On fractional tempered stable processes and their governing differential equations

We derive the governing equation of the Tempered Stable Subordinator (hereafter TSS), which generalizes the space-fractional differential equation satisfied by the law of the α  -stable subordinator itself. This equation is expressed in terms of the shifted fractional derivative of order α∈(0,1)α∈(0,1) coinciding with the stability parameter. We then generalize this equation by introducing a time-fractional derivative of order β∈(0,1)β∈(0,1) (resp. 1/β>11/β>1) and we prove that it is satisfied by the law of a TSS time-changed by the inverse of a β-stable subordinator (resp. by the stable subordinator itself). The corresponding processes can therefore be called “fractional TS processes”. Finally we provide fractional extensions of the relativistic stable processes, which we define as a Brownian motion with a random time argument represented by independent fractional TS processes of order β   (resp. 1/β1/β).