Stopping Times Occurring Simultaneously

TL;DR

This paper introduces a new framework for modeling stopping times that can be dependent and even equal, expanding beyond the traditional assumption of conditional independence, with applications in epidemiology, engineering, and finance.

Contribution

It develops a novel family of dependent stopping times using a modified Cox construction and bivariate exponential, allowing for positive and negative dependence, including simultaneous stopping.

Findings

Constructed a family of dependent stopping times with positive dependence.

Proposed a joint distribution enabling negative dependence and simultaneous stopping.

Illustrated applications in epidemic modeling, civil engineering, and credit risk.

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.