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