Since a -dimensional Brownian motion is a -rough path for , we know how to give a sense to the signature of the Brownian motion.

In particular, the iterated integrals at any order of the Brownian motion are well defined using rough path theory. It turns out that these iterated integrals do not coincide with iterated Ito’s integrals but with iterated Stratonovitch integrals.

We start with some reminders about Stratonovitch integration. Let be a one dimensional Brownian motion defined on a filtered probability space . Let be a adapted process such that . The Stratonovitch integral of against can be defined as the limit in probability of the sums

where is a sequence of subdivisions whose mesh goes to 0. This limit is denoted and does not depend on the choice of the subdivision. It is an easy exercise to see that the relation between Ito’s integral and Stratonovitch’s is given by:

where is the quadratic covariation between and .

If is dimensional Brownian motion, we can then inductively define the iterated Stratonovitch integrals . The next theorem proves that the signature of the Brownian rough path is given by multiple Stratonovitch integrals.

**Theorem:*** If is a -dimensional Brownian motion, the signature of as a rough path is the formal series:
*

**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 lecture, we denote by the piecewise linear process which is obtained from by interpolation along the subdivision , that is for ,

We know from the previous lecture that converges to in the -rough paths topology . In particular all the iterated integrals converge. We claim that actually,

Let us denote

We are going to prove by induction on that . We have

By taking the limit when , we deduce therefore that . In the same way, we have for , . Assume now by induction, that for every and , . Let us denote

From the Chen’s relations, we immediately see that

Moreover, it is easy to estimate

where and , being the lift of in the free Carnot group of step 2. Indeed, the bound

comes from the continuity of Lyons' lift and the bound

easily comes from the Garsia-Rodemich-Rumsey inequality. As a conclusion, we deduce that which proves the induction

We finish this lecture by a very interesting probabilistic object, the expectation of the Brownian signature.

If

is a random series, that is if the coefficients are real random variables defined on a probability space, we will denote

as soon as the coefficients of are integrable, where stands for the expectation.

**Theorem:*** For ,
*

**Proof:**

An easy computation shows that if is the set of words with length obtained by all the possible concatenations of the words , then, if then

and if then

Therefore,