Analysis of variations for self-similar processes : a stochastic calculus approach / Ciprian A. Tudor

This monograph is a profound survey of recent developments in the fields of (Gaussian and non-Gaussian) self-similar processes and their calculus of variations. The first part of this book provides a long list of examples of self-similar processes: (bi- and multi-) fractional Brownian motion, the mild solutions of the stochastic heat and wave equation with white and colored noise in space and time, the family of Hermite processes, with the special case of the Rosenblatt process and finally multiparameter Gaussian processes, such as the fractional Brownian sheet. The second part of the book treats the quadratic (or other) variations and, as an application, the respective rates of convergence in central and non-central limit theorems. These are applied to Stein’s method and provide the tool for the construction of a statistical estimator for the self-similarity parameter of self-similar processes, which is of eminent importance in applications.

The book is completed with a number of appropriate, interesting and at times nontrivial exercises at the end of each chapter. It may serve as an excellent basis for research seminars or special classes on Gaussian processes and Malliavin’s calculus and as a starting point for applied mathematicians with interest in self-similar processes. (zbMath)