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

TL;DR

This paper introduces a novel set-valued Young integral for H"older multifunctions, enabling numerical approximation and existence results for stochastic differential inclusions driven by fractional Brownian motion.

Contribution

It develops a new Aumann-like integral for H"older multifunctions, extending Young integration to set-valued functions with applications to stochastic differential inclusions.

Findings

Established continuity properties for the new integral.

Proved an existence theorem for stochastic differential inclusions.

Applied the theory to fractional Brownian motion-driven systems.

Abstract

We present a new Aumann-like integral for a H\"older multifunction with respect to a H\"older signal, based on the Young integral of a particular set of H\"older selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.