Boundedness of set-valued stochastic integrals

Michał Kisielewicz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2015)

  • Volume: 35, Issue: 2, page 197-207
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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Michał Kisielewicz. "Boundedness of set-valued stochastic integrals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 35.2 (2015): 197-207. <http://eudml.org/doc/276653>.

@article{MichałKisielewicz2015,
abstract = {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.},
author = {Michał Kisielewicz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {set-valued mapping; Itô set-valued integral; set-valued stochastic process; integrably boundedness of set-valued integral},
language = {eng},
number = {2},
pages = {197-207},
title = {Boundedness of set-valued stochastic integrals},
url = {http://eudml.org/doc/276653},
volume = {35},
year = {2015},
}

TY - JOUR
AU - Michał Kisielewicz
TI - Boundedness of set-valued stochastic integrals
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2015
VL - 35
IS - 2
SP - 197
EP - 207
AB - 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.
LA - eng
KW - set-valued mapping; Itô set-valued integral; set-valued stochastic process; integrably boundedness of set-valued integral
UR - http://eudml.org/doc/276653
ER -

References

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  1. [1] F. Hiai, Multivalued stochastic integrals and stochastic inclusions, Division of Applied Mathematics, Research Institute of Applied Electricity, Sapporo 060 Japan (not published). 
  2. [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 Zbl0368.60006
  3. [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 Zbl0887.47001
  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 Zbl1049.60048
  5. [5] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Discuss. Math. Diff. Incl. 15 (1) (1995), 61-74. 
  6. [6] M. Kisielewicz, Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl. 15 (5) (1997), 783-800. doi: 10.1080/07362999708809507 Zbl0891.93070
  7. [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 Zbl1235.60058
  8. [8] M. Kisielewicz, Stochastic Differential Inclusions and Applications (Springer, New York, 2013). doi: 10.1007/978-1-4614-6756-4 Zbl1277.93002
  9. [9] M. Kisielewicz, Properties of generalized set-valued stochastic integrals, Discuss. Math. DICO 34 (1) (2014), 131-147. doi: 10.7151/dmdico.1155 Zbl1329.60163
  10. [10] M. Kisielewicz and M. Michta, Integrably bounded set-valued stochastic integrals, J. Math. Anal. Appl. (submitted to print). 
  11. [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 Zbl1303.60044

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