Andô–Douglas type characterization of optional projections and predictable projections

Optional projections and predictable projections of stochastic processes play important roles in the general theory of stochastic processes, semimartingale theory and stochastic calculus. They share some important properties with ordinary conditional expectations and generalized conditional expectations. While the characterization of ordinary conditional expectations and generalized conditional expectations has been studied by several authors, no similar work has been done for optional projections and predictable projections. This paper aims at undertaking this task by giving Andô–Douglas type characterization theorem for both.