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2014 | 34 | 1 | 131-147
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Properties of generalized set-valued stochastic integrals

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The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals defined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not integrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions.
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra
Bibliografia
  • [1] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977) 149-182. doi: 10.1016/0047-259X(77)90037-9
  • [2] W. Hildenbrand, Core and Equilibria of a Large Economy (Princeton University Press, 1974).
  • [3] Sh. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis I, (Kluwer Academic Publishers, 1997). doi: 10.1007/978-1-4615-6359-4
  • [4] E.J. Jung and J.H. Kim, On the set-valued stochastic integrals, Stoch. Anal. Appl. 21 (2)(2003) 401-418. doi: 10.1081/SAP-120019292
  • [5] M. Kisielewicz, Viability theorems for stochastic inclusions, Discuss. Math. 15 (1995) 61-74.
  • [6] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997) 783-800. doi: 10.1080/07362999708809507
  • [7] M. Kisielewicz, Some properties of set-valued stochastic integrals, J. Math. Anal. Appl. 388 (2012) 984-995. doi: 10.1016/j.jmaa.2011.10.050
  • [8] M. Kisielewicz, Stochastic Differential Inclusions and Applications (Springer, New York, 2013). doi: 10.1007/978-1-4614-6756-4
  • [9] M. Kisielewicz, Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations, Comm. Math. 53 (2) (2013) 213-226.
  • [10] M. Kisielewicz, Martingale representation theorem for set-valued martingales, J. Math. Anal. Appl. 409 (2014) 111-118. doi: 10.1016/j.jmaa.2013.06.066
  • [11] M. Michta, Remarks on unboundedness of set-valued Itô stochastic integrals, J. Math. Anal. Appl. (presented to print).
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1155
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