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

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

2009-07-01

arxiv.org/abs/0907.0253v3

Mathematics

Probability

17 pages. Text rewritten in a succinct form; v2 contains a more detailed preliminary section (Section 2) as well as the proof

Scientific paper

10.1007/s10959-010-0289-4

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.

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