The Zili Generalized Fractional Brownian Motion

This chapter introduces the generalized fractional Brownian motion. It summarizes some fundamental properties of the generalized fractional Brownian motion, derived directly from its definition and the known characteristics of two-sided fractional Brownian motion. The chapter investigates a crucial characteristic of the ZgfBm: its non-Markovian nature, which distinguishes it from classical Brownian motion and highlights its long-memory properties for certain parameter configurations. For the generalized fractional Brownian motion, understanding the covariance of its increments is essential, particularly in discerning its departure from traditional processes, such as the Brownian motion. The chapter establishes a crucial integral representation for ZgfBm in terms of the two-sided Brownian motion, commonly known as a moving average expression. This representation is fundamental for understanding its underlying structure and is often used in stochastic analysis. The ZgfBm can also be expressed in an alternative form, linking it to the odd and even fractional Brownian motions.