Boundedness of set-valued stochastic integrals

• Autorzy: Michał Kisielewicz
• Języki publikacji: EN

Abstrakty

EN

The paper deals with integrably boundedness of Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4], where has not been proved that this integral is integrably bounded. The problem of integrably boundedness of the above set-valued stochastic integrals has been considered in the paper [7] and the monograph [8], but the problem has not been solved there. The first positive results dealing with this problem due to M. Michta, who showed (see [11]) that there are bounded set-valued 𝔽-nonanticipative mappings having unbounded Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim. The present paper contains some new conditions implying unboundedness of the above type set-valued stochastic integrals.

Słowa kluczowe

EN

set-valued mapping; Itô set-valued integral; set-valued stochastic process; integrably boundedness of set-valued integral

Kategorie tematyczne

• 47H04: Set-valued operators
• 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections
• 60H05: Stochastic integrals

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2015
• Tom: 35
• Numer: 2
• Strony: 197-207

Daty

wydano

2015

otrzymano

2015-11-14

Twórcy

autor

Michał Kisielewicz

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

Bibliografia

• [1] F. Hiai, Multivalued stochastic integrals and stochastic inclusions, Division of Applied Mathematics, Research Institute of Applied Electricity, Sapporo 060 Japan (not published).
• [2] 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
• [3] Sh. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis I (Kluwer Academic Publishers, Dordrecht, London, 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, Set-valued stochastic integrals and stochastic inclusions, Discuss. Math. Diff. Incl. 15 (1) (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, Properties of generalized set-valued stochastic integrals, Discuss. Math. DICO 34 (1) (2014), 131-147. doi: 10.7151/dmdico.1155
• [10] M. Kisielewicz and M. Michta, Integrably bounded set-valued stochastic integrals, J. Math. Anal. Appl. (submitted to print).
• [11] M. Michta, Remarks on unboundedness of set-valued Itô stochastic integrals, J. Math. Anal. Appl 424 (2015), 651-663. doi: 10.1016/j.jmaa.2014.11.041

Identyfikatory

DOI

10.7151/dmdico.1173