Properties of set-valued integrals and set-valued stochastic equations driven by two-parameter martingales

Abstract

In the paper we study properties of Aumann's stochastic integral driven by a two-parameter increasing process and set-valued Itô's integral with respect to the two-parameter martingale. Both types of integrals are understood as set-valued processes. Next, the existence, uniqueness and convergence properties of solutions to set-valued stochastic integral equations with respect to such integrators are investigated. We present new types of such equations that generalize those studied earlier. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in single-valued deterministic and stochastic cases.