Occupation times, drawdowns, and drawups for one-dimensional regular diffusions

TL;DR

This paper analyzes the behavior of drawdown and drawup processes in one-dimensional regular diffusions, providing probability calculations, occupation time laws, and applications to risk and option pricing, with specific examples including Brownian motion with drift.

Contribution

It introduces new methods to compute probabilities and occupation times for drawdown and drawup processes in diffusions, with applications to risk and financial modeling.

Findings

Derived probability that drawdown exceeds drawup in exponential time

Established laws for occupation times of drawdown and drawup processes

Provided examples with Brownian motion with drift and Bessel processes

Abstract

The drawdown process of an one-dimensional regular diffusion process $X$ is given by $X$ reflected at its running maximum. The drawup process is given by $X$ reflected at its running minimum. We calculate the probability that a drawdown proceeds a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.