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Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations

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  • Kei Kobayashi

    (Tufts University)

Abstract

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Itô formula is derived. When a standard Brownian motion is the original semimartingale, classical Itô stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.

Suggested Citation

  • Kei Kobayashi, 2011. "Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 24(3), pages 789-820, September.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:3:d:10.1007_s10959-010-0320-9
    DOI: 10.1007/s10959-010-0320-9
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    References listed on IDEAS

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    1. 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.
    2. Bondesson, Lennart & Kristiansen, Gundorph K. & Steutel, Fred W., 1996. "Infinite divisibility of random variables and their integer parts," Statistics & Probability Letters, Elsevier, vol. 28(3), pages 271-278, July.
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    Cited by:

    1. Bareche, Aîcha & Bibi, Abdelouahab, 2023. "On inverse-Gamma distribution delayed by Poisson process," Statistics & Probability Letters, Elsevier, vol. 195(C).

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