Before revision:  To overcome this obstacle, we elaborate our optional martingale representation results, which state that any local martingale in the large filtration stopped at the death time can be decomposed into several and precise orthogonal local martingales .
After revision:  To overcome this obstacle, we elaborate our optional martingale representation results, which state that any local martingale in the large filtration stopped at the death time can be decomposed into several and precise orthogonal local martingales (i.e. local martingales whose product remains a local martingale).