Before revision:  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:  If, additionally, all sub-vectors of the survival indicator process are also Markovian, this constitutes a new characterization of the Marshall-Olkin distribution, and hence of multi-variate lack-of-memory.