Hazard processes and martingale hazard processes

TL;DR

This paper addresses open problems in credit risk modeling by analyzing hazard processes and martingale hazard processes, proving new properties, and clarifying their relationship for different classes of random times.

Contribution

It proves that continuous Azéma's supermartingales imply the default time avoids stopping times and corrects a conjecture about hazard processes equality, identifying pseudo-stopping times as key cases.

Findings

Continuous Azéma's supermartingales imply default times avoid stopping times.

The conjecture about hazard process equality is disproved and corrected.

Pseudo-stopping times are the most general class where hazard and martingale hazard processes are equal.

Abstract

In this paper, we provide a solution to two problems which have been open in default time modeling in credit risk. We first show that if $\tau$ is an arbitrary random (default) time such that its Az\'ema's supermartingale $Z_t^\tau=\P(\tau>t|\F_t)$ is continuous, then $\tau$ avoids stopping times. We then disprove a conjecture about the equality between the hazard process and the martingale hazard process, which first appeared in \cite{jenbrutk1}, and we show how it should be modified to become a theorem. The pseudo-stopping times, introduced in \cite{AshkanYor}, appear as the most general class of random times for which these two processes are equal. We also show that these two processes always differ when $\tau$ is an honest time.