Generalization of Doob decomposition Theorem

TL;DR

This paper generalizes the classical Doob decomposition theorem to local regular supermartingales relative to a convex set of equivalent measures in the discrete case, expanding its applicability.

Contribution

It introduces the concept of local regular supermartingales relative to convex measure sets and proves an optional Doob decomposition for them.

Findings

Generalization of Doob decomposition to convex measure sets

Introduction of local regular supermartingales concept

Proof of optional Doob decomposition in discrete case

Abstract

In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it an optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of equivalent measures.