Stopping times occurring simultaneously

¹ Department of Statistics, Columbia University, New York, NY 10027, USA
² Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706, USA

^* Corresponding author: alejandra.quintos@wisc.edu

Received: 9 May 2023
Accepted: 15 February 2024

Abstract

Stopping times are used in applications to model random arrivals. A standard assumption in many models is that they are conditionally independent, given an underlying filtration. This is a widely useful assumption, but there are circumstances where it seems to be unnecessarily strong. We use a modified Cox construction along with the bivariate exponential introduced by Marshall and Olkin (1967) to create a family of stopping times, which are not necessarily conditionally independent, allowing for a positive probability for them to be equal. We show that our initial construction only allows for positive dependence between stopping times, but we also propose a joint distribution that allows for negative dependence while preserving the property of non-zero probability of equality. We indicate applications to modeling COVID-19 contagion (and epidemics in general), civil engineering, and to credit risk.