Stopping times are hitting times: a natural representation

TL;DR

This paper establishes a clear, didactic correspondence between stopping times and certain non-increasing adapted processes, demonstrating that all stopping times can be viewed as hitting times, which simplifies their understanding and representation.

Contribution

It introduces a natural, one-to-one representation linking stopping times with specific adapted processes, providing a new perspective on their structure.

Findings

Stopping times are equivalent to hitting times under a natural representation.

A one-to-one relationship exists between stopping times and non-increasing adapted processes.

This representation simplifies the conceptual understanding of stopping times.

Abstract

There exists a simple, didactically useful one-to-one relationship between stopping times and adapted c\`agl\`ad (LCRL) processes that are non-increasing and take the values 0 and 1 only. As a consequence, stopping times are always hitting times.