Generalized conditional expectations, 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. While the characterization of ordinary conditional expectations has been studied by several authors, no similar work seems to have been done for these three concepts. This paper aims at undertaking this task by giving Andô-Douglas type characterization theorem for each of them.
Introduction, notation, and setupProperties of (ordinary) conditional expectation operators have been extensively studied in the literature. In particular, various authors have characterized conditional expectations. Earliest works along this line of research include [7], [21], [25], and [26]. [13] first characterized conditional expectations as contractive projections on L 1 spaces. [2] provided a simple proof of the main theorem in [13]. [6] extended the results of [13] to L p spaces. [23] gave two characterizations of conditional expectations using expectation invariance. [17] applied the theory of Riesz spaces to characterize conditional expectations as order-continuous projections. [12] derived a general characterization theorem of conditional expectation operators. The same result was derived in [18] and [19] independently. A refined account is given in [1]. [8], [9] and [14] studied and characterized conditional expectations with respect to a σ-lattice. Recently, [28] generalized the Andô-Douglas theorem to the Riesz spaces.To our best knowledge, no similar efforts have been made for the generalized conditional expectation (cf. Section I.4 of [15]). This paper aims at filling this gap by proving an Andô-Douglas type characterization theorem for it. It seems that