In the previous lecture we introduced the signature of a bounded variation path as the formal series
If now , the iterated integrals can only be defined as Young integrals when . In this lecture, we are going to derive some estimates that allow to define the signature of some (not all) paths with a finite variation when . These estimates are due to Terry Lyons in his seminal paper and this is where the rough paths theory really begins.
For that can be writen as
It is quite easy to check that for
Let . For , we denote
where is the set of subdivisions of the interval . Observe that for , in general
Actually from the Chen’s relations we have
It follows that needs not to be the -variation of .
The first major result of rough paths theory is the following estimate:
Proposition: Let . There exists a constant , depending only on , such that for every and ,
By , we of course mean . Some remarks are in order before we prove the result. If , then the estimate becomes
which is immediately checked because
We can also observe that for , the estimate is easy to obtain because
So, all the work is to prove the estimate when . The proof is split into two lemmas. The first one is a binomial inequality which is actually quite difficult to prove:
Lemma: For , , and ,
The second one is a lemma that actually already was essentially proved in the Lecture on Young’s integral, but which was not explicitly stated.
Lemma: Let . Let us assume that:
- There exists a control such that
- There exists a control and such that for ,
Then, for all ,
See the proof of the Young-Loeve estimate or Lemma 6.2 in the book by Friz-Victoir
We can now turn to the proof of the main result.
Let us denote
We claim that is a control. Indeed for , we have from Holder’s inequality
It is clear that for some constant which is small enough, we have for ,
Let us now consider
From the Chen’s relations, for ,
On the other hand, we have
We deduce from the previous lemma that
with . The general case is dealt by induction. The details are let to the reader