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Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations

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Kisielewicz M.

Commentationes Mathematicae

2013

tom Vol. 53, No. 2

113--126

The paper contains new properties of set-valued stochastic integrals defined as multifunctions with subtrajectory integrals equal to closed decomposable hulls of functional set-valued integrals defined in the author paper [8]. In particular, it is proved that such defined integrals for set-valued predictable square integrably bounded processes having finite Castaing representations are square integrably bounded. Up to now this property has not been proved. Unfortunately, in the general case the above boundedness problem is still open.

Properties of generalized set-valued stochastic integrals

100%

Kisielewicz M.

nr 1

131-147

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.