On the characterisation of honest times that avoid all stopping times

TL;DR

This paper characterizes honest times that avoid all stopping times as the last time of maximum of certain continuous local martingales, providing a clear criterion and illustrative examples.

Contribution

It offers a concise proof linking honest times avoiding all stopping times to the last maximum of continuous local martingales with specific properties.

Findings

Honest times avoiding all stopping times are characterized by the last maximum of certain local martingales.

The paper provides a self-contained proof of this characterization.

Examples include local martingales with discontinuous paths.

Abstract

We present a short and self-contained proof of the following result: a random time is an honest time that avoids all stopping times if and only if it coincides with the (last) time of maximum of a nonnegative local martingale with zero terminal value and no jumps while at its running supremum, where the latter running supremum process is continuous. Illustrative examples involving local martingales with discontinuous paths are provided.