Abstract
The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.
Highlights
Since the pioneering work of Aumann [6], the notion of set-valued integrals for multivalued functions has attracted the interest of many authors both from theoretical and practical points of view
Concepts of set-valued integrals, both deterministic and stochastic, were used to define the notion of fuzzy integrals applied in the theory of fuzzy differential equations, e.g., [16,21,27,32]
It seems reasonable to investigate differential inclusions driven by a fractional Brownian motion and Young type integrals
Summary
Since the pioneering work of Aumann [6], the notion of set-valued integrals for multivalued functions has attracted the interest of many authors both from theoretical and practical points of view. The theory has been developed extensively, among others, with applications to optimal control theory, mathematical economics, theory of differential inclusions and set-valued differential equations, see e.g. The notion of the integral for set-valued functions has been extended to a stochastic case where set-valued Itoand Stratonovich integrals have been studied and applied to stochastic differential inclusions and set-valued stochastic differential equations, see e.g., 164 Page 2 of 22
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