Time Reversal and Last Passage Time of Diffusions with Applications to Credit Risk Management

TL;DR

This paper explores the mathematical properties of diffusions related to time reversal and last passage times, applying these insights to develop a novel risk management framework for companies based on leverage processes.

Contribution

It introduces new analytical tools for diffusions with killing and applies them to optimize alarming levels in credit risk management.

Findings

Derived probability density of last passage times for diffusions with killing

Analyzed distribution of time until killing after last passage

Proposed an optimization framework for alarming levels

Abstract

We study time reversal, last passage time, and $h$-transform of linear diffusions. For general diffusions with killing, we obtain the probability density of the last passage time to an arbitrary level and analyze the distribution of the time left until killing after the last passage time. With these tools, we develop a new risk management framework for companies based on the leverage process (the ratio of a company asset process over its debt) and its corresponding alarming level. We also suggest how a company can determine the alarming level for the leverage process by constructing a relevant optimization problem.