SDEs driven by a time-changed Lévy process and their associated time-fractional order pseudo-differential equations

It is known that if a stochastic process is a solution to a classical Itô stochastic differential equation (SDE), then its transition probabilities satisfy in the weak sense the associated Cauchy problem for the forward Kolmogorov equation. The forward Kolmogorov equation is a parabolic partial differential equation with coefficients determined by the corresponding SDE. Stochastic processes which are scaling limits of continuous time random walks have been connected with time-fractional differential equations. However, the class of SDEs that is associated with time-fractional Kolmogorov type equations is unknown. 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 Lévy processes where the independent time-change is given respectively by the inverse of a stable subordinator or the inverse of a mixture of independent stable subordinators.