SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

Marjorie Hahn (), Kei Kobayashi () and Sabir Umarov ()
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Marjorie Hahn: Tufts University
Kei Kobayashi: Tufts University
Sabir Umarov: Tufts University

Journal of Theoretical Probability, 2012, vol. 25, issue 1, 262-279

Abstract: Abstract It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients 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 Lévy 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.

Keywords: Time-change; Stochastic differential equation; Semimartingale; Kolmogorov equation; Fractional order differential equation; Pseudo-differential operator; Lévy process; Stable subordinator; 60H10; 35S10; 60G51 (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-010-0289-4

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