Remarks on unboundedness of set-valued Itô stochastic integrals

The paper deals with set-valued stochastic integrals which are defined as a random multifunctions. Integrable boundedness of a such integral is one of the most important features in potential applications. Unfortunately, up to now there were no correct proofs of such property. Surprisingly, also a negative answer to this problem is still not explained correctly. Hence the problem seems to be still undetermined. We shall show that in general the answer is negative. We shall provide several relatively simple examples both in convex and nonconvex-valued case. As a consequence, we will show that under some restrictions on the class of selections of the integrand the integrable boundedness of set-valued Itô's integral is equivalent to its single valuedness.