On a dyadic approximation of predictable processes of finite variation

TL;DR

This paper demonstrates that predictable processes of finite variation can be approximated by elementary processes, leading to classical results about predictable stopping times and increasing processes.

Contribution

It introduces a dyadic approximation method for predictable processes of finite variation, providing new insights into their structure and properties.

Findings

Predictable stopping times can be approximated from below by finitely-valued predictable stopping times.

Predictable processes of finite variation are limits of elementary predictable processes.

Classical theorems on predictability and naturalness of increasing processes are derived.

Abstract

We show that any cadlag predictable process of finite variation is an a.s. limit of elementary predictable processes; it follows that predictable stopping times can be approximated `from below' by predictable stopping times which take finitely many values. We then obtain as corollaries two classical theorems: predictable stopping times are announceable, and an increasing process is predictable iff it is natural.