Aumann Type Set-valued Lebesgue Integral and Representation Theorem

In this paper, we shall firstly illustrate why we should discuss the Aumann type set-valued Lebesgue integral of a set-valued stochastic process with respect to time t under the condition that the set-valued stochastic process takes nonempty compact subset of d-dimensional Euclidean space. After recalling some basic results about set-valued stochastic processes, we shall secondly prove that the Aumann type set-valued Lebesgue integral of a set-valued stochastic process above is a set-valued stochastic process. Finally we shall give the representation theorem, and prove an important inequality of the Aumann type set-valued Lebesgue integrals of set-valued stochastic processes with respect to t, which are useful to study set-valued stochastic differential inclusions with applications in finance.