Geodesic
Properties of generalized set-valued stochastic integrals
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 131-147.

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: set-valued mappings, set-valued integrals, set-valued stochastic processes 
@article{DMDICO_2014_34_1_a6,
     author = {Kisielewicz, Micha{\l}},
     title = {Properties of generalized set-valued stochastic integrals},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {131--147},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {2014},
     zbl = {1329.60163},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a6/}
}
TY  - JOUR
AU  - Kisielewicz, Michał
TI  - Properties of generalized set-valued stochastic integrals
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2014
SP  - 131
EP  - 147
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a6/
LA  - en
ID  - DMDICO_2014_34_1_a6
ER  - 
%0 Journal Article
%A Kisielewicz, Michał
%T Properties of generalized set-valued stochastic integrals
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2014
%P 131-147
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a6/
%G en
%F DMDICO_2014_34_1_a6
Kisielewicz, Michał. Properties of generalized set-valued stochastic integrals. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 34 (2014) no. 1, pp. 131-147. http://geodesic.mathdoc.fr/item/DMDICO_2014_34_1_a6/

[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).