This book is not a research monograph about Malliavin calculus with the latest results and the most sophisticated proofs. It does not contain all the results which are known even for the basic subjects which are addressed here. The goal was to give the largest possible variety of proof techniques. For instance, we did not focus on the proof of concentration inequality for functionals of the Brownian motion, as it closely follows the lines of the analog result for Poisson functionals. This book grew from the graduate courses I gave at Paris-Sorbonne and Paris-Saclay universities, during the last few years. It is supposed to be as accessible as possible for students who have knowledge of Itô calculus and some rudiments of functional analysis.
Author(s): Laurent Decreusefond
Series: Bocconi & Springer Series 10
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 172
City: Cham, Switzerland
Tags: Wiener Space, Wiener Chaos, Fractional Brownian Motion, Poisson Space
Preface
Contents
About the Author
1 Wiener Space
1.1 Gaussian Random Variables
1.2 Wiener Measure
1.3 Wiener Integral
A Quick Refresher About Hilbert Spaces
Self-reproducing Hilbert Spaces
Compact Maps in Hilbert Spaces
1.4 Problems
1.5 Notes and Comments
References
2 Gradient and Divergence
2.1 Gradient
2.2 Divergence
Banach Spaces
Dual Spaces
Dunford–Pettis Integral
Tensor Products of Banach Spaces
Convergence, Strong, and Weak
2.3 Problems
2.4 Notes and Comments
References
3 Wiener Chaos
3.1 Chaos Decomposition
3.2 Ornstein–Uhlenbeck Operator
3.3 Problems
3.4 Notes and Comments
References
4 Fractional Brownian Motion
4.1 Definition and Sample-Paths Properties
4.2 Cameron–Martin Space
4.3 Wiener Space
4.4 Gradient and Divergence
4.5 Itô Formula
Deterministic Fractional Calculus
4.6 Problems
4.7 Notes and Comments
References
5 Poisson Space
5.1 Point Processes
5.2 Poisson Point Process
5.3 Finite Poisson Point Process
5.3.1 Operations on Configurations
5.4 Stochastic Analysis
5.4.1 Discrete Gradient and Divergence
5.4.2 Functional Calculus
5.5 A Quick Refresher About the Poisson Process on the Line
5.6 Problems
5.7 Notes and Comments
Reference
6 The Malliavin–Stein Method
6.1 Principle
6.2 Fourth Order Moment Theorem
6.3 Poisson Process Approximation
6.4 Problems
6.5 Notes and Comments
References
Index
