Set-Valued Stochastic Lebesgue Integral And Representation Theorems

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued stochastic process with respect to the time t. Finally we shall prove the presentation theorem of set-valued stochastic integral and discuss further properties that will be useful to study set-valued stochastic differential equations with their applications.