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Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

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Malinowski M. T.

Open Mathematics

2015

tom 13

nr 1

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.

Ograniczanie wyników

1 Open Mathematics

1 Malinowski M. T.

1 2015

Wyniki wyszukiwania

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition