof. Since F is differentiable in space and ∇ x F admits a modulus ω , by meanvalue theorem and the local boundedness of ( ∇ x F ) − , we obtain | F ( t, x nt − + a I [ t, ∞ ) ) − F ( t, x nt − + b I [ t, ∞ ) ) | ≤ ( ω ( r ) + const ) | a − b | . Definition 4.15 ( C , functional) . A continuous functional F is called C , if F is once differentiable in time andtwice differentiable in space; 17i) D F is right continuous,(ii) ( ∇ x F ) − , ( ∇ x F ) − are left-continuous,(iii) D F , ( ∇ x F ) − are locally bounded;(iv) ∇ x F admits a (vertical) modulus of continuity.If in addition, ( ∇ x F ) − is locally bounded, then F is called C , b . Definition 4.16 (class X ) . A continuous and differentiable functional F is of class X if D F is right contin-uous and locally bounded, ∇ x F is left continuous and strictly causal. Definition 4.17 (class M ) . A class X functional F is of class M if D F vanishes. If in addition, ∇ x F islocally bounded, then F is of class M b . Remark . Every functional of class M is infinitely differentiable (Prop. 4.6). Proposition 4.19. C , (Λ) , X (Λ) , M (Λ) , M b (Λ) are vector spaces; C , b (Λ) is an algebra.Proof. It is a consequence of Rem. 3.15, 3.17, 4.3, 4.8, 4.13, Lem. 4.10 & 4.14. Lemma 4.20. Let Ω ⊂ QV m and φ : Λ R m × m . If φ − is left continuous and locally bounded,then F ( t, x t ) := Z t φ ( t, x t − ) d [ x ] is a continuous functional.Proof. Since t φ ( t, x t − ) is left continuous and locally bounded (Lem. 3.18(i))and that t [ x i , x j ]( t ) is in BV , càdlàg with ∆[ x i , x j ] ≡ ∆ x i ∆ x j (Prop. 2.2),it follows F is a finite sum of Lebesgue-Stieltjes integrals and satisfies conditionsDef. 3.10.1(a),(b) & 2(a),(b). For the other conditions in Def. 3.10, it is sufficeto assume t n −→ t ; t n ≥ t ′ n (i.e. the other criteria follow similar lines). Define φ n ( s ) := φ ( t , x nt − )1 I { } ( s ) + X t i ∈ π n φ ( t i , x nt i − )1 I ( t i ,t i +1 ] ( s ) , which is a R m × m -valued left continuous function on R + . By the local bound-edness of φ − , we see that ∃ n ≥ N ( x ) ; ( φ n ) n ≥ n is locally bounded on R + and18onverges pointwise to s φ ( s, x s − ) on R + . By Cor. 2.8(ii), we obtain F ( t n , x nt n )= Z t n φ ( s, x ns − ) d [ x n ]= X π n ∋ t i For the if part: We can write f ( t, u ) = α + β · u and hence F ( t, x t ) = α + βx ( t ) on Λ for some contants α ∈ R , β ∈ R m . By Example 3.19(i) and computing thederivatives of F , we see that F is of class M . The only if part: By Def. 4.17and Prop. 4.6, we first obtain(i) ∂ t f ( t, x ( t )) = D F ( t, x t ) = 0 ,(ii) ∇ f ( t, x ( t )) = ∇ x F ( t, x t ) = 0 , ∀ t ≥ , x ∈ Ω . Since S m ⊂ Ω , we have R := { ( t, x ( t )) | t ∈ R + , x ∈ Ω } = R + × R m , hence ∂ t f ≡ ∇ f ≡ on R + × R m . By the mean value theorem, we deduce that ∇ f ≡ β on R , for some β ∈ R m . Remark . The condition S m ⊂ Ω may be weaken to simply requiring that R be a convex region in R + × R m , in this case, the only if part holds up to R . Example 4.23 (Non-Markovian) . Let Ω ⊂ QV m , φ : Λ R m × m ; φ − is leftcontinuous and locally bounded, f ∈ C ( R m ) , f i ∈ C ( R ) , then the functionals19i) F ( t, x t ) := R t φ ( t, x t − ) d [ x ] ,(ii) F ( t, x t ) := R t ∇ f ◦ xdx ,(iii) F ( t, x t ) := P i (cid:16)R t ( x i ( t ) − x i ) f i ◦ x i dx i − R t f i ◦ x i d [ x i ] (cid:17) ,are all C , b . In particular, (ii) & (iii) are of class M b . Proof. The functional in (iii) is well defined, since F ( t, x t ) = X i (cid:18) x i ( t ) Z t f i ◦ x i dx i − Z t x i f i ◦ x i dx i − Z t f i ◦ x i d [ x i ] (cid:19) , (15)the integrals in (15) are either Lebesgue-Stieltjes or Föllmer . Continuity of F in (i), (ii) & (iii) follows from Lem. 4.20, Example 3.19 & Rem. 3.15. Since D F ≡ in all cases, let us first compute ∇ kx F for k = 1 , and demonstrate that F possesses the required properties. In case of (i), we have ∇ x F ( t, x t ) = ( φ + φ ′ )( t, x t − )∆ x ( t ) , ∇ x F ( t, x t ) = ( φ + φ ′ )( t, x t − ) , which are left continuous, locally bounded and ∇ x F ( t, x t ) is strictly causal, byProp. 4.6, Lem. 4.10(ii) & 4.14, F is C , b . In case of (ii), we obtain ∇ x F ( t, x t ) = ∇ f ( x ( t − )) , which is left continuous, locally bounded and strictly causal, hence F is of class M b . In case of (iii), we apply ∇ x to (15) and verify that ∇ x i F ( t, x t ) = Z t f i ◦ x i dx i − f i ( x i ( t − ))∆ x i ( t )= (cid:18)Z f i ◦ x i dx i (cid:19) ( t − ) . (16)Applying f ( x ) := R x i f i ( λ ) dλ ; x ∈ R m to (ii) and by Prop. 4.9 & Lem. 4.10(i),we see that each ∇ x i F is left continuous and locally bounded and so is ∇ x F .Since ∇ x F is strictly causal, F is of class M b . In this section, we extend the definition, give existence conditions and ex-amples of pathwise integral. We obtain change of variable formulas and ananalogue to the classical Fundamental theorem of calculus (FTC). For pathsthat possess quadratic variation, we obtain Itô ’s formula.In particular, we show that pathwise integral is of class M and that func-tionals of class M are primitives , a new fact that facilitates the computation ofpathwise integrals, as in classical calculus. We relate class M to the notion of martingale . 20 emma 5.1. Let F be a left continuous functional, differentiable in time, if D F is rightcontinuous and locally bounded, then F ( t, x s ) − F ( s, x s ) = Z ts D F ( u, x s ) du, (17) for all x ∈ Ω , t ≥ s ≥ .Proof. Put z := x s ∈ Ω , then z t = x s for t ≥ s and z t − = x s for t > s . Define f ( t ) := F ( t, x s ) for t ≥ s , then f ( t ) = F ( t, z t ) on [ s, ∞ ) and f ( t ) = F ( t, z t − ) on ( s, ∞ ) . Since F is differentiable in time, f is right differentiable (henceright continuous) on [ s, ∞ ) and the right derivative f ′ ( t ) is D F ( t, x s ) on [ s, ∞ ) .Since F is left continuous, it follows from Lem. 3.18 that f ( t ) = F ( t, z t − ) is leftcontinuous on ( s, ∞ ) , hence we have first established that f is continuous on [ s, ∞ ) . Next, we observe that f ′ ( u ) = D F ( u, x s ) = D F ( u, z u ) on [ s, ∞ ) . The right continuity of D F and Lem. 3.18 implies that f ′ is rightcontinuous on [ s, ∞ ) . Since D F is right continuous and locally bounded, itfollows from Lem.4.10(ii) that u −→ D F ( u, z u ) is locally bounded. Hence, f ′ is right continuous and bounded on [ s, T ] , henceRiemann integrable. By a stronger version [2] of the Fundamental theorem ofcalculus, the proof is complete. Lemma 5.2. Let φ be a right continuous and locally bounded on Λ , then X π n ∋ t i ≤ T Z t i +1 t i φ ( t, x nt i ) dt −→ Z T φ ( t, x t ) dt, for all x ∈ Ω , T ≥ .Proof. Define φ n ( t ) := X π n ∋ t i ≤ T φ ( t, x nt i )1 I [ t i ,t i +1 ) ( t ) = X π n ∋ t i ≤ T φ ( t, x nt )1 I [ t i ,t i +1 ) ( t ) . By the local boundedness of φ , we see that ∃ n ≥ N ( x ) ; ( φ n ) n ≥ n is locallybounded on [0 , T ] . Since φ is right continuous, it follows from Lem. 3.18that t φ n ( t ) is right continuous (hence measurable) on [0 , T ] and fromDef.3.10.2(c) that φ n converges to t φ ( t, x t ) pointwise on [0 , T ] . and (i)follows from dominated convergence. 21 orollary 5.3. Let φ be a right continuous and locally bounded Λ , then ( t, x t ) Z t φ ( s, x s ) ds is continuous.Proof. The path t R t φ ( s, x s ) ds is continuous. The rest follows from thelocal boundedness of φ and Lem. 5.2. Definition 5.4 (pathwise integral) . Let φ : Λ R m ; φ − be left continuous. For every x ∈ Ω , define I ( t, x nt ) := X π n ∋ t i ≤ t φ ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) (18)for all n sufficiently large and I ( t, x t ) := lim n I ( t, x nt ) . If I is continuous, then φ is called integrable and I is called the pathwise integral . Theorem 5.5 (existence) . Let φ : Λ R m ; φ − be left continuous and I bedefined as in (18). If for every x ∈ Ω , T > and all n sufficiently large, thesequence of step functions on [0 , T ] I n ( t ) := I ( t, x nt ) , is Cauchy in ( D [0 , T ] , d J ) , then φ is integrable.Proof. Since ( I n ) is Cauchy and D is complete with regard to the said metric,there exists a G ∈ D ; I n J G , hence I n ( t ) G ( t ) for every continuity pointof G on [0 , T ] . Observe that ∆ I n ( t ) = φ ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) , if t = t i ∈ π n . , otherwise . (19)If ∆ G ( t ) > , there exists [5, VI.2.1(a)] a sequence t ∗ n → t ; ∆ I n ( t ∗ n ) → ∆ G ( t ) .Using the fact that φ − is left continuous, x is càdlàg and (19), we see that lim n ∆ I n ( t ∗ n ) = φ ( t, x t − ) · ∆ x ( t ) = lim n φ ( t ′ n , x nt ′ n − ) · ∆ x n ( t ′ n ) = lim n ∆ I n ( t ′ n ) , (20)else we will contradict ∆ G ( t ) > . Applying [5, VI.2.1(b)], we deduce that ( t ∗ n ) must coincide with ( t ′ n ) for all n sufficiently large and by [5, VI.2.1(b.3)], wehave established that I n ( t ) −→ G ( t ) , (21)hence we can define I ( t, x t ) := G ( t ) on [0 , T ] .Put t ′′ n := min { t i > t ′ n | t i ∈ π n } , z := x t − ∈ Ω , it follows from (18), (20) &(21) that I ( t, x t − ) = lim n I ( t, z nt ) = lim n (cid:16) I ( t, x nt ) − φ ( t ′ n , x nt ′ n − ) · ( x ( t ′′ n ) − x ( t − )) (cid:17) = G ( t − ) , t I ( t, x t ) is càdlàg and its jump at time t is I ( t, x t ) − I ( t, x t − ) . If t n −→ t , the limits of I n ( t n ) and I n ( t n − ) are readily determined according to(20) and [5, VI.2.1(b)]. The continuity criteria in Def. 3.10 are all satisfied. Proposition 5.6. Let φ be integrable, then D I = 0 and ∇ x I = φ − .Proof. Put z := x + e I [ t, ∞ ) , then I ( t, z t ) − I ( t, x t ) = lim n ( I ( t, z nt ) − I ( t, x nt ))= lim n φ ( t ′ n , z nt ′ n − ) · e = lim n φ ( t ′ n , x nt ′ n − ) · e = φ ( t, x t − ) · e, by the continuity of I and left continuity of φ − . Theorem 5.7 (Change of variable formula) . Let x ∈ Ω , F be a functional of class X , then F ( T, x T ) = F (0 , x ) + Z T D F ( t, x t ) dt + Z T ∇ x F ( t, x t − ) dx, where Z T ∇ x F ( t, x t − ) dx := lim n X π n ∋ t i ≤ T ∇ x F ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) (22) exists.Proof. Appendix § 7. Remark . By Prop. 5.6, we see that all pathwise integrals are functionals ofclass M , hence by Thm. 5.7, we can write I = Z φdx. As we shall see, the converse is also true, all integrals that may be definedby (22) are pathwise integral in the sense of Def. 5.4: Corollary 5.9 (Decomposition) . Let F be of class X , then M ( t, x t ) := F ( t, x t ) − F (0 .x ) − Z t D F ( s, x s ) ds is of class M and ∇ x M = ∇ x F . In particular, M is a pathwise integral.Proof. By differentiating M , we obtain D M = 0 and ∇ x M = ∇ x F . Continuityof M follows from Rem. 3.15, Cor. 5.3 and Thm. 5.7, hence by (22), M is apathwise integral. 23n fact, all functionals of class M are primitives . We obtain as a corollary,the analogue of the classical Fundamental theorem of calculus (FTC): Corollary 5.10 (FTC) . (i) Let φ be integrable, then R φdx is continuous, differentiable and ∇ x Z φdx = φ − . (ii) Let φ be a function on Λ , if F is any functional of class M ; ∇ x F = φ − ,then φ is integrable and Z t φdx = F ( t, x t ) − F (0 , x ) . Proof. (i) is due to Prop. 5.6 & Rem. 5.8. (ii) is due to (22) & Cor. 5.9.Using class M as primitives , we now demonstrate how to apply FTC(ii) tocompute pathwise integral. Example 5.11. Let Ω ⊂ QV m , f i ∈ C ( R ) , then Z T (cid:18)Z f ◦ x dx , . . . , Z f m ◦ x m dx m (cid:19) ′ dx = X i Z T ( x i ( T ) − x i ) f i ◦ x i dx i − Z T f i ◦ x i d [ x i ] ! , (23)by an application of Cor. 5.10(ii) to the RHS of (23), Example. 4.23(iii) & (16).A strategy is a pair ( φ, ψ ) of continuous function on Λ , where φ is R m -valuedand ψ is real valued. The value is V := φ − · x + ψ − . A strategy is self-financing if (i) ∆ φ ( t, x t ) · x ( t ) + ∆ ψ ( t, x t ) = 0 ,(ii) ( φ ( t + h, x t ) − φ ( t, x t )) · x ( t ) + ψ ( t + h, x t ) − ψ ( t, x t ) = 0 . Example 5.12 (self-financing strategy) . If ( φ, ψ ) is self-financing, then φ is integrable and Z t φdx = V ( t, x t ) − V (0 , x ) Proof. We first obtain ∇ x V = φ − , which is left continuous and strictly causal.(i) implies that V = φ · x + ψ which is continuous and by (ii), D V vanishes,hence V ∈ M . The claim follows by an application of Cor. 5.10(ii). Remark . Differentiability is not imposed on V and ( φ, ψ ) .24he term martingale was originally introduced to prove the impossibility ofsuccessful betting strategies (Jean Ville among others). The term martingale issynonymous to the term fair game . Theorem 5.14 (fair game) . Let Ω be a generic set of paths and F ∈ M . If there exists T > such that ∀ x ∈ Ω , F ( T, x T ) − F (0 , x ) ≥ then F ( T, x T ) = F (0 , x ) for all x ∈ Ω .Proof. Appendix § 7.We remark here that arbitrage does not exist regardless what the variationindex a collection of paths may possess. We now conclude this section with Itô’s formula. Theorem 5.15 (Itô ’s formula) . Let x ∈ Ω ∩ QV m . For any F be C , (Λ) we have the Itô formula F ( T, x T ) = F (0 , x ) + Z T D F ( t, x t ) dt + Z T ∇ x F ( t, x t − ) dx (24) + 12 Z T ∇ x F ( t, x t − ) d [ x ] c + X t ≤ T (∆ F ( t, x t ) − ∇ x F ( t, x t − ) · ∆ x ( t )) , where the series is absolute convergent and Z T ∇ x F ( t, x t − ) dx := lim n X π n ∋ t i ≤ T ∇ x F ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) , (25) exists.Proof. Appendix § 7.Note (25) may only be defined up to Ω . In the case Ω ⊂ QV m (so that Ω = Ω ), all integrals that may be defined by (25) are pathwise integrals. Proposition 5.16. Let Ω ⊂ QV m and F be C , , then I : Λ R ( t, x t ) I ( t, x t ) := Z t ∇ x F dx is continuous. In particular, ∇ x F is integrable and I is a pathwise integral.Proof. We apply Itô ’s formula (Thm. 5.15) to F . Rearranging the terms in (24)(i.e. isolating (25) in one side), we see that t I ( t, x t ) is càdlàg whose jumpat time t is I ( t, x t ) − I ( t, x t − ) . It remains to show that I satisfies the continuitycriteria Def. 3.10.1(c),(d) & 2(c),(d). It is suffice to assume t n −→ t ; t n ≥ t ′ n x is right continuous,we first obtain I ( t n , x nt n ) = Z t n ∇ x F ( t, x nt − ) dx n = X π n ∋ t i 26y Prop. 5.16, the integral operator ∫ : L (Λ) R Λ φ φ := Z φdx is a well defined linear operator. Example 6.1 (A non-Markovian 1-form) . Let f i ∈ C ( R ) , i = 1 , . . . , m then φ ( t, x t ) := (cid:18)(cid:18)Z f ◦ x dx (cid:19) ( t − ) , . . . , (cid:18)Z f m ◦ x m dx m (cid:19) ( t − ) (cid:19) ′ are contained in L b (Λ) . Proof. See Example 4.23(16). Lemma 6.2. (i) If φ ∈ L (Λ) then ∫ φ ∈ M (Λ) and ∇ x ( ∫ φ ) = φ − .(ii) If φ ∈ L b (Λ) then ∫ φ ∈ M b (Λ) and ∇ x ( ∫ φ ) = φ − .(iii) If φ ∈ L (Λ) then ∫ φ ∈ M (Λ) and ∇ x ( ∫ φ ) = φ .(iv) If φ ∈ L b (Λ) then ∫ φ ∈ M b (Λ) and ∇ x ( ∫ φ ) = φ .Proof. It is due to Prop. 5.16 & Cor.5.10(i). Corollary 6.3. Define M (Λ) := { F ∈ M b (Λ) | F (0 , x ) ≡ } , then the integral operator ∫ : L b (Λ) (Λ) is an isomorphism and the inverse of ∫ is the differential operator ∇ x .Proof. Injective follows from Lem. 6.2(iv). Surjective is due to Cor. 5.10(ii).We now obtain an analytic analogue of Itô ’s isometry, in the spirit of thepathwise Burkholder-Davis-Gundy inequality in [9] and give an example of ap-plication. Definition 6.4 (generator) . A subspace E in L b (Λ) is called a generator if (cid:18) ψ Z φdx + φ Z ψdx (cid:19) ( t − ) ∈ E , whenever φ, ψ ∈ E . 27 heorem 6.5 (Itô ’s isometry) . (cid:18)Z φdx (cid:19) (cid:18)Z ψdx (cid:19) = Z φψ ′ d [ x ] + Z (cid:18) ψ Z φdx + φ Z ψdx (cid:19) dx holds for all φ, ψ ∈ L b (Λ) . In particular, κ ( t, x t ) := (cid:18) ψ Z φdx + φ Z ψdx (cid:19) ( t − ) ∈ L b (Λ) . Proof. By Prop. 4.19, C , b is an algebra. Let φ, ψ ∈ L b , put F := R φdx, G := R ψdx , then F, G ∈ M b by Lem. 6.2(iv). Since M b ⊂ C , b , it follows F G ∈ C , b . Apply Itô ’s formula (Thm. 5.15) to F G and by Lem.6.2(ii), the proof iscomplete. Corollary 6.6. L b (Λ) is a generator.Proof. It is due to Thm. 6.5. Corollary 6.7. Let E be a generator and denote Im ( E ) the image of E under ∫ . If E is anypositive element of the algebraic dual C ∗ (Λ) such that Im ( E ) ⊂ ker ( E ) , then (cid:28)Z φdx, Z ψdx (cid:29) Im ( E ) := E (cid:18)Z φdx Z ψdx (cid:19) = E (cid:18)Z φψ ′ d [ x ] (cid:19) =: h φ, ψ i E holds for all φ, ψ ∈ E .In particular, the brackets are semi-inner products (each induces a semi-norm). Denote ˜ E the quotient space induced by the semi-norm, then the integraloperator ˜ ∫ : ˜ E 7−→ Im ( ˜ E )˜ φ ˜ ∫ ˜ φ := ∫ φ is an isometric isomorphism between the pre-Hilbert spaces ˜ E and Im ( ˜ E ) . Theinverse of ˜ ∫ is the differential operator ˜ ∇ x : Im ( ˜ E ) ˜ E ˜ F ˜ ∇ x ˜ F := ∇ x F Proof. It is a direct consequence of Cor. 6.3, Thm. 6.5 & Def. 6.4. Remark . If E is sublinear (instead of linear), then ˜ E and Im ( ˜ E ) are isomet-rically isomorphic normed vector spaces.We conclude this article with a discussion on the relation between class M functionals and harmonic functionals and illustrate the universal nature of class M as canonical solutions to path-dependent heat equations. Let Σ be a rightcontinuous function on Λ taking values in positive-definite symmetric m × m matrices and Ω Σ := { x ∈ Ω | d [ x ] = Σ dt } ⊂ Ω . efinition 6.9. F ∈ C , (Λ) is called Σ -harmonic if D F ( t, x t ) + 12 h∇ x F ( t, x t ) , Σ( t, x t ) i = 0 (27)for all t ≥ and x ∈ Ω Σ .If F is Σ -harmonic, then Itô ’s formula gives F ( t, x t ) = F (0 , x ) + Z t ∇ x F ( s, x s − ) dx (28)for all t ≥ and x ∈ Ω Σ . Equality in (28) then holds up to Ω Σ but not theentire Ω . Every functional of class M satisfies (27), hence is Σ -harmonic for all Σ . Theorem 6.10. If F is Σ -harmonic, then there exists a class M functional M such that M | Ω Σ ≡ F. In particular, such M is determined uniquely by (28) up to Ω Σ .Proof. Let F ∈ C , (Λ) be Σ -harmonic, we can define a new functional on Λ M ( t, x t ) := F (0 , x ) + Z t ∇ x F ( s, x s − ) dx. (29)By Lem. 6.2(i), we see that M is of class M and ∇ x M = ( ∇ x F ) − . By (28) and(29), the proof is complete. For every α ∈ R + , define w α ( t ) := 1 I [ α, ∞ ) ( t ) ∈ D =: Ω ,where D denotes the Skorokhod space. We assign to the collection ( w α ) α ∈ R + ,a normalized Lebesgue measure P ( { w α | α ∈ A } ) := X n ≥ λ ( A ∩ [0 , n ])2 n +1 , then P ( { w α | α ∈ R + } )) = 1 and X t ( w ) := w ( t ) is a finite variation process(i.e. a semi-martingale) under P . Since lim inf n π n is countable, it follows that P ( { w α | α ∈ lim inf n π n } ) = 0 for all π and therefore P ( { ω ∈ Ω | X · ( ω ) ∈ QV π } ) =0 , for all π . Proof of Thm. 5.7 & 5.15. By the right continuity of F (Def. 3.10.2(d)), wehave F ( T, x T ) − F (0 , x ) = lim n X π n ∋ t i ≤ T F ( t i +1 , x nt i +1 − ) − F ( t i , x nt i − ) , (30)29here for all n sufficiently large, we can decompose each F ( t i +1 , x nt i +1 − ) − F ( t i , x nt i − )= F ( t i +1 , x nt i +1 − ) − F ( t i , x nt i +1 − ) + F ( t i , x nt i +1 − ) − F ( t i , x nt i − )= (cid:0) F ( t i +1 , x nt i ) − F ( t i , x nt i ) (cid:1)| {z } time + (cid:0) F ( t i , x nt i ) − F ( t i , x nt i − ) (cid:1)| {z } space into the sum of a time and a space increment.Since F is left continuous and differentiable in time, D F is right continuousand locally bounded, by Lem. 5.1, each time increment F ( t i +1 , x nt i ) − F ( t i , x nt i ) = Z t i +1 t i D F ( t, x nt i ) dt. By Lem.5.2, we obtain lim n X π n ∋ t i ≤ T F ( t i +1 , x nt i ) − F ( t i , x nt i ) = Z T D F ( t, x t ) dt, which in light of (30), implies that the sum of the space increments convergesto lim n X π n ∋ t i ≤ T F ( t i , x nt i ) − F ( t i , x nt i − ) | {z } ∆ F ( t i ,x nti ) = F ( T, x T ) − F (0 , x ) − Z T D F ( t, x t ) dt. (31)If F is of class X , then ∇ x F is strictly causal and by Prop. 4.6, ∇ x F vanishes,we obtain F ( t i , x nt i ) − F ( t i , x nt i − ) = ∇ x F ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) by Taylor’s Theorem and Thm. 5.7 follows. If F is C , then, by Taylor’sTheorem, each space increment admits the following second order expansion ∆ F ( t i , x nt i ) = F (cid:0) t i , x nt i − + ∆ x n ( t i )1 I [ t i , ∞ ) (cid:1) − F ( t i , x nt i − )= ∇ x F ( t i , x nt i − ) · ∆ x n ( t i ) + 12 h∇ x F ( t i , x nt i − ) , ∆ x n ( t i )∆ x n ( t i ) ′ i + R nt i , (32)where ∆ x n ( t i ) = ( x ( t i +1 ) − x ( t i )) and R nt i = 12 h∇ x F ( t i , x nt i − + α ni ∆ x n ( t i )1 I [ t i , ∞ ) ) − ∇ x F ( t i , x nt i − ) , ∆ x n ( t i )∆ x n ( t i ) ′ i where α ni ∈ (0 , . Since x ∈ Ω ⊂ QV m , by Cor. 2.8 and Rem. 2.9 lim n X π n ∋ t i ≤ T h∇ x F ( t i , x nt i − ) , ∆ x n ( t i )∆ x n ( t i ) ′ i = Z T ∇ x F ( t, x t − ) d [ x ]= Z T ∇ x F ( t, x t − ) d [ x ] c + X t ≤ T h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i . (33)30et δ > , r := sup t ∈ [0 ,T ] | ∆ x ( t ) | , r δ := δ + sup t ∈ [0 ,T + δ ] | ∆ x ( t ) | . By a resulton càdlàg function [8, Lemma 8], we see that | ∆ x n ( t i ) | ≤ r δ for all n sufficientlylarge. By Rem. 3.2, we see that α ni ∆ x n ( t i ) ∈ U t i − ( x n ) ∩ B r δ (0) . Since ∇ x F admits a modulus, it follows from Def. 4.11 that there exists a modulus ofcontinuity ω such that | R nt i | ≤ ω ( r δ ) | ∆ x n ( t i )∆ x n ( t i ) ′ | for all n sufficiently large, hence by an application of Cor. 2.8(i), we obtain lim sup n X π n ∋ t i ≤ T | R nt i | ≤ ω ( r δ ) ≤ ω ( r δ ) tr ([ x ]( T )) . Send δ ↓ , and by the right continuity of x , we have established that lim sup n X π n ∋ t i ≤ T | R nt i | ≤ ω ( r +) tr ([ x ]( T )) , (34)which, in reminiscent of (31)-(34) and that ω (0+) = 0 , we remark here that theFunctional Itô ’s formula would have already been proven if x were continuous(i.e. r = 0 ). Let < ǫ < r , define the following finite sets on [0 , T ] J ( ǫ ) := { t ≤ T || ∆ x ( t ) | > ǫ } ,J n ( ǫ ) := { π n ∋ t i ≤ T |∃ t ∈ ( t i , t i +1 ] , | ∆ x ( t ) | > ǫ } . We can decompose X π n ∋ t i ≤ T R nt i = X t i ∈ J n ( ǫ ) R nt i + X t i ∈ ( J n ( ǫ )) c R nt i . (35)into two partial sums. By (32), the right continuity (resp. left continuity) of F (resp. ( ∇ x F ) − , ( ∇ x F ) − ) and that x is càdlàg we obtain X t i ∈ J n ( ǫ ) (cid:0) R nt i (cid:1) ± n −→ X t ∈ J ( ǫ ) (cid:18) ∆ F ( t, x t ) − ∇ x F ( t, x t − ) · ∆ x ( t ) − h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i (cid:19) ± ≤ ω ( r +) tr ([ x ]( T )) , (36)where the inequality follows from (34) and (35). Observe that J ( ǫ ) ↑ J (0) as ǫ ↓ , by monotone convergence, we obtain lim n X t i ∈ J n ( ǫ ) (cid:0) R nt i (cid:1) ± ǫ −→ X t ≤ T (cid:18) ∆ F ( t, x t ) − ∇ x F ( t, x t − ) · ∆ x ( t ) − h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i (cid:19) ± ≤ ω ( r +) tr ([ x ]( T )) . (37)31n the other hand, since w is monotonic, by (34) and (35), it follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim sup n X t i ∈ ( J n ( ǫ )) c R nt i − lim inf n X t i ∈ ( J n ( ǫ )) c R nt i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ω ( ǫ ) tr ([ x ]( T )) , (38)and by (31)-(33), (35),(36) and (38), so is (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim sup n X π n ∋ t i ≤ T ∇ x F nt i · ∆ x n ( t i ) − lim inf n X π n ∋ t i ≤ T ∇ x F nt i · ∆ x n ( t i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ω ( ǫ ) tr ([ x ]( T )) , where we have denoted ∇ x F nt i := ∇ x F ( t i , x nt i − ) . Send ǫ ↓ , we obtain Z T ∇ x F ( t, x t − ) dx := lim n X π n ∋ t i ≤ T ∇ x F ( t i , x nt i − ) · ( x ( t i +1 ) − x ( t i )) . (39)Upon a second look at (31)-(33), (35),(36) and in light of (39), we immedi-ately see that lim n X t i ∈ ( J n ( ǫ )) c R nt i =: o ( ǫ ) also exists and by (34), | o ( ǫ ) | ≤ ω ( ǫ ) tr ([ x ]( T )) ǫ −→ which, combines with(35) and (37) imply lim n X π n ∋ t i ≤ T R nt i = X t ≤ T (cid:18) ∆ F ( t, x t ) − ∇ x F ( t, x t − ) · ∆ x ( t ) − h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i (cid:19) . (40)In view of (31)-(33), (39) and (40), it remains to show that X t ≤ T (cid:18) ∆ F ( t, x t ) − ∇ x F ( t, x t − )∆ x ( t ) − h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i (cid:19) = X t ≤ T (cid:18) ∆ F ( t, x t ) − ∇ x F ( t, x t − )∆ x ( t ) (cid:19) − X t ≤ T h∇ x F ( t, x t − ) , ∆ x ( t )∆ x ( t ) ′ i , (41)and that the series are absolute convergent.Since ( ∇ x F ) − is left continuous and locally bounded, we see from Lem. 4.10(ii)that the map t x F ( t, x t − ) is also bounded on [0 , T ] , hence by (3) X t ≤ T |∇ x F ( t, x t − ) || ∆ x ( t )∆ x ( t ) ′ | ≤ const X i X t ≤ T (∆ x i ( t )) ≤ const · tr ([ x ]( T )) , which, combines with (37) imply (41) and that the series are absolute conver-gent, hence Theorem 5.15 is proven. 32 roof of Thm. 5.14. Since D M vanishes, by Lem. 5.1 we obtain M ( t, x t ) = M ( t, x t ) + Z Tt D M ( s, x t ) ds = M ( T, x t ) ≥ (42)for all t ≤ T , where the last inequality is due to x t ∈ Ω . Suppose there exists z ∈ Ω ; M ( T, z T ) > . By Thm. 5.7 and the continuity of M , it follows M ( T, z nT ) = X π n ∋ t i ≤ T ∇ x M ( t i , z nt i − )( z ( t i +1 ) − z ( t i )) > (43)for all n sufficiently large. Define t ∗ n := min { t i ∈ π n | M ( t i , z nt i ) > } , then t ∗ n ≤ T . By (42), (43), the left continuity of M and the fact that z n ∈ Ω , weobtain M ( t ∗ n , z nt ∗ n ) > M ( t ∗ n , z nt ∗ n − ) = 0 , hence M ( t ∗ n , z nt ∗ n ) = ∇ x M ( t ∗ n , z nt ∗ n − )∆ z ( t ∗ n ) > . Def. 3.1(ii) implies that thereexists ǫ > ; z ∗ := z nt ∗ n − − ǫ ∆ z ( t ∗ n )1 I [ t ∗ n , ∞ ) ∈ Ω , hence M ( t ∗ n , z ∗ t ∗ n ) = ∇ x M ( t ∗ n , z nt ∗ n − )( − ǫ ∆ z ( t ∗ n )) < , which is a contradiction to(42). References [1] Föllmer , H. (1981) Calcul d’Ito sans probabilitiés . Séminaire de probabilités(Strasbourg), 15:143-150.[2] Dotsko, M.W., Gosser, R.A. (1986) Stronger Versions of the FundamentalTheorem of Calculus . The American Mathematical Monthly, 93(4):294-296.[3] Doob, J.L. (1994) Measure Theory . 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