Optional and predictable projections of normal integrands and convex-valued processes

TL;DR

This paper investigates the existence, uniqueness, and duality properties of optional and predictable projections of integrands and convex-valued processes, extending classical results to set-valued and convex cases without relying on reference measures.

Contribution

It establishes general conditions for projections of integrands and convex processes, including duality relations and set-valued extensions, broadening the theoretical framework.

Findings

Existence and uniqueness of projections under general conditions

Duality correspondences between projections and epigraphs

Extensions to set-valued integrands and processes

Abstract

This article studies optional and predictable projections of integrands and convex-valued stochastic processes. The existence and uniqueness are shown under general conditions that are analogous to those for conditional expectations of integrands and random sets. In the convex case, duality correspondences between the projections and projections of epigraphs are given. These results are used to study projections of set-valued integrands. Consistently with the general theory of stochastic processes, projections are not constructed using reference measures on the optional and predictable sigma-algebras.