In this lecture we 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