Representation theorems, set-valued and fuzzy set-valued Ito integral

In this paper, we shall present representation theorems of set-valued martingales and set-valued processes of finite variation with continuous time. We shall also obtain a representation theorem of a predictable set-valued stochastic process. We shall give a new definition of Ito integral of a set-valued stochastic process with respect to a Brownian motion based on the work [E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stochastic Anal. Appl. 21(2) (2003) 401–418.]. We shall also discuss some properties of set-valued Ito integral, especially the presentation theorem of set-valued Ito integral. Finally, we extend some of above results to the fuzzy set-valued case.