Generalized Fractional Brownian Motion

著者名：Zili, Mounir

Wiley-ISTE（2026/02/13発売）

• 言語：ENG
• ISBN：9781786309631
• eISBN：9781394441532

Description

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.

Table of Contents

Introduction ix

Chapter 1. Gaussian Processes 1

Chapter 2. Key Fractional Gaussian Processes: Building Blocks for the Generalized Fractional Brownian Motion 35

Chapter 3. The Zili Generalized Fractional Brownian Motion: Definition and Core Properties 63

Chapter 4. Parameter Estimation for the Generalized Fractional Brownian Motion 93

Chapter 5. Stochastic Calculus with Generalized Fractional Brownian Motion 123

Chapter 6. Stochastic Partial Differential Equations Driven by a ZgfBm Noise 171

Appendix 203

References 225

Index 231