Representation theorems, set-valued and fuzzy set-valued Itô integral (Q878973)

The authors begin by discussing quite usual properties of set-valued martingales with continuous time, in particular their Castaing representations by means of martingale selections. This is used to define a variant of the integral of a set-valued square integrable process with respect to the Brownian motion. This definition relies on taking integrals of predictable selections of a set-valued process. The stochastic integral then becomes a set-valued process, whose basic properties (like selection representation and linearity) are established. Finally, the authors provide a rather straightforward generalisation for the case of fuzzy set-valued stochastic processes (i.e. monotonic families of set-valued processes).