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.
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 ,
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