Set-valued stochastic integrals for convoluted Lévy processes

Abstract

In this study, we consider set-valued Lévy-driven Volterra-type stochastic integrals, defined as the closed decomposable hull of convoluted integral functionals. In addition to the well-established results for set-valued Itô integrals, we show that while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are integrably bounded for bounded integrands, this is not true for infinite-variation random measures. For indefinite integrals, we prove that kernel singularity and jumps can lead to the possible explosiveness of set-valued integrals. These results have important implications for the construction of set-valued fractional dynamical systems. Two classes of set-monotone processes are studied for economic and financial modeling.