In this lecture, it is now time to harvest the fruits of the two previous lectures. This will allow us to finally define the notion of -rough path and to construct the signature of such path.

A first result which is a consequence of the theorem proved in the previous lecture is the following continuity of the iterated iterated integrals with respect to a convenient topology. The proof uses very similar arguments to the previous two lectures, so we let it as an exercise to the student.

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

and

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

This continuity result naturally leads to the following definition.

**Definition:*** Let and . We say that is a -rough path if there exists a sequence such that in -variation and such that for every , there exists such that for ,*

The space of -rough paths will be denoted .

From the very definition, is the closure of inside for the distance

If and is such that in -variation and such that for every , there exists such that for ,

then we define for as the limit of the iterated integrals . However it is important to observe that may then depend on the choice of the approximating sequence . Once the integrals are defined for , we can then use the previous theorem to construct all the iterated integrals for . It is then obvious that if , then

implies that

In other words the signature of a -rough path is completely determinated by its truncated signature at order :

For this reason, it is natural to present a -rough path by this truncated signature at order in order to stress that the choice of the approximating sequence to contruct the iterated integrals up to order has been made. This will be further explained in much more details when we will introduce the notion of geometric rough path over a rough path.

The following results are straightforward to obtain from the previous lectures by a limiting argument.

**Lemma:** * Let , . For , and ,*

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

If , the space is not a priori a Banach space (it is not a linear space) but it is a complete metric space for the distance

The structure of will be better understood in the next lectures, but let us remind that if , then is the closure of inside for the variation distance it is therefore what we denoted . As a corollary we deduce

**Proposition:*** Let . Then if and only if*

where is the set of subdivisions of . In particular, for ,

We are now ready to define solutions of linear differential equations driven by -rough paths, and present the Lyons’ continuity theorem in this setting. Let be a -rough path with truncated signature and let be an approximating sequence such that

Let us consider matrices . We have the following theorem:

**Theorem:** * Let be the solution of the differential equation*

Then, when , converges in the -variation distance to some . is called the solution of the rough differential equation

**Proof:** It is a classical result that the solution of the equation

can be expanded as the convergent Volterra series:

Therefore, in particular, for ,

which implies that

with . From the theorems of the previous lectures, there exists a constant depending only on and

such that for and big enough:

As a consequence, there exists a constant such that for big enough:

This already proves that converges in the supremum topology to some . We now have

and we can bound

Again, from the theorems of the previous lectures, there exists a constant , depending only on and

such that for and big enough

where is a control such that . Consequently, there is a constant , such that

This implies the estimate

and thus gives the conclusion

With just a little more work, it is possible to prove the following stronger result whose proof is let to the reader.

**Theorem:** * Let be the solution of the differential equation*

and be the solution of the rough differential equation:

Then, and when ,

We can get useful estimates for solutions of rough differential equations. For that, we need the following analysis lemma:

**Proposition:*** For and ,*

**Proof:** For , we denote

This is a special function called the Mittag-Leffler function. From the binomial inequality

Thus we proved

Iterating this inequality, times we obtain

It is known (and not difficult to prove) that

By letting we conclude

This estimate provides the following result:

**Proposition:** * Let be the solution of the rough differential equation:*

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

where .

**Proof:** We have

Thus we obtain

,

and we conclude by using estimates on iterated integrals of rough paths together with the previous lemma