Fokker-Planck-Kolmogorov equations associated with SDEs driven by time-changed fractional Brownian motion

TL;DR

This paper derives Fokker-Planck-Kolmogorov equations for SDEs driven by time-changed fractional Brownian motion, introducing two equivalent forms and analyzing the associated operators' semigroup properties.

Contribution

It introduces two equivalent forms of the Fokker-Planck-Kolmogorov equations for SDEs with time-changed fractional Brownian motion and studies the semigroup property of related operators.

Findings

Derived two equivalent forms of the equations.

Identified the semigroup property of the operator family.

Analyzed equations with different time-change processes.

Abstract

In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is either the first hitting time process for a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property.