Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals

Abstract

The paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These 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.

Properties of set-valued Young integrals and Young differential inclusions generated by sets of Hölder functions

“…Thus, it seems reasonable to investigate also differential inclusions driven by a fractional Brownian motion and Young type integrals as well. Recently, in [6,11,32,33] the authors studied properties of set-valued Young type integrals and the existence of solutions of differential inclusions with such integrals. Once having conditions assuring the existence of solutions, it is natural to ask about their properties.…”

On a Set-Valued Young Integral with Applications to Differential Inclusions

“…In the case of a deterministic signal w with possibly infinite variation, Michta and Motyl [26,27] are the only references so far defining a set-valued Young integral à la Aumann of the form (1), for convex as well as nonconvex-valued multifunctions. In their approach, the set of selections S(F ) is large, namely, in the case of our setting, S(F ) is the set of all α-Hölder continuous selections of F .…”