# Properties of generalized set-valued stochastic integrals

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2014)

- Volume: 34, Issue: 1, page 131-147
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topMichał Kisielewicz. "Properties of generalized set-valued stochastic integrals." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 34.1 (2014): 131-147. <http://eudml.org/doc/270400>.

@article{MichałKisielewicz2014,

abstract = {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.},

author = {Michał Kisielewicz},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {set-valued mappings; set-valued integrals; set-valued stochastic processes; set-valued stochastic integrals; stochastic differential equations; set-valued solutions},

language = {eng},

number = {1},

pages = {131-147},

title = {Properties of generalized set-valued stochastic integrals},

url = {http://eudml.org/doc/270400},

volume = {34},

year = {2014},

}

TY - JOUR

AU - Michał Kisielewicz

TI - Properties of generalized set-valued stochastic integrals

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2014

VL - 34

IS - 1

SP - 131

EP - 147

AB - 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.

LA - eng

KW - set-valued mappings; set-valued integrals; set-valued stochastic processes; set-valued stochastic integrals; stochastic differential equations; set-valued solutions

UR - http://eudml.org/doc/270400

ER -

## References

top- [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 Zbl0368.60006
- [2] W. Hildenbrand, Core and Equilibria of a Large Economy (Princeton University Press, 1974). Zbl0351.90012
- [3] Sh. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis I, (Kluwer Academic Publishers, 1997). doi: 10.1007/978-1-4615-6359-4 Zbl0887.47001
- [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] M. Kisielewicz, Viability theorems for stochastic inclusions, Discuss. Math. 15 (1995) 61-74. Zbl0844.93072
- [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] 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] M. Kisielewicz, Stochastic Differential Inclusions and Applications (Springer, New York, 2013). doi: 10.1007/978-1-4614-6756-4 Zbl1277.93002
- [9] M. Kisielewicz, Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations, Comm. Math. 53 (2) (2013) 213-226. Zbl1303.60043
- [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 Zbl1306.60044
- [11] M. Michta, Remarks on unboundedness of set-valued Itô stochastic integrals, J. Math. Anal. Appl. (presented to print). Zbl1303.60044

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.