The first attempt on the stochastic calculus on time scale. (English) Zbl 1238.60061
In this article, the authors present a first attempt on the theory of stochastic calculus on time scale, that is a nonempty closed subset on \(\mathbb{R}\). To this end, they begin with a survey on some basic notions and properties on the analysis on time scale, introduced by S. Hilger [Ein Masskettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten. Würzburg: Univ. Würzburg, Diss. (1988; Zbl 0695.34001)], including the construction of a Lebesgue-Stieltjes measure on time scale and the integral associated to it, called Lebesgue-Stieltjes \(\nabla\)-integral (see e.g. [A. Deniz and Ü. Ufuktepe, Turk. J. Math. 33, No. 1, 27–40 (2009; Zbl 1179.28005)]). Using these notions, they first present a Doob-Meyer decomposition theorem for a semimartingale indexed by a time scale. In the rest of the paper, they construct a stochastic integral (called \(\nabla\)-stochastic integral) and they show some properties. Finally, they prove an Itô formula followed by some examples.
Reviewer: Stavros Vakeroudis (Paris)
MSC:
| 60H05 | Stochastic integrals |
| 60G44 | Martingales with continuous parameter |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60J60 | Diffusion processes |
| 26E70 | Real analysis on time scales or measure chains |
| 39A50 | Stochastic difference equations |
Keywords:
Doob-Mayer decomposition; Itô’s formula; martingale; Lebesgue-Stieltjes measure on time scale; Lebesgue-Stieltjes \(\nabla\)-integral; natural increasing process; stochastic integration; time scale; \(\nabla\)-stochastic integralReferences:
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