Chapter 7 - Stochastic integration

Summary

The theory of Gaussian Hilbert spaces developed in this book has strong connections to stochastic integration, in particular to Itô integrals with respect to Brownian motion. We treat these Itô integrals in the first section, and some extensions and related results in the following ones: stochastic integrals over general measure spaces in Section 2, the Skorohod integral in Section 3, and complex stochastic integrals and measures in Section 4.

Our treatment is self-contained, and we do not require that the reader has any prior knowledge of stochastic integration. On the other hand, such a knowledge would certainly be useful; we treat only those parts of stochastic integration theory that are directly relevant to the subject of this book, and many important topics are not included. For example, we consider only stochastic integrals with respect to Gaussian processes. Moreover, even for Brownian motion we do not include the fundamental Itô formula.

Hence, this chapter will perhaps be best understood in connection and comparison with other, more direct and complete, treatments of stochastic integration; see for example McKean (1969) and Protter (1990).

Brownian motion and Itô integrals

In this section, we assume that B_t, 0 ≤ t < ∞, is a standard Brownian motion and consider, as in Example 1.10, the Gaussian Hilbert space H = H(B) spanned by {B_t}_t≥0.