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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.

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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