Parameter Estimation for the Generalized Fractional Brownian Motion

This chapter presents a comprehensive methodology for the estimation of all key parameters of the Zili generalized fractional Brownian motion (ZgfBm): the Hurst parameter H and the scaling coefficients a and b . It addresses the problem of estimating the Hurst parameter H for the ZgfBm under the foundational assumption that the deterministic coefficients a and b are known constants. The chapter defines a statistic based on the squared increments of the ZgfBm. It focuses on an approach to estimate the deterministic coefficients a and b from discrete observations of the ZgfBm process, assuming that H is a known quantity. This is particularly relevant in applications where the underlying long-range dependence structure is well established, and the goal is to quantify the specific asymmetric properties of the process.