Set-valued stochastic integrals with respect to Poisson processes in a Banach space

Abstract

In a separable Banach space X, at first we study X-valued stochastic integrals with respect to the Poisson random measure N(dsdz) and the compensated Poisson random measure N∼(dsdz) generated by a stationary Poisson stochastic process p. When the characteristic measure ν of p is finite, both N(dsdz) and N∼(dsdz) are of finite variation a.s. Then the set-valued integrals with respect to the Poisson random measure and the compensated Poisson random measure are integrably bounded. The set-valued integral with respect to the compensated Poisson random measure is a right continuous (under Hausdorff metric) set-valued martingale.