This book offers an introduction to the field of stochastic analysis of Hermite processes. These selfsimilar stochastic processes with stationary increments live in a Wiener chaos and include the fractional Brownian motion, the only Gaussian process in this class.
Using the Wiener chaos theory and multiple stochastic integrals, the book covers the main properties of Hermite processes and their multiparameter counterparts, the Hermite sheets. It delves into the probability distribution of these stochastic processes and their sample paths, while also presenting the basics of stochastic integration theory with respect to Hermite processes and sheets.
The book goes beyond theory and provides a thorough analysis of physical models driven by Hermite noise, including the Hermite Ornstein-Uhlenbeck process and the solution to the stochastic heat equation driven by such a random perturbation. Moreover, it explores up-to-date topics central to current research in statistical inference for Hermite-driven models.
Author(s): Ciprian Tudor
Series: SpringerBriefs in Probability and Mathematical Statistics
Publisher: Springer-ISI
Year: 2023
Language: English
Pages: 109
City: Voorburg
Preface
About This Book
Contents
1 Multiple Stochastic Integrals
1.1 Isonormal Processes
1.1.1 The Wiener Integral with Respect to the Wiener Process
1.1.2 The Wiener Integral with Respect to the Brownian Sheet
1.2 Multiple Wiener-Itô Integrals
1.2.1 Definition and Basic Properties
1.2.2 A First Product Formula
1.2.3 The Wiener Chaos
1.2.4 The General Product Formula
1.3 Random Variables in the Second Wiener Chaos
2 Hermite Processes: Definition and Basic Properties
2.1 The Kernel of the Hermite Process
2.2 Definition of the Hermite Process and Some Immediate Properties
2.2.1 Self-similarity and Stationarity of the Increments
2.2.2 Moments and Hölder Continuity
2.2.3 The Hermite Noise and the Long Memory
2.2.4 pp-Variation
2.2.5 Approximation by Semimartingales
2.3 Some Particular Hermite Processes: Fractional Brownian Motion …
2.3.1 Fractional Brownian Motion
2.3.2 The Rosenblatt Process
2.4 Alternative Representation
2.5 On the Simulation of the Rosenblatt Process
3 The Wiener Integral with Respect to the Hermite Process and the Hermite Ornstein-Uhlenbeck Process
3.1 Wiener Integral
3.2 The Cases q equals 1q=1 and q equals 2q=2
3.3 Wiener Integral in the Riemann-Stieltjes Sense
3.4 The Hermite Ornstein-Uhlenbeck Process
3.4.1 Definition and Properties
3.4.2 The Stationary Hermite Ornstein-Uhlenbeck Process
4 Hermite Sheets and SPDEs
4.1 Definition of the Hermite Sheet
4.2 Basic Properties
4.3 Wiener Integral with Respect to the Hermite Sheet
4.4 The Stochastic Heat Equation with Hermite Noise
4.4.1 Existence of the Solution
4.4.2 Self-similarity
4.4.3 Regularity of Sample Paths
4.4.4 A Decomposition Theorem
4.4.5 pp-Variation
5 Statistical Inference for Stochastic (Partial) Differential Equations with Hermite Noise
5.1 Parameter Identification for the Hermite Ornstein-Uhlenbeck Process
5.1.1 Quadratic Variation
5.1.2 Estimation of the Hurst Parameter
5.1.3 Estimation of sigmaσ
5.2 Drift Estimation for the Stochastic Heat Equation with Hermite Noise
Appendix Bibliography
