Before revision:  The stochastic innovation, that we propose herein, consists of singling out three classes of martingales in the large filtration. One of these classes is generated by a new process, up to our knowledge, that has nice features. The three orthogonal martingale classes are vital pillars for establishing our optional martingale representation theorem, when (local ) martingales of the large filtration are stopped at the death time  .
After revision:  The challenge in this setting, when no model specification for these securities nor for the death time is given, lies in finding the dynamics and the structures for the mortality/longevity securities used in the securitization. To overcome this obstacle, we elaborate our optional martingale representation theorem, when (local ) martingales of the large filtration are stopped at the death time .