Before revision:  Thereby, we derive analytically the dynamics of the default process by set-valued Markov chain approach. Applying the new framework, we give analytical pricing formulaes for synthetic collateralized debt obligations (CDOs).
After revision:  Thereby, we derive analytically the dynamics of the default process by set-valued Markov chain to model the default process, which is the set of all defaulted obligors in the group. We obtain analytic characterizations for the default process, and use them to derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs).