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

we define

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 ,
*

**Proof:** See Lemma 2.2.2 in the article by Lyons or this proof for the sharp constant

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 ,
*

**Proof:**

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.

**Proof:**

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 ,

Therefore,

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

There is a misprint in the definition of the p-variation of iterated integral.

Could you please clarify, how you use Holder inequality in the part of proof the main theorem, that corresponds to the statement that smth is control? I believe, you want to prove that , however I do not understand how it follows from Holder inequality. Thanks!

Thanks. This inequality actually follows from the reverse Minkowski inequality, where and .

First of all, thanks a lot for this lecture, it’s helping me quite a bit.

Can you please verify, that in the definition of the p-variation the running index of the sum on the right hand side should be i instead of k?

\left\| \int dx^{\otimes k}\right\|_{p-var, [s,t]}=\left( \sup_{ \Pi \in \mathcal{D}[s,t]} \sum_{k=0}^{n-1} \left\| \int_{\Delta^k [t_i,t_{i+1}]} dx^{\otimes k} \right\|^p \right)^{1/p},

Thanks. This is now corrected.