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This invaluable research monograph presents a unified and fascinating theory of generalized functionals of Brownian motion and other fundamental processes such as fractional Brownian motion and Levy process — covering the classical Wiener–Ito class including the generalized functionals of Hida as special cases, among others. It presents a thorough and comprehensive treatment of the Wiener–Sobolev spaces and their duals, as well as Malliavin calculus with their applications. The presentation is lucid and logical, and is based on a solid foundation of analysis and topology. The monograph develops the notions of compactness and weak compactness on these abstract Fock spaces and their duals, clearly demonstrating their nontrivial applications to stochastic differential equations in finite and infinite dimensional Hilbert spaces, optimization and optimal control problems.
Readers will find the book an interesting and easy read as materials are presented in a systematic manner with a complete analysis of classical and generalized functionals of scalar Brownian motion, Gaussian random fields and their vector versions in the increasing order of generality. It starts with abstract Fourier analysis on the Wiener measure space where a striking similarity of the celebrated Riesz–Fischer theorem for separable Hilbert spaces and the space of Wiener–Ito functionals is drawn out, thus providing a clear insight into the subject.
Sample Chapter(s)
Chapter 1: Background Material (375 KB)
Contents:
- Background Material
- Regular Functionals of Brownian Motion
- Generalized Functionals of the First Kind I
- Functional Analysis on $\{G,{\cal G}\}$ and Their Duals
- L2-Based Generalized Functionals of White Noise II
- Lp-Based Generalized Functionals of White Noise III
- ${\cal W}^{p,m}$-Based Generalized Functionals of White Noise IV
- Some Elements of Malliavin Calculus
- Evolution Equations on Fock Spaces
Readership: Graduate students in stochastic analysis, stochastic differential equations, stochastic systems; individual researchers in the field of stochastic processes, stochastic differential equations in finite and dimensional Hilbert spaces.
