On pathwise quadratic variation for cadlag functions

We revisit H. Foellmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition of quadratic variation which implies the Lebesgue decomposition as a result, rather than requiring it as an extra condition.


Quadratic variation along a sequence of partitions
In his seminal paper Calcul d'Itô sans probabilités [14], Hans Föllmer introduced a pathwise concept of quadratic variation and used it to provide a pathwise proof of the Itô formula. Föllmer showed that if a function x ∈ D([0, T ], R) has quadratic variation along a sequence π n = (t n 0 = 0 < .. < t n j < ... < t n m(n) = T ) of time partitions of [0, T ] in the sense that for each t ∈ [0, T ] the limit converges weakly to a Radon measure µ π such that [x] c π : t → µ π ([0, t]) − s≤t |∆x(s)| 2 is a continuous increasing function, (L) then a pathwise Itô formula may be obtained for functions of x [14]: for any f ∈ C 2 (R, R), where t 0 f ′ (x)d π x is a pathwise integral defined as a limit of left Riemann sums computed along the partition: g(x(t n j )). x(t n j+1 ) − x(t n j ) .
The quantity is called the quadratic variation of x along π. This result has many interesting applications and has been extended to less regular functions [2,8,11,19] and path-dependent functionals [1,4,5,6,8]. With the exception of [6,22,18], these extensions have focused on continuous paths. Föllmer's definition [14] contains the condition (L) on the Lebesgue decomposition of the limit µ π : the atoms of µ should correspond exactly to the jumps of x and their mass should be |∆x(t)| 2 or, equivalently, the discontinuity points of [x] π should coincide with those of x, with ∆[x] π (t) = |∆x(t)| 2 . This condition can not be removed: as shown by Coquet et al. [9], there are counterexamples of continuous functions x such that (1) converges to a limit with atoms. Conversely, one can give examples of discontinuous functions for which (1) converges to an atomless measure. If this condition is not satisfied, then the pathwise integral (3) fails to satisfy at each t where condition (L) is not met. On the other hand, this condition (L) is not easy to check and seems to require a link between the path x and the sequence of partitions π, making it difficult to apply to sample paths of stochastic processes.
In this work, we revisit Föllmer's concept of pathwise quadratic variation along a sequence of partitions and show that it has hitherto unsuspected links with the Skorokhod topology. In particular, we show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of the quadratic variation as a limit, in Skorokhod topology, of discrete approximations defined along the partition. This leads to a simpler definition of pathwise quadratic variation which holds in any dimension and, rather than requiring the Lebesgue decomposition of the pathwise quadratic variation as an extra condition, yields it as a consequence.
Outline We begin by recalling Föllmer's definition of pathwise quadratic variation and variations of it which have been used in the literature. We then introduce a new definition of quadratic variation for real-valued càdlàg functions based on the Skorokhod topology and prove equivalence among the various definitions. Section 3 extends the results to vector-valued functions: we show that, unlike Föllmer's original definition in which the one dimensional case plays a special role, our definition applies regardless of dimension, thus simplifying various statements regarding quadratic variation for vector-valued functions. Finally, in Section 4, we show that our approach leads to simple proofs for various properties of pathwise quadratic variation.
Proof. Let f ∈ C K ([0, ∞)) be a compactly supported continuous function. Since we may take f ≥ 0 and define the following extensions: Since v n → v vaguely and T / ∈ J, thus, as n → ∞ we obtain Proposition 2.1. If x ∈ Q π 0 , then the pointwise limit s of exists, s = [x] π and s admits the Lebesgue decomposition: Proof.
the distribution function of µ n in (4 Thus, we may take any subsequence (n k ) such that lim k q n k (t) =: q(t). Since x ∈ Q π 0 and the Lebesgue decomposition If t ± ǫ ∈ I,π k := π n k and t (k) j by the fact that x is càdlàg and that t / ∈ I. We see from (8) Since the choice of the convergent subsequence is arbitrary, we conclude that q n → [x] pointwise on [0, ∞). Observe that the pointwise limits of (s n ) and (q n ) coincide i.e.
converges to 0 by the right-continuity of x, where t Denote Q π 1 the set of x ∈ D such that (s n ) defined in (6) has a pointwise limit s with Lebesgue decomposition given by (7). Then Q π 0 ⊂ Q π 1 and we have: exists. Furthermore q = s and admits the Lebesgue decomposition: Proof. Since the pointwise limits of (s n ) and (q n ) coincide by (9). Prop. 2.2 now follows from x ∈ Q π 1 .
Denote Q π 2 the set of x ∈ D such that the quadratic sums (q n ) defined in (10) have a pointwise limit q with Lebesgue decomposition (11). Then Q π 0 ⊂ Q π 1 ⊂ Q π 2 and we have: Proof. Since x ∈ Q π 2 , we have q n → q pointwise on [0, ∞) and that (q n ), q are elements in D + 0 . By [16,Thm.VI.2.15], it remains to show that Since x is càdlàg and that |π n | ↓ 0 on compacts, the first sum in (12) converges to J ǫ ∋s≤t (∆x s ) 4 and the second sum in (12) By the Lebesgue decomposition (11), we observe J ǫ ∋s≤t (∆x s ) 4 ≤ q(t) 2 and that lim n s≤t as ǫ → 0.
Theorem 2.1. Let • Q π 0 be the set of x ∈ D satisfying Definition 2.1.
• Q π 1 the set of x ∈ D such that (s n ) defined in (6) has a pointwise limit s with Lebesgue decomposition given by (7).
• Q π 2 the set of x ∈ D such that the quadratic sums (q n ) defined by (9) have a pointwise limit q with Lebesgue decomposition (10).
• Q π the set of càdlàg functions x ∈ D such that the limitq of (q n ) exists in (D, d). Then: (iii) x has finite quadratic variation along π if and only if converges in (D, d).
(iv) If (q n ) defined by (9) converges in (D, d), the limit is equal to [x]. Corollary 2.1. Let x ∈ Q π , s n defined as in (6), (q n ) defined by (10). by uniform convergence. Put t ′ n := max{t i < t|t i ∈ π n }, since q n → [x] in the Skorokhod topology, we also have by (13) 1(b)] implies t ′ n must coincide with t for all n large enough, but t ′ n < t for all n, hence ∆[x](t) = 0 which implies ∆x(t) = 0 by Prop. 2.4 and (5). Since t is arbitrary, we conclude that x ∈ C.
Remark 2.1. The converse of (ii) is not true in general.
Remark 2.2. We note that some references have used the pointwise limit of the sequence together with the Lebesgue decomposition (5), to define [x]. To see why this is not the correct choice, take t 0 / ∈ π, put x(t) := 1I [t0,∞) (t) then obviously [x](t 0 ) = lim s n (t 0 ) = lim q n (t 0 ) = 1 but lim p n (t 0 ) = 0. This requirement of exhausting all jumps can always be satisfied for a given càdlàg path by adding all discontinuity points to the sequence of partitions. However if one is interested in applying this definition to a process, say a semimartingale, then in general there may exist no sequence of partitions satisfying this condition. And, if this requirement of exhausting all discontinuity points is removed, then Vovk's definition differs from Föllmer's (and therefore, fails to satisfy the Ito formula (2) in general).
By contrast, our definition does not require such a condition and easily carries over to stochastic processes without requiring the use of random partitions (see Theorem 4.1). It is known that x, y ∈ Q π does not imply x + y ∈ Q π [7, 20] so one cannot for instance define a quadratic covariation [x, y] π of two such functions in the obvious way. This prevents a simple componentwise definition of the finite quadratic variation property for vector-valued functions. Therefore, the notion of quadratic variation in the multidimensional setting was originally defined in [14] as follows: Föllmer1981). We say that x := (x 1 , . . . , x m ) T ∈ D m has finite quadratic variation along π if all x i , x i + x j (1 ≤ i, j ≤ m) have finite quadratic variation.

Quadratic variation for multidimensional functions
The quadratic (co)variation [x i , x j ] is then defined as which admits the following Lebesgue decomposition: The function [x] π := ([x i , x j ]) 1≤i,j≤m , which takes values in the cone of symmetricsemidefinite positive matrices, is called the quadratic (co)variation of x.
Note Def. 3.1 requires first introducing the case m = 1. The following definition, by contrast, avoids this and directly defines the concept of multidimensional quadratic variation in any dimension: We shall now prove the equivalence of these definitions. Define, for u, v, w ∈ D and q Note that the sequence t n is chosen from the partition points of π n , independently of u, v ∈ D.

Some applications
We now show that our approach yields simple proofs for some properties of pathwise quadratic variation, which turn out to be useful in the study of pathwise approaches to Ito calculus. Denote D := D([0, ∞), R) and D d×d := D([0, ∞), R d×d ) to be the Skorokhod spaces, each of which equipped with a complete metric δ which induces the corresponding Skorokhod (a.k.a J 1 ) topology. Denote F to be the J 1 Borel sigma algebra of D (a.k.a the canonical sigma algebra generated by coordinates). Recall that Q π 0 is the set of paths with finite quadratic variation along π in the sense of Def. 2.1.
We now give a criterion for x ∈ D to have finite quadratic variation without any reference to the Lebesgue decomposition (5) on the limit measure µ: 0 if and only if (q n ) defined by (9) is a Cauchy sequence in (D, δ).
Proof. This is a consequence of Thm. 2.1 and that (D, δ) is complete.
One of the main advantages of having convergence in the J 1 topology is that it ensures convergence of jumps in a regulated manner. It comes in handy when accessing the limit of q n (t n ) as n → ∞.
In particular, the sequence (t ′ n ) is asymptotically unique in the sense that any other sequence (t ′′ n ) meeting the above properties coincides with (t ′ n ) for n sufficiently large.
Proof. This is a consequence of Thm. Given a càdlàg process X (i.e. a (D, F )-measurable random variable), a natural quantity to consider is P(X ∈ Q π 0 ). This only makes sense however if Q π 0 is F -measurable. This 'natural' property, not easy to show using the original definition (Def. 2.1), becomes simple thanks to Theorem. 2.1: Property 3 (Measurability of Q π 0 ). Q π 0 is F -measurable.
Proof. By Thm. 2.1, Q π 0 = Q π and by definition, Q π is the J 1 convergence set of n ≥ 1 on D. Since D is completely metrisable, the claim follows from [12, V.3].
The pathwise Itô formula (2) can be applied to this class of processes, which is strictly larger than the class of semimartingales [10].
Theorem 4.1. Let X be an R d -valued càdlàg process, define a sequence of (D d×d , δ)-valued random variables (q n ) by q n (t) := πn∋ti≤t (X(t i+1 ) − X(t i ))(X(t i+1 ) − X(t i )) T then the following properties are equivalent: (i) X is a process with finite quadratic variation.
(iii) (q n ) is a Cauchy sequence in probability.
In addition, iv If (q n ) converges in probability, the limit is [X].
v The convergence of (q n ) to [X] is UCP if and only if X is a continuous process of quadratic variation [X].
vi (q n ) converges (resp. is a Cauchy sequence) in probability if and only if each component sequence of (q n ) converges (resp. is a Cauchy sequence) in probability.
Proof. We first remark that (D d×d , δ) is a complete separable metric space [16], hence by [