  We investigate under which conditions a single simulation of joint default
times at a final time horizon can be decomposed into a set of simulations of
joint defaults on subsequent adjacent sub-periods leading to that final
horizon. Besides the theoretical interest, this is also a practical problem as
part of the industry has been working under the misleading assumption that the
two approaches are equivalent for practical purposes. As a reasonable trade-off
between realistic stylized facts, practical demands, and mathematical
tractability, we propose models leading to a Markovian multi-variate
survival--indicator process, and we investigate two instances of static models
for the vector of default times from the statistical literature that fall into
this class. On the one hand, the "looping default" case is known to be equipped
with this property, and we point out that it coincides with the classical
"Freund distribution" in the bivariate case. On the other hand, if all
sub-vectors of the survival indicator process are Markovian, this constitutes a
new characterization of the Marshall--Olkin distribution, and hence of
multi-variate lack-of-memory. A paramount property of the resulting model is
stability of the type of multi-variate distribution with respect to elimination
or insertion of a new marginal component with marginal distribution from the
same family. The practical implications of this "nested margining" property are
enormous. To implement this distribution we present an efficient and unbiased
simulation algorithm based on the L\'evy-frailty construction. We highlight
different pitfalls in the simulation of dependent default times and examine,
within a numerical case study, the effect of inadequate simulation practices.
