Interval-valued Stochastic Processes and Stochastic Integrals

In this paper, we study interval-valued stochastic processes and stochastic integrals with respect to real-valued Brownian motion. Especially for interval-valued martingale we obtain several equivalent propositions based on measurable selections. By using Castaing representation of set-valued random variables we prove that an interval-valued integral may be not an interval-valued martingale but an interval-valued submartingale, which is different from single valued stochastic integrals.