Chapter 2 - Integrators and Martingales

Summary

Now that the basic notions of filtration, process, and stopping time are at our disposal, it is time to develop the stochastic integral ∫ X dZ, as per Itô's ideas explained on page 5. We shall call X the integrand and Z the integrator. Both are now processes.

For a guide let us review the construction of the ordinary Lebesgue–Stieltjes integral ∫ x dz on the half-line; the stochastic integral ∫ X dZ that we are aiming for is but a straightforward generalization of it. The Lebesgue–Stieltjes integral is constructed in two steps. First, it is defined on step functions themselves, restrictions must be placed on the integrator: z must be right-continuous and must have finite variation. This chapter discusses the stochastic analog of these restrictions, identifying the processes that have a chance of being useful stochastic integrators.

Given that a distribution function z on the line is right-continuous and has finite variation, the second step is one of a variety of procedures that extend the integral from step functions to a much larger class of integrands. The most efficient extension procedure is that of Daniell; it is also the only one that has a straightforward generalization to the stochastic case. This is discussed in chapter 3.