Mounir Zili

This comprehensive book establishes the Zili generalized fractional Brownian motion (ZgfBm) as a powerful new foundation in the mathematical theory of stochastic processes. Generalized Fractional Brownian Motion provides the first rigorous and systematic stochastic analysis of the ZgfBm, a versatile Gaussian process that uniquely extends both the classic fractional Brownian motion with stationary increments and the sub-fractional Brownian motion with nonstationary increments. Defined by three tunable parameters, the ZgfBm offers unprecedented flexibility for modeling complex phenomena across diverse fields, overcoming the limitations of single-parameter models. The book carefully builds from foundational Gaussian theory and key fractional processes to advanced topics, including a complete methodology for parameter estimation, the development of a rigorous stochastic calculus with generalized Itô formulas and an investigation into the regularity of solutions to stochastic heat equations. This essential resource provides researchers, practitioners and graduate students with a unified and in-depth perspective on advanced fractional Gaussian processes.

This book is entirely devoted to discrete time and provides a detailed introduction to the construction of the rigorous mathematical tools required for the evaluation of options in financial markets. Both theoretical and practical aspects are explored through multiple examples and exercises, for which complete solutions are provided. Particular attention is paid to the Cox, Ross and Rubinstein model in discrete time. The book offers a combination of mathematical teaching and numerous exercises for wide appeal. It is a useful reference for students at the master’s or doctoral level who are specializing in applied mathematics or finance as well as teachers, researchers in the field of economics or actuarial science, or professionals working in the various financial sectors. Martingales and Financial Mathematics in Discrete Time is also for anyone who may be interested in a rigorous and accessible mathematical construction of the tools and concepts used in financial mathematics, or in the application of the martingale theory in finance

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.