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In an M-type 2 Banach space, firstly we explore some properties of theset-valued stochastic integral associated with the stationary Poisson pointprocess. By using the Hahn decomposition theorem and bounded linear functional,we obtain the main result: the integral of a set-valued stochastic process withrespect to the compensated Poisson measure is a set-valued submartingale butnot a martingale unless the integrand degenerates into a single-valued process.Secondly we study the strong solution to the set-valued stochastic integralequation, which includes a set-valued drift, a single-valued diffusion drivenby a Brownian motion and the set-valued jump driven by a Poisson process.

Submartingale property of set-valued stochastic integration associated with Poisson process and related integral equations on Banach spaces