Stochastic Calculus with Generalized Fractional Brownian Motion

Building on the introduction of Zili generalized fractional Brownian motion (ZgfBm), this chapter develops the stochastic calculus for this more general class of Gaussian processes. For a Hurst index H ∈ (1/2, 1), It examines the unique characteristics of its covariance function and proves key probabilistic properties, such as exponential integrability. The chapter is dedicated to deriving generalized Itô formulas for ZgfBm, utilizing both Skorokhod and symmetric integrals. It first establishes key properties of the covariance function R H,a,b (s,t) of ZgfBm. The chapter defines a first type of integral with respect to the ZgfBm. It provides a second type of integration with respect to Z H .(a, b), which under certain conditions, can coincide with the divergence integral. The chapter presents a crucial lemma that relates the symmetric approximation of an integral to the divergence operator and establishes the convergence of the smoothed process.