Operations on controlled rough paths

Abstract

At first sight, the notation $\int {Y\;dX}$ introduced in Chapter 4 is ambiguous since the resulting controlled rough path depends in general on the choices of both the second-order process ${\mathbb{X}}$ and the derivative process Y′. Fortunately, this “lack of completeness” in our notations is mitigated by the fact that in virtually all situations of interest, Y is constructed by using a small number of elementary operations described in this chapter. For all of these operations, it turns out to be intuitively rather clear how the corresponding derivative process is constructed.