Abstract
We develop a fractional calculus approach to rough path analysis, introduced by Y. Hu and D. Nualart [6], and show that our integration can be generalized so that it is consistent with the rough path integration introduced by M. Gubinelli [5].
Funding statement: This work was partially supported by JSPS Research Fellowships for Young Scientists.
A Appendix
In this appendix, we prove some statements used in Sections 2 and 3. Note that almost all statements follow from slight modifications of the proofs in the preceding studies [6, 8, 9, 14].
First, Proposition A.2 shows that
Lemma A.1 ([8, Lemma 3.1]).
For
For
Proposition A.2.
Let
Proof.
Take the real numbers s and t such that
Then, by using (A.1), we have
and, from the equality
Next, with regard to the second term of
Then,
and
By using the change of variables
and, from the equality
Therefore, by combining these estimates, we obtain the statement of the proposition. ∎
Corollary A.3.
Let
Proof.
We set
We next prove the following proposition used in the proof of Theorem 2.3.
Proposition A.4.
In the setting of Definition 2.1, for each
where the limits are taken over all finite partitions
Proof.
We prove only that (A.5) holds, since (A.3) and (A.6) follow from
[14, Theorem 4.1.1], and (A.4) follows from [9, Proposition 2.4].
The proof of (A.5) here is based on the proofs of [14, Theorem 4.1.1] and [9, Proposition 2.4]. For
where
for
holds for
where
for
Then, from the equality
where we use that
and so
Therefore, we have
where C is a positive constant that does not depend on
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[2]
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