Properties of generalized set-valued stochastic integrals

Michał Kisielewicz

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

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

Abstract

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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.

How to cite

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Michał 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

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  1. [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. [2] W. Hildenbrand, Core and Equilibria of a Large Economy (Princeton University Press, 1974). Zbl0351.90012
  3. [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. [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, Viability theorems for stochastic inclusions, Discuss. Math. 15 (1995) 61-74. Zbl0844.93072
  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, Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations, Comm. Math. 53 (2) (2013) 213-226. Zbl1303.60043
  10. [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. [11] M. Michta, Remarks on unboundedness of set-valued Itô stochastic integrals, J. Math. Anal. Appl. (presented to print). Zbl1303.60044

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