Three essays on stopping

TL;DR

This paper provides new formulas and conditions for the distribution of stopping times and maximums in stochastic processes like reflected Brownian motion and spectrally negative Lévy processes, extending existing theoretical results.

Contribution

It introduces a closed-form formula for first passage times of reflected Brownian motion with drift and characterizes when the maximum before a fixed drawdown is exponentially distributed.

Findings

Closed-form formula for first passage time of reflected Brownian motion with drift.

Exponential distribution of maximum before drawdown if and only if diffusion coefficient is constant.

Alternative proof that maximum at fixed drawdown is exponentially distributed for spectrally negative Lévy processes.

Abstract

First, we give a closed-form formula for first passage time of a reflected Brownian motion with drift. This modifies a formula by Perry et al (2004). Second, we show that the maximum before a fixed drawdown is exponentially distributed for any drawdown threshold, if and only if the diffusion characteristic mu/sigma^2 is constant. This complements the sufficient condition formulated by Lehoczky (1977). Third, we give an alternative proof for the fact that the maximum at a fixed drawdown threshold is exponentially distributed for any spectrally negative L\'evy process, a result due to Mijatovic and Pistorius (2012).