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Set valued measures and integral representation

Xiao Ping Xue, Cheng Lixin, Goucheng Li, Xiao Bo Yao (1996)

Commentationes Mathematicae Universitatis Carolinae

The extension theorem of bounded, weakly compact, convex set valued and weakly countably additive measures is established through a discussion of convexity, compactness and existence of selection of the set valued measures; meanwhile, a characterization is obtained for continuous, weakly compact and convex set valued measures which can be represented by Pettis-Aumann-type integral.

Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Marek T. Malinowski (2015)

Open Mathematics

We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect...

Set-valued random differential equations in Banach space

Mariusz Michta (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider the problem of the existence of solutions of the random set-valued equation: (I) D H X t = F ( t , X t ) P . 1 , t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space K c ( E ) , of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

Some applications of Girsanov's theorem to the theory of stochastic differential inclusions

Micha Kisielewicz (2003)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The Girsanov's theorem is useful as well in the general theory of stochastic analysis as well in its applications. We show here that it can be also applied to the theory of stochastic differential inclusions. In particular, we obtain some special properties of sets of weak solutions to some type of these inclusions.

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