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
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
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 ,