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Generalized conditional expectations, optional projections and predictableprojections of stochastic processes play important roles in the general theoryof stochastic processes, semimartingale theory and stochastic calculus. Theyshare some important properties with ordinary conditional expectations. Whilethe characterization of ordinary conditional expectations has been studied byseveral authors, no similar work seems to have been done for these threeconcepts. This paper aims at undertaking this task by givingAnd$\hat{o}$-Douglas type characterization theorem for each of them.

And$\hat{o}$-Douglas type characterization of generalized conditional expectations, optional projections and predictable projections