An Analytical Foundation for Emerging Methods in Stochastic Fractional Analysis

Abstract. Stochastic processes governed by a new fractional Brownian motion are thoroughly examined in this paper. Building on the foundational work of El-Borai and El-Nadi, we offer advanced analytical techniques to study these processes, which are distinguished by their memory effects and long-range interdependence. Our approach uses fractional stochastic calculus, fractional stochastic analysis, and fixed-point theorems to provide a solid mathematical basis to deal with these complicated systems. The study makes a substantial contribution to the subject by explicitly solving fractional stochastic differential equations, analyzing their statistical properties with characteristic and moment-generating functions, and proving the existence and uniqueness of solutions within this new framework.