Stochastic Calculus with a Special Generalized Fractional Brownian Motion
Authors: Zili Mounir 
Published: January 2025

This work is a first step toward developing a stochastic calculus theory with respect to the generalized fractional Brownian motion, which a recently introduced Gaussian process is extending both fractional and sub-fractional Brownian motions. A Malliavin divergence operator and a stochastic symmetric integral with respect to this process are defined, and sufficient integrability conditions are provided. Moreover, corresponding Ito formulas are established, then applied to introduce a generalized version of the fractional Black–Scholes option pricing model.
MSC: 60G15, 60G22, 60H05.
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Communicated Editor: Mezerdi BrahimManuscript received Dec 03, 2023; revised Feb 02, 2024; accepted Feb 02, 2024; published May 11, 2024