Before revision:  The well-known "looping default" case is shown to be equipped with this property, to be linked to the classical "Freund distribution" , and to allow for a new construction with immediate multi-variate extensions. If, additionally, all sub-vectors of the default indicator process are also Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory.
After revision:  The well-known "looping default" case is shown to be equipped with this property, to be linked to the classical "Freund distribution" in the bivariate case. On the other hand, if all sub-vectors of the default indicator process are also Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory.