Our next goal in this course is to define an integral that can be used to integrate rougher paths than bounded variation. As we are going to see, Young’s integration theory allows to define as soon as has finite -variation and and has a finite -variation with . This integral is simply is a limit of Riemann sums as for the Riemann-Stiletjes integral. In this lecture we present some basic properties of the space of continuous paths with a finite -variation. We present these results for valued paths but most of the results extend without difficulties to paths valued in metric spaces (see chapter 5 in the book by Friz-Victoir).
Definition. A path is said to be of finite -variation, if the -variation of on , which is defined as
is finite. The space of continuous paths with a finite -variation will be denoted by .
The notion of -variation is only interesting when .
Proposition: Let be a continuous path of finite -variation with . Then, is constant.
Proof: We have for ,
Since is continuous, it is also uniformly continuous on . By taking a sequence of subdivisions whose mesh tends to 0, we deduce then that
so that is constant
The following proposition is immediate:
Proposition: Let , be a continuous path. If then
As a consequence
We remind that a continuous map that vanishes on the diagonal is called a control f if for all ,
Proposition: Let . Then is a control such that for every ,
Proof: It is immediate that
so we focus on the proof that is a control. If and , then . As a consequence, we obtain
The proof of the continuity is left to the reader (see also Proposition 5.8 in the book by Friz-Victoir)
In the following sense, is the minimal control of a path .
Proposition: Let and let be a control such that for ,
Proof: We have
The next result shows that the set of continuous paths with bounded -variation is a Banach space.
Theorem: Let . The space endowed with the norm is a Banach space.
Proof: The proof is identical to the case , so we let the careful reader check the details
Again, the set of smooth paths is not dense in for the -variation convergence topology. The closure of the set of smooth paths in the -variation norm shall be denoted by . We have the following characterization of paths in .
Proposition: Let . if and only if
Proof: See Theorem 5.31 in the book by Friz-Victoir
The following corollary shall often be used in the sequel:
Corollary: If , then .
Proof: Let whose mesh is less than . We have
As a consequence, we obtain