EXISTENCE OF MEASURABLE ADAPTED SELECTORS OF SET-VALUED FUNCTIONS
Abstract
The present article is devoted to considering measurable set-valued random functions that are adapted to a fixed filtration of σ-algebras and the values of which are closed subsets of some complete separable metric space. For such functions, a criterion of measurability and adaptation is proved, which is analogous to Castain’s well-known criterion of measurability of set-valued functions. A theorem on existence of measurable and adapted selectors of set-valued random functions, which approximate some measurable adapted random function, is obtained. This theorem is improved in the case of set-valued functions with compact values. The generalization of Filippov’s theorem about the inverse function to the set-valued measurable random functions is proved. The obtained results can be useful both for proving the existence and for considering the properties of the solutions of stochastic differential inclusions.
About the Authors
A. A. Levakov
Belarusian State University, Minsk
Belarus
D. Sc. (Physics and Mathematics), Professor, Professor of the Department of Higher Mathematics
Belarus
D. Sc. (Physics and Mathematics), Professor, Professor of the Department of Higher Mathematics
Y. B. Zadvorny
Belarusian State University, Minsk
Belarus
Postgraduate
Belarus
Postgraduate
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