Selected Topics in Malliavin Calculus - Chaos, Divergence and So Much More

Description

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. 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