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

Let . Since

is a control, the estimate

easily implies that for ,

We stress that it does not imply a bound on the 1-variation of the path . What we can get for this path, are bounds in -variation:

**Proposition:*** Let . There exists a constant , depending only on , such that for every and ,*

where

**Proof:** This is an easy consequence of the Chen’s relations. Indeed,

and we conclude with the binomial inequality

We are now ready for a second major estimate which is the key to define iterated integrals of a path with -bounded variation when .

**Theorem:*** Let , and such that*

and

Then there exists a constant depending only on and such that for and

where is the control

**Proof:** We prove by induction on that for some constants ,

For , we trivially have

and

.

Not let us assume that the result is true for with . Let

From the Chen’s relations, for ,

Therefore, from the binomial inequality

where

We deduce

with . A correct choice of finishes the induction argument