Based on the results of the previous Lecture, it should come as no surprise that differential equations driven by the Brownian rough path should correspond to Stratonovitch differential equations. In this Lecture, we prove that it is indeed the case. Let us first remind to the reader the following basic result about existence and uniqueness for solutions of stochastic differential equations.

Let be a -dimensional Brownian motion defined on some filtered probability space that satisfies the usual conditions.

**Theorem:** * Assume that are vector fields with bounded derivatives up to order 2. Let . On , there exists a unique continuous and adapted process such that for ,
*

Thanks to Ito’s formula the corresponding Ito’s formulation is

where for , is the vector field given by

The main result of the Lecture is the following:

**Theorem:** *Let and let be -Lipschitz vector fields on . Let . The solution of the rough differential equation
is the solution of the Stratonovitch differential equation:
*

**Proof:** Let us work on a fixed interval and consider a sequence of subdivisions of such that and whose mesh goes to 0 when . As in the previous lectures, we denote by the piecewise linear process which is obtained from by interpolation along the subdivision , that is for ,

Let us then consider the process that solves the equation

and the process , which is piecewise linear and such that

We can write

Now,

From Davie’s estimate, we have, with ,

We deduce that, almost surely when ,

On the other hand,

We deduce that in probability,

We conclude that in probability,

Up to an extraction of subsequence, we can assume that almost surely

We now know that from the Lyons’ continuity theorem, almost surely where is the solution of the rough differential equation

Thus almost surely, we have that . On the othe hand, by definition, we have

which easily implies that converges in probability to . This proves that