Book contents
- Frontmatter
- Contents
- Introduction
- Chapter 1 Gaussian spaces
- Chapter 2 Wiener chaos
- Chapter 3 Wick products
- Chapter 4 Tensor products and Fock space
- Chapter 5 Hypercontractivity
- Chapter 6 Variables with finite chaos decompositions
- Chapter 7 Stochastic integration
- Chapter 8 Gaussian stochastic processes
- Chapter 9 Conditioning
- Chapter 10 Pairs of Gaussian subspaces
- Chapter 11 Limit theorems for generalized U-statistics
- Chapter 12 Applications to operator theory
- Chapter 13 Some operators from quantum physics
- Chapter 14 The Cameron—Martin shift
- Chapter 15 Malliavin calculus
- Chapter 16 Transforms
- Appendices
- References
- Index of notation
- Index
Chapter 7 - Stochastic integration
Published online by Cambridge University Press: 21 October 2009
- Frontmatter
- Contents
- Introduction
- Chapter 1 Gaussian spaces
- Chapter 2 Wiener chaos
- Chapter 3 Wick products
- Chapter 4 Tensor products and Fock space
- Chapter 5 Hypercontractivity
- Chapter 6 Variables with finite chaos decompositions
- Chapter 7 Stochastic integration
- Chapter 8 Gaussian stochastic processes
- Chapter 9 Conditioning
- Chapter 10 Pairs of Gaussian subspaces
- Chapter 11 Limit theorems for generalized U-statistics
- Chapter 12 Applications to operator theory
- Chapter 13 Some operators from quantum physics
- Chapter 14 The Cameron—Martin shift
- Chapter 15 Malliavin calculus
- Chapter 16 Transforms
- Appendices
- References
- Index of notation
- Index
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 Bt, 0 ≤ t < ∞, is a standard Brownian motion and consider, as in Example 1.10, the Gaussian Hilbert space H = H(B) spanned by {Bt}t≥0.
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- Gaussian Hilbert Spaces , pp. 86 - 116Publisher: Cambridge University PressPrint publication year: 1997