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.
Recenzijas
The book is written in a clear and self-contained manner, thanks to the expertise of the author in related topics, and could serve as a nice self-studying reference for graduate students. (Yizao Wang, Mathematical Reviews, December, 2024)
Introduction.
Chapter
1. Multiple Stochastic Integrals.
Chapter
2. Hermite processes: Definition and basic properties.
Chapter
3. The Wiener integral with respect to the Hermite process and the Hermite Ornstein-Uhlenbeck process.
Chapter
4. Hermite sheets and SPDEs.
Chapter
5. Statistical inference for stochastic (partial) differential equations with Hermite noise.- References.
Ciprian Tudor is Full Professor at the University of Lille, France. He graduated from the University of Bucharest and earned his PhD degree in probability from Université de La Rochelle in 2002. Following his doctorate, he held positions at the Université Pierre et Marie Curie (Paris 6) and at the Université de Panthéon-Sorbonne (Paris 1), where he obtained his Habilitation in 2006. With a research focus on stochastic processes, particularly Malliavin calculus, self-similar processes, and their applications to statistics, he has authored three monographs and more than 150 scientific publications in well-regarded journals.
