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<p>We define a fractional Itô stochastic integral with respect to a randomly scaled fractional Brownian motion via an <i>S</i>-transform approach. We investigate the properties of this stochastic integral, prove an Itô formula for functions of such stochastic integrals and apply this Itô formula to the investigation of related generalized time-fractional evolution equations. We show that the constructed stochastic integrals serve as stochastic solutions for a large class of integro-differential evolution equations containing non-local operators acting on the time variable. Our results cover such special cases of randomly scaled fractional Brownian motion as gamma-grey Brownian motion, generalized grey Brownian motion, superstatistical fractional Brownian motion with random diffusion coefficient having generalized gamma distribution, generalized Fox-<i>H</i> processes and some other processes discussed in the literature.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月2日

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  • Yana A. Butko,
  • Merten Mlinarzik

摘要

We define a fractional Itô 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 an Itô formula for functions of such stochastic integrals and apply this Itô formula to the investigation of related generalized time-fractional evolution equations. We show that the constructed stochastic integrals serve as stochastic solutions for a large class of integro-differential evolution equations containing non-local operators acting on the time variable. Our results cover such special cases of randomly scaled fractional Brownian motion as gamma-grey Brownian motion, generalized grey Brownian motion, superstatistical fractional Brownian motion with random diffusion coefficient having generalized gamma distribution, generalized Fox-H processes and some other processes discussed in the literature.