Stopping Times Occurring Simultaneously
Published in ESAIM: PS, 2024
Recommended citation: P. Protter, A. Quintos, “Stopping Times Occurring Simultaneously”, ESAIM: PS, 28, 110-131 (2024). https://doi.org/10.1051/ps/2024001
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.