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 ris