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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 {\it 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. [{\it A. Deniz} and {\it 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.

MSC:
60H05Stochastic integrals
60G44Martingales with continuous parameter
60H10Stochastic ordinary differential equations
60J60Diffusion processes
26E70Real analysis on time scales or measure chains
39A50Stochastic difference equations
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