Set-valued functions of bounded generalized variation and set-valued Young integrals

TL;DR

This paper studies set-valued functions with bounded variation, introduces Young-type integrals for them, and explores their properties, including applications to stochastic differential inclusions driven by fractional Brownian motion.

Contribution

It introduces set-valued Young integrals for functions with bounded Riesz p-variation and analyzes their properties and applications in stochastic differential inclusions.

Findings

Set-valued Young integrals generalize stochastic integrals with fractional Brownian motion.

Selection results for set-valued integrals are established.

Properties of these integrals are crucial for stochastic differential inclusion analysis.

Abstract

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