Andô–Douglas type characterization of optional projections and predictable projections

Abstract

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.

Law of total probability and Bayes' theorem in Riesz spaces

“…The study of stochastic processes in an abstract space can date back to as early as [2]. Works along this line include [4], [5], [15], [26] and [27]. About a decade ago, [17] initiated the study of stochastic processes in a measure-free setting.…”

“…Since then this order-theoretic approach to stochastic processes has been developing fast. For order-theoretic investigation of conditional expectations and related concepts, we refer to [14], [15], [20], [31]; for discrete-time processes in Riesz spaces, we refer to [16], [17], [18], [19], [21], [22], [23], [28]; for continuous-time processes in Riesz spaces, we refer to [6], [7], [8], [9], [10], [13], [29], [30]; for stochastic integrals in Riesz spaces, we refer to [11], [12], [24]. [22] extended the notation of independence to Riesz spaces; [9] gave a slightly more general definition of independence in Riesz spaces.…”