Stochastic Analysis of Mixed Fractional Gaussian Processes

Stochastic Analysis of Mixed Fractional Gaussian Processes presents the main tools necessary to characterize Gaussian processes. The book focuses on the particular case of the linear combination of independent fractional and sub-fractional Brownian motions with different Hurst indices. Stochastic integration with respect to these processes is considered, as is the study of the existence and uniqueness of solutions of related SDE's. Applications in finance and statistics are also explored, with each chapter supplying a number of exercises to illustrate key concepts.

"The normal distribution, the Brownian motion process and the bigger class of Gaussian processes are well known in both theory and applications. Chapter 1 of this book contains the main notions and results on Gaussian processes. Then, in Chapter 2, the authors turn to the analysis of fractional and sub-fractional Brownian motions. It is shown their existence and analytical properties which are used in constructing stochastic integrals. The main subject in the book is Chapter 3 dealing with mixed fractional and mixed sub-fractional Brownian motions. The authors establish path properties, series expansions and semi-martingale properties. Proofs are provided for some of the results, otherwise, appropriate references are given. There are illustrative examples in the text. At the end of each chapter there are useful exercises. The book ends with an appendix, bibliography and an index. The book will be useful to PhD students and researchers in modern probability theory, especially to those interested in modelling phenomena which are dynamic and of a fractional nature.

Two final comments. The first is to the authors. There are typos, fortunately not too many, they are easy to detect and correct. There are a few fundamental sources missing in the bibliography. The second comment is addressed to the publisher. The practice to use italics font for the full text of all examples and all exercises is unusual and quite strange. To follow such a style puts you far from the well-established traditions of the best international publishers."—zbMATH Open

"The book under review deals with two classes of specific Gaussian processes, namely, (mixed) fractional Brownian motion and (mixed) sub fractional Brownian motion."—Mathematical Reviews Clippings