Set-valued stochastic integrals for convoluted Lévy processes

Abstract

In this paper we study set-valued Volterra-type stochastic integrals driven by Lévy processes. Upon extending the classical definitions of set-valued stochastic integral functionals to convoluted integrals with square-integrable kernels, set-valued convoluted stochastic integrals are defined by taking the closed decomposable hull of the integral functionals for generic time. We show that, aside from well-established results for set-valued Itô integrals, while set-valued stochastic integrals with respect to a finite-variation Poisson random measure are guaranteed to be integrably bounded for bounded integrands, this is not true when the random measure is of infinite variation. For indefinite integrals, we prove that it is a mutual effect of kernel singularity and jumps that the set-valued convoluted integrals are possibly explosive and take extended vector values. These results have some important implications on how set-valued fractional dynamical systems are to be constructed in general. Two classes of set-monotone processes are studied for practical interests in economic and financial modeling.