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.

Mounir Zili is Full Professor of Mathematics and a key member of the "Analysis, Probability and Fractals" laboratory at the University of Monastir, Tunisia. He holds a PhD in Mathematics from Sorbonne University in Paris, France.