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

Author

Listed:
  • Marjorie Hahn

    (Tufts University)

  • Kei Kobayashi

    (Tufts University)

  • Sabir Umarov

    (Tufts University)

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.

Suggested Citation

  • Marjorie Hahn & Kei Kobayashi & Sabir Umarov, 2012. "SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations," Journal of Theoretical Probability, Springer, vol. 25(1), pages 262-279, March.
    • Handle: RePEc:spr:jotpro:v:25:y:2012:i:1:d:10.1007_s10959-010-0289-4
    DOI: 10.1007/s10959-010-0289-4
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    References listed on IDEAS

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    1. Gorenflo, Rudolf & Mainardi, Francesco & Vivoli, Alessandro, 2007. "Continuous-time random walk and parametric subordination in fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 87-103.
    2. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2008. "Triangular array limits for continuous time random walks," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1606-1633, September.
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    Cited by:

    1. Alessandro Gregorio & Francesco Iafrate, 2024. "Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3246-3280, November.
    2. Oraby, T. & Suazo, E. & Arrubla, H., 2023. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).

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