Selection theorems for set-valued stochastic integrals

Abstract

The paper is devoted to some selection theorems for set-valued stochastic integrals considered in papers by Kisielewicz et al. In particular, the Itô set-valued stochastic integrals are considered for absolutely summable countable subsets of the space ${\mathbb{L}}^2([0,T]\times \Omega ,{\Sigma }_{\mathbb{F}},{\mathbb{R}}^{d\times m})$ of all square integrable $\mathbb{F}$-nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded. Selection theorems, presented in the paper, cannot be considered for integrals defined by Jung and Kim in their paper for $\mathbb{F}$-nonanticipative integrably bounded set-valued mappings $G:[0,T]\times \Omega \to {\mathbb{R}}^{d\times m}$, because such integrals are not in the general case integrably bounded.

Keywords:

• Set-valued mappings
• set-valued integrals
• set-valued stochastic processes
• decomposable sets

2001 MATHEMATICS SUBJECT CLASSIFICATION:

• 60H05
• 28B20
• 47H04