SDEs driven by a time-changed L\'evy process and their associated time-fractional order pseudo-differential equations

TL;DR

This paper explores the connection between time-fractional Kolmogorov equations and SDEs driven by time-changed Lévy processes, expanding the understanding of stochastic models for complex systems.

Contribution

It establishes that SDEs driven by time-changed Lévy processes correspond to time-fractional and distributed order differential equations, broadening the class of models linked to these equations.

Findings

SDEs driven by inverse stable subordinators model time-fractional processes.

The framework includes time-distributed order differential equations.

A fractional Feynman-Kac formula is derived.

Abstract

It is known that the transition probabilities of a solution to a classical It\^o stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coeffcients determined by the corresponding SDE. Time-fractional Kolmogorov type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed L\'evy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman-Kac formula.