Semimartingales and Decomposable Processes

Abstract

In Chap. II we defined a semimartingale as a good integrator and we developed a theory of stochastic integration for integrands in L, the space of adapted processes with left continuous, right-limited paths. Such a space of integrands suffices to establish a change of variables formula (or “Itô’s formula”), and it also suffices for many applications, such as the study of stochastic differential equations. Nevertheless the space L is not general enough for the consideration of such important topics as local times and martingale representation theorems. We need a space of integrands analogous to measurable functions in the theory of Lebesgue integration. Thus defining an integral as a limit of sums—which requires a degree of smoothness on the sample paths—is inadequate. In this chapter we lay the groundwork necessary for an extension of our space of integrands, and the stochastic integral is then extended in Chap. IV.