On isomorphism of the space of -Hölder continuous functions with finite -th variation.
Abstract
We study the concept of (generalized) -th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the -th variation of a given function is closely related to the finiteness of -norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to -norm of the Schauder coefficients. This result provides an isomorphism between the space of -Hölder continuous functions with finite (generalized) -th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski.
Keywords— -th variation, Hölder regularity, Ciesielski’s isomorphism, Schauder basis, Variation index, Refining partition sequences
1 Introduction
In the seminal paper [14], Föllmer derived the pathwise Itô’s formula for a class of real functions with a finite quadratic variation. In particular, for a twice differentiable function and a one-dimensional continuous function with finite quadratic variation along a partition sequence , the pathwise Itô formula is given as
| (1.1) |
Here, the first integral is defined as a left Riemann sum
and the integrator of the second integral is the quadratic variation of along the partition sequence , defined as the following uniform limit in :
| (1.2) |
This pathwise Itô’s formula has been generalized in several aspects [1, 4, 9, 11, 12, 17, 22]. Among these, Cont and Perkowski [11] defined the notion of -th variation of continuous functions along by raising the exponent in (1.2) to any even integers , and derived high-order pathwise change-of-variable formula; more recently, Cont and Jin [10] developed fractional pathwise Itô formula for functions with -th variation for any , with a fractional Itô remainder term. These pathwise calculus formulae, including Föllmer’s original one (1.1), require the continuous function to have finite -th variation along . In other words, the existence of the limit
| (1.3) |
is the crucial assumption when applying these formulae. It is then natural to study a class of functions such that the limit (1.3) exists for a fixed partition sequence and .
In this regard, Schied [21] showed that the space is not a vector space by constructing an example of two continuous functions and on such that and exist, but does not exist, along the dyadic partition sequence with . These two functions and belong to a class of so-called generalized Takagi functions, constructed via the Schauder representation of continuous functions. From the Schauder representation of and along , one can obtain explicit expressions of both terms in the following strict inequality to show that does not exist:
Since Schied’s example implies that requiring the existence of the limit (1.3) restricts the function space too much, in this paper we study a larger space of functions that satisfy
| (1.4) |
but does not require the limit to exist. With an appropriate norm, we prove that the space is a Banach space (see definition (2.7) and Proposition 2.5 below).
Even though we may not apply the aforementioned pathwise change-of-variable formulae to every function in , we shall study the Banach space , instead of , because the notion of variation index, i.e., the infimum number such that the condition (1.4) holds (see Definition 2.3 below), can be used for measuring ‘roughness’ of a given function (or a path of a stochastic process) [2, 6]. It is well known that (almost every path of) a fractional Brownian motion (fBM) with Hurst index , has Hölder exponent equal to , whereas its variation index along ‘reasonable’ partition sequences (e.g., dyadic partition sequence ) is equal to . These facts are closely related to the self-similarity property of fBMs, but it is generally not true for general continuous functions that the reciprocal of the variation index is equal to (the supremum of) Hölder exponent. In a recent work [2], a specific example of -Hölder continuous function with variation index along the dyadic partition sequence equal to is constructed, thus, the variation index should be considered as an alternative way of measuring function’s roughness.
With the help of Schauder representation along a general class of partition sequences, our main result provides a necessary and sufficient condition for elements of the Banach space , in terms of their Schauder coefficients (see Theorem 4.3). More specifically, the condition (1.4) is equivalent to the -finiteness of the sequence composed of -norm of Schauder coefficients of functions along each partition , scaled by a -power of the mesh size of .
When the Schauder coefficients of functions are arranged in an infinite dimensional matrix, this result gives rise to an isomorphism between the space of -Hölder continuous functions with finite (generalized) -th variation along a partition sequence and a subspace of infinite-dimensional matrices with an appropriate matrix norm (see Theorem 5.3). Our isomorphism result reminds that of Ciesielski’s in 1960 [5], between the space of -Hölder continuous functions and the space of bounded real sequences, using Schauder representation along the dyadic partition sequence , which has been generalized recently by [2] along a wider class of partition sequences.
Preview: This paper is organized as follows. Section 2 introduces the notion of variation index and defines the Banach space . Section 3 provides some notations and reviews preliminary results regarding Schauder representation of continuous functions. Section 4 states and proves our main result, the characterization of generalized -th variation in terms of a function’s Schauder coefficients. Section 5 includes the isomorphism, as an important consequence of the result. Finally, Appendix A provides an explicit expression of the -th variation in terms of Schauder coefficients, for a limited case of even integers along the dyadic partition sequence, which is of independent interest.
2 Variation index and the Banach space
2.1 -th variation and variation index
First, we introduce some relevant notations and definitions for partition sequences. For a fixed , we shall consider a (deterministic) sequence of partitions of
where we denote the number of intervals in the partition . By convention, . For example, the dyadic partition sequence, denoted by , contains partition points for , .
Definition 2.1 (Refining sequence of partitions).
A sequence of partitions is said to be refining (or nested), if implies for every . In particular, we have .
For a partition sequence , we write
| (2.1) |
the size of the smallest and the largest interval of , respectively. In the following, we denote the collection of all refining partition sequences of with vanishing mesh, i.e., as .
Let us denote the space of real-valued continuous functions defined on . In this subsection, we fix a partition sequence and . For , we denote
| (2.2) |
the -th variation of along a partition for each level .
Remark 2.2.
If there exists a continuous, non-decreasing function such that
| (2.3) |
then we say admits finite -th variation along , and the above convergence is uniform in ([11, Definition 1.1 and Lemma 1.3]). We write the space of such functions admitting finite -th variation along . In the particular case of (then is often denoted as ) and given as the dyadic partition sequence , it is shown in [21, Proposition 2.7] that is not a vector space.
Even though the -th variation of along a given sequence defined in Remark 2.2 may not exist, one can always define its variation index along as the following.
Definition 2.3 (Variation index along a partition sequence, Definition 2.3 of [6]).
The variation index of along is defined as
| (2.4) |
Thanks to the continuity of , it is straightforward to show
| (2.5) |
Therefore, the definition (2.4) can be formulated as
Moreover, since if and only if , we also have
| (2.6) |
Now that the quantity in (2.2) can be recognized as the -th power of -norm of the real sequence , we provide the following definition.
Definition 2.4.
For , , and , we denote
and consider the subspace of :
| (2.7) |
We say is the class of continuous functions with finite (generalized) -th variation along .
The space turns out to be a Banach space, in contrast to the space .
Proposition 2.5.
The mapping is a norm, and the space is a Banach space.
Proof.
We first prove that the mapping is a norm. For any scalar , the identity is straightforward. Thanks to Minkowski’s inequality, it is also easy to prove the subadditive property (triangle inequality). These imply, in particular, that is a vector space. Finally, if , then has zero value on every partition point of for all . Since as , the set of all partition points of is dense in , and the continuity of with concludes . This shows that is a norm.
To prove the space is a Banach space, we fix a Cauchy sequence of , i.e., for any , there exists such that for all . In particular, for every , we have and
| (2.8) |
holds for each . Since is a real Cauchy sequence, its limit exists. Moreover, we fix an arbitrary , then for all indices such that belongs to , we have
for every , in other words, is a Cauchy sequence in for each . Again by the completeness of , the limit exists for each index and .
Let us recall the set of all partition points of , and define a function on
Since is a dense subset of and a function defined on a dense set can be extended to a continuous function, there exists such that holds for all points of . Furthermore, we have as well as
thus for each .
In line with Proposition 2.5, it is well-known that the space of -Hölder continuous functions, is also a Banach space. We next note the inclusion
| (2.10) |
due to the straightforward inequality for every . We conclude this subsection with the following property that adding a function with vanishing -th variation does not affect the variation index.
Lemma 2.6.
For , , , and , suppose that
holds. Then, we have
therefore . In particular, the identity holds, provided that the limit exists in the sense of Remark 2.2.
Proof.
Applying Minkowski’s inequality twice yields
Taking or respectively gives the result. ∎
2.2 Variation index along different partition sequences
A continuous function can have different -th variations, and , along two different refining partition sequences and . In this subsection, we study the variation index of along different partition sequences. We first introduce Proposition 2.8, inspired by Freedman [15], whose proof needs a preliminary result.
Lemma 2.7.
For any given numbers , , and , there exists a finite set in such that the -th variation of along is less than , i.e.,
Proof.
If , then we just take . Thus, we suppose that ; the other case can be handled by applying the same argument to .
We assume without loss of generality that , , and . For given and , we choose large enough so that , and define for . Let if , or otherwise. Now it is simple to check . ∎
Proposition 2.8.
For any , we have
Proof.
Let us fix . For any , we shall show that there exists a sequence satisfying
| (2.11) |
Then, the identity (2.11), together with (2.5), implies that for any there exists satisfying , which in turn proves the result.
We choose a decreasing real sequence , and set . We shall inductively define for each . Suppose is defined, and let be a partition of satisfying and . Suppose that has points, dividing into subintervals. From Lemma 2.7, we construct a partition of with , such that for each pair of consecutive points of we have
where and is the -th variation along on the interval . Then, we obtain and , therefore, satisfies condition (2.11). ∎
On the other hand, the rough path theory asserts that an -Hölder continuous function has finite -variation, i.e., , with
where the supremum is taken over all partitions of . This implies that for a given refining partition sequence with vanishing mesh, the variation index of should be bounded above by the reciprocal of its Hölder exponent (see Lemma 4.3 of [2] for the proof), namely
We formalize the above arguments into the following theorem.
Theorem 2.9.
For any , we have
Moreover, for any , we have
| (2.12) |
This result implies that an -Hölder continuous function can have any variation index between and , along a given partition sequence . Moreover, the inclusion (2.10) shows that for any .
Example 1.
The inequality (2.12) can be strict. Consider the increasing function defined on , which is -Hölder continuous. The function has finite -variation along any partition sequence , thus , as it is an increasing function. ∎
Example 2.
A uniformly continuous function defined on
is not -Hölder continuous for any . However, it is a decreasing function on the compact support, thus of bounded variation. As in the previous example, for every , which implies the left-hand side of (2.12) for is . ∎
In what follows, we shall characterize conditions for to belong to the Banach space , in terms of the Schauder coefficients of along .
3 Schauder representation along a general class of partition sequences
In this section, we provide several definitions and preliminary results, mostly taken from [7, 8], regarding Schauder representation of continuous functions along a general class of partition sequences. This type of representation was originally introduced by Schauder [20]. After that, we shall provide our results in the next sections.
3.1 Properties of partition sequence
Let us recall Definition 2.1 and the notations (2.1). We introduce a subclass of refining sequence of partitions with a ‘finite branching’ property at every level .
Definition 3.1 (Finitely refining sequence of partitions).
A sequence of partitions in is said to be finitely refining, if there exists a positive integer such that the number of partition points of within any two consecutive partition points of is always bounded above by , irrespective of . In particular, we have .
The following definition provides a condition that the ratio of the biggest step size to the smallest step size at each level is bounded.
Definition 3.2 (Balanced sequence of partitions).
A sequence of partitions is said to be balanced, if there exists a constant such that
| (3.1) |
holds for every .
We now give two conditions of refining partition sequences involving the biggest step sizes of two consecutive levels.
Definition 3.3 (Complete refining sequence of partitions).
A finitely refining sequence of partitions is said to be complete refining, if there exist positive constants and such that
| (3.2) |
holds for every .
Definition 3.4 (Convergent refining sequence of partitions).
A complete refining sequence of partitions is said to be convergent refining, if the following limit exists:
| (3.3) |
3.2 Generalized Haar basis and Schauder representation
This subsection recalls some relevant definitions of generalized Haar and Schauder functions, which were introduced in [7].
Let us fix and denote . Since is refining, we have the following inequality for every
| (3.4) |
With the notation , we now define the generalized Haar basis associated with .
Definition 3.6 (Generalized Haar basis).
The generalized Haar basis associated with a finitely refining sequence of partitions is a collection of piecewise constant functions defined as follows:
| (3.5) |
We note that the function values of are chosen to satisfy and so that the collection is an orthonormal basis in . The Schauder functions are obtained by integrating the generalized Haar basis:
To further simplify the notations in what follows, we introduce
Definition 3.7 (Generalized Schauder function).
For every index of Definition 3.6, the following function is called generalized Schauder function associated with :
| (3.6) |
Note that generalized Schauder functions are continuous, triangle-shaped (and not differentiable) functions. The following result shows that any continuous function defined on admits a unique Schauder representation along a given partition sequence .
Proposition 3.8 (Theorem 3.8 of [7]).
Let be a finitely refining partition sequence of . Then, every continuous function has a unique Schauder representation along :
| (3.7) |
with a closed-form representation of the Schauder coefficient
| (3.8) |
Remark 3.9.
A family of Schauder functions in Definition 3.7 can be reordered as , such that for each the values of run from to after reordering. We shall frequently use this reordering to simplify the notation and denote the index set
| (3.9) |
for each . The corresponding Schauder coefficients in Proposition (3.8) can be reordered as for and in the same manner.
4 Characterization of variation index
In this section, we characterize the variation index of along , in terms of the Schauder coefficients introduced in Section 3.2. We recall the definition (2.2) of the -th variation, as well as Definitions 3.1-3.4.
Remark 4.1.
Any can be translated to with , by adding a linear function. For any , the -th variation of a linear function along any element of is zero, i.e., . Moreover, the subadditive property of the norm in Definition 2.4 implies if and only if . Since we are only interested in the conditions regarding the finiteness of -norm (or ), we shall assume without loss of generality in what follows. Then, the Schauder representation (3.7) of any becomes simpler:
| (4.1) |
The above triple sum can be expressed as a double sum after re-indexing as in Remark 3.9.
4.1 Results
We provide Proposition 4.2 and Theorem 4.3 below, and their proofs are given in the next subsection.
Proposition 4.2.
For any , , and a balanced, complete refining partition sequence of , we denote
| (4.2) |
Then, we have
| (4.3) |
For any balanced, complete refining partition sequence , Proposition 4.2 immediately provides the sufficient and necessary condition for to belong to the Banach space in (2.7), in terms of its Schauder coefficients through the sequence :
Moreover, it also yields the equivalent formulation of the variation index in (2.4):
| (4.4) |
Thus, the -finiteness of the sequence can provide useful path property of along any balanced, complete refining partition sequences, and each term contains the Schauder coefficients of up to level , namely . However, with nominal additional conditions on the partition sequence, we have a much simpler condition involving Schauder coefficients.
Theorem 4.3.
For any , , and a balanced, convergent refining partition sequence of , we denote
| (4.5) |
Then, we have
| (4.6) |
Thus, we also have
In the definition (4.5), the quantity only contains the Schauder coefficients of that belong to the -th level, for each . Theorem 4.3 also provides a similar equivalent formulation of the variation index in (2.4).
Corollary 4.4.
Let be a balanced, convergent refining partition sequence. Then, we have
| (4.7) |
Remark 4.5.
In all of the previous results, we considered the (generalized) -th variation up to the terminal time . However, we can derive similar results for any partition points . For , let us recall the definition (1.3) of such that the mapping is nondecreasing. We also introduce the notations
| (4.8) | ||||
| (4.9) |
Then, the results (4.3) and (4.6) can be replaced by
| (4.10) | ||||
| (4.11) |
To show (4.10) and (4.11), we first define a ‘stopped function’ for . Furthermore, we define
and
For , the two functions and differ only by a finite sum of piecewise linear functions, say , which hence satisfies for every . Lemma 2.6 therefore yields that . Now applying Proposition 4.2 and Theorem 4.3 to with the quantities (4.8) and (4.9), proves (4.10) and (4.11).
For , we can choose a point which is sufficiently close and bigger than , and check the finiteness of , or , to conclude the finiteness .
4.2 Proofs
Lemma 4.6.
Let and be real sequences such that , for every , and the limit exists. Then, we have the inequality
| (4.12) |
Proof of Lemma 4.6.
Taking to the both sides of the following identity
| (4.13) |
with the following properties for any real sequences proves the result:
| (4.14) | |||
∎
Lemma 4.7.
Let and be real sequences such that is strictly increasing and . Then, we have the following inequalities
| (4.15) |
Proof of Lemma 4.7.
The middle inequality is obvious. We shall show the last inequality; the first inequality then follows from (4.14). If the right-most term of (4.15) diverges to infinity, there is nothing to show. Thus, we assume
For any , there exists such that
holds for every . Fix an arbitrary integer greater than , and sum up the last inequalities for to obtain
Sending to infinity and using the fact yields the inequality
Since this should hold for any , we conclude that the last inequality of (4.15) holds. ∎
Lemma 4.8.
Let be an infinite-dimensional matrix satisfying the following properties:
-
(i)
for every ;
-
(ii)
;
-
(iii)
.
Then, for any real sequence with nonnegative terms, i.e., for all , we have
| (4.16) |
Remark 4.9.
Proof of Lemma 4.8.
If , then there is nothing to prove; thus, we assume . This implies that there exists such that for all . We denote in condition (iii), and fix an arbitrary . Then, there exists such that
| (4.18) |
Condition (i) implies that there exist constants such that
Set , then
On the other hand, we have from (4.18)
Combining the last two inequalities,
| (4.19) |
We now claim that is an absolutely convergence sequence
thanks to condition (iii). Therefore, taking the limit as in (4.19), together with condition (ii), we conclude
Since is chosen arbitrarily, this proves the result. ∎
Proof of Proposition 4.2.
Using the Schauder representation (4.1), we expand the -th variation of along for each
| (4.20) | ||||
Since is finitely refining, for each fixed pair with and , the cardinality of the set has an upper bound . Also, in Definition 3.7, we note that
as is a length of an interval containing at most many consecutive intervals of , whereas is a length of a single interval of . From the balanced and complete refining property, we have
Thus, we have from (4.20)
We now set and expand the -th power to obtain
Here, the inequality follows from generalized Hölder inequality with . We further derive
Here, the second inequality uses the fact that for a fixed there are at most many partition points of sharing the same , and this number is bounded by due to the balanced condition Therefore, we obtain
| (4.21) | ||||
from the definition (4.2) (after re-indexing into as in Remark 3.9).
On the other hand, using the expression (3.8) of the Schauder coefficients, we obtain the following bound on the -th power of , thanks to the balanced condition
| (4.22) | ||||
Here, note that and are consecutive partition points of , but and may not be. Recalling the notations in (3.4), we use the telescoping sum
with the bound , and apply Jensen’s inequality to the right-hand side of (4.22) to obtain
We note that the quantities inside the last big parenthesis is the -th variation of along the partition points of that belong to the interval , and these intervals are disjoint for different values of . We now derive the following inequality
since the largest value can take is and the first -th power increment (which has been most repeatedly added) has been added at most many times.
Plugging the last expression into (4.2) with the complete refining property, we obtain
| (4.23) |
We now define an infinite-dimensional matrix with entries
and we shall show that the matrix satisfies properties (i) - (iii) of Lemma 4.8. First, condition (i) is obvious. In order to show (ii), we use the geometric series to derive
Condition (iii) is also obvious from (ii); .
Proof of Theorem 4.3.
For fixed , and satisfying the conditions of Theorem 4.3, let us define
such that
Moreover, from the notation (4.2), we have
| (4.25) |
and the complete refining property provides the bounds
| (4.26) |
We further define
| (4.27) |
then, the limit exists, thanks to the convergent refining property of . Applying (4.12) of Lemma 4.6 with the bounds (4.26), (4.2) yields
This implies .
5 Isomorphism on
In this section, we shall use several function norms and matrix norms, thus we note that Table 1 at the end of this section lists all the norms with their definitions for the convenience of readers.
Recall the space of -Hölder continuous functions with the norm
| (5.1) |
Ciesielski [5] proved that the following mapping is an isomorphism between and the space of all bounded real sequences, equipped with the supremum norm :
Here, ’s are the Schauder coefficients of along the dyadic partition sequence , and the double-indexed set can be identified as a real sequence by flattening it. A recent work [2] extends this isomorphism to any balanced, complete refining partition sequence :
| (5.2) |
We may arrange each element of the sequence in a matrix without flattening it. Let us denote the space of infinite-dimensional matrices and fix a partition sequence of . For each , recall the index set of (3.9) corresponding to , and consider the subspace
| (5.3) |
composed of infinite-dimensional matrices whose -th row vector can take nonzero values only for the first components. We now construct a ‘Schauder coefficient matrix’ in to arrange the Schauder coefficients:
We also define a diagonal matrix with each -th entry equal to :
| (5.4) |
From this construction, we have the identity
| (5.5) |
where is the supremum norm for matrices; in the mapping of (5.2), the condition is then equivalent to .
We now restate the isomorphism in (5.2) along any balanced and complete refining partition sequence.
Proposition 5.1.
For any balanced, complete refining partition sequence and , the mapping
| (5.6) |
is an isomorphism, where
Moreover, we have the following bounds for the operator norms:
| (5.7) |
where and with the constants in Remark 3.5.
Proof of Proposition 5.1.
From [2, Theorem 3.4] and the identity (5.5), it is easy to show that the mapping is bijective. We note that the notation in the bounds [2, Equation (3.2)] represents the Hölder semi-norm ( in (5.1) of this paper).
The bound for operator norm is also straightforward from [2, Theorem 3.4] and (5.5):
The same theorem also yields the inequality
| (5.8) |
Furthermore, we can derive that
Here, the second inequality and the last inequality follow from [2, bound (2.4) and Lemma 3.2], respectively. Combining this with (5.8) yields the bound for . ∎
Let us fix and , and recall from Theorem 2.9 that belongs to for some . In what follows, we shall characterize such functions in terms of its Schauder coefficients.
We now fix and define a diagonal matrix in such that every -th entry is equal to :
| (5.9) |
With the matrix norm
| (5.10) |
we define
| (5.11) |
Recalling the definition (4.5), we obtain the identity from (5.11)
| (5.12) |
Therefore, the condition (4.6) of Theorem 4.3 is also equivalent to . We are now ready to provide the following results regarding the intersection space .
Proposition 5.2.
For any , , and , the space is a Banach space.
Proof of Proposition 5.2.
Since and are Banach spaces (Proposition 2.5), it is obvious that is a norm in the intersection space, and it is enough to show the completeness of . Fix any Cauchy sequence in -norm. Then, is also Cauchy in -norm, thus it has a limit such that as ; in particular, is a Cauchy sequence in , and as for each . Moreover, since is also a Cauchy sequence in -norm, there exists a limit such that as . As in the proof of Proposition 2.5, we have for every partition point of . In other words, and coincide on the dense set , thus the unique continuous extension of must be , thus converges to in -norm. ∎
In addition to Ciesielski’s isomorphism, we have the following isomorphism from the intersection space.
Theorem 5.3 (Isomorphism on the Banach space ).
For any , , and a balanced, convergent refining partition sequence , the mapping
| (5.13) |
is an isomorphism. Furthermore, we have the following bounds for the operator norms:
| (5.14) | ||||
| (5.15) |
Proof of Theorem 5.3.
We shall prove the result in the following parts.
Part 1: For any , we shall prove .
We fix . Proposition 5.1 proves , thus we need to show , which is equivalent to from (5.12).
Recalling the inequality (4.23) and computing the geometric series, we have for each
Furthermore, recalling the notations (4.25) and (4.27) with the identity (4.13), we derive
Here, the last inequality uses the fact that has an upper bound from the complete refining property.
Combining the last two inequalities, we obtain for each
| (5.16) |
Since , we have , which shows .
Part 2: For any , we shall prove .
We fix . Using the entries of as Schauder coefficients along , we can construct an -Hölder continuous function from Proposition 5.1. The identity (5.12) with Corollary 4.4 and (2.6) imply .
Part 3: We shall prove that the mapping is bounded.
For any , consider . From (5.12) and (5.16), we have
Moreover, from Proposition 5.1, we have . Combining the two bounds concludes (5.14).
Part 4: We shall prove that the inverse mapping is bounded.
For any , we write and consider its Schauder coefficients . Recalling the inequality (4.21) and the notation (4.5), we obtain for any
From the complete refining property and computing the geometric series, we have for each
Combining the last two inequalities,
Moreover, thanks to (5.12), we have
Also, Proposition 5.1 yields a bound . Combining these bounds proves (5.15). ∎
Remark 5.4.
From Proposition 5.1 and Theorem 5.3, one may expect that the following mapping would also be an isomorphism:
However, this is not an isomorphism, since is a subclass of continuous functions, and the continuity is not guaranteed without additional conditions if one constructs a function from Schauder coefficients. In the following, we provide an example of function constructed from a given Schauder matrix , satisfying the condition , but .
Let us consider the dyadic partition sequence on a unit interval and a matrix such that for each the components of -th row are given by and for all . Then, it is easy to verify that . We now construct a function on . It turns out that is not continuous at ; we take for each , then we have
thus , so .
| Function norm | Definition |
|---|---|
| in Definition (2.4) | |
| in (5.1) | |
| Matrix norm | Definition |
| where is the matrix defined in (5.4) | |
| in (5.10) | |
| where is the matrix defined in (5.9) |
In the following table, represents a (continuous) function defined on , and represents an infinite dimensional matrix.
Appendix A The case of even integers, , along the dyadic sequence
The concept of pathwise quadratic variation, that is, the limit in (1.3), was introduced in [14], and was extended in [11] to even integers . However, as mentioned earlier, the existence of the limit is a strong assumption, indicated by the fact that the class is not a vector space in general. Moreover, a closed-form formula of the -th variation is known only for the quadratic case (along the dyadic partition sequence [18] and along general finitely refining partition sequences [7]). In this appendix, we provide a generalized closed-form expression of the -th variation for even integers along the dyadic partition sequence, which can be of independent interest.
We first write the dyadic partition sequence as in the beginning of Section 2.1. From Propositions 4.1 and 4.4 of [7], the quadratic variation of along the -th dyadic partition has a simple expression in terms of its Faber-Schauder coefficients:
| (A.1) |
Here, the Schauder coefficients along the dyadic sequence are often called ‘Faber-Schauder’ coefficients, as Faber [13] earlier constructed a basis by integrating the orthonormal basis along the dyadic partitions introduced by Haar [16] in 1910.
This expression (A.1) can be generalized to any even integers along the dyadic partitions in the following.
Proposition A.1.
For a fixed , the -th variation of along the -th dyadic partition can be expressed as:
| (A.2) |
Proof of Proposition A.1.
We recall the identity (4.20) with the fact that for any dyadic partition there is a unique such that , to derive
| (A.3) |
where is the Haar basis associated with the Faber-Schauder function (Definition 3.6).
The coefficient of the -th power term for each pair is
Here, the number of indices of the set is equal to , and the absolute values for such ’s are all equal to .
In order to handle the coefficients of the cross-terms like in (A.3), we fix pairs , , such that at least one pair among the pairs is different, and consider the following two cases.
Case 1. Suppose that there exist two pairs with disjoint support, i.e., such that . Then, for any , thus the coefficient of the cross-term in this case is zero.
Case 2. The only remaining case is , after some re-numbering of the indices. This is because if we have two pairs such that but , then the supports of and should be disjoint, which is of Case 1. Thus, the values of should be all different. The coefficient of the cross-term in (A.3) is given by
where is the left-end point of the support of . Now, the values of take positive values for exactly half of the indices in the set ; for the remaining half of the indices of the set, take the same absolute, but negative values. Therefore, the last summation is zero.
Remark A.2.
For an odd integer , the argument in the proof of Proposition A.1 does not work in general, so we don’t expect such a simple expression of the -th variation in terms of Faber-Schauder coefficients. For an odd integer , the identity (A.3) becomes
After expanding the -th power inside the parenthesis, we can argue as before to conclude that the coefficients of the cross-terms of Case 1 still vanish. However, the -th power terms and Case 2 cross-terms don’t vanish, because the outermost summation and the absolute value symbol cannot be exchanged in the following equation.
Theorem A.3.
For in Theorem 4.3, has finite -th variation along , i.e., the limit exists, if and only if the limit exists as . In particular, we have the identity
| (A.4) |
Proof.
Lemma A.4 (Theorems 1.22, 1.23 of [19]).
Let and be real sequences such that is strictly monotone, divergent, and satisfies . Then, we have the following equivalence
| (A.5) |
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