Fractional Ito Calculus for Randomly Scaled Fractional Brownian Motion and its Applications to Evolution Equations

TL;DR

This paper introduces a fractional Ito stochastic integral for randomly scaled fractional Brownian motion, establishes its properties and Ito formula, and applies these results to analyze generalized time-fractional evolution equations.

Contribution

It presents a novel fractional Ito integral for scaled fractional Brownian motion and derives an Ito formula applicable to related evolution equations.

Findings

Established properties of the fractional Ito integral

Proved Ito formula for functions of the integral

Applied the formula to time-fractional evolution equations

Abstract

We define a fractional Ito stochastic integral with respect to a randomly scaled fractional Brownian motion via an $S$-transform approach. We investigate the properties of this stochastic integral, prove the Ito formula for functions of such stochastic integrals and apply this Ito formula for investigation of related generalized time-fractional evolution equations.