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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA, 02155, USA

    Marjorie Hahn, Kei Kobayashi & Sabir Umarov

Authors
  1. Marjorie Hahn
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  2. Kei Kobayashi
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  3. Sabir Umarov
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Correspondence to Marjorie Hahn.

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Hahn, M., Kobayashi, K. & Umarov, S. SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations. J Theor Probab 25, 262–279 (2012). https://doi.org/10.1007/s10959-010-0289-4

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  • DOI: https://doi.org/10.1007/s10959-010-0289-4

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