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We consider a market model where there are two levels of information. Thepublic information generated by the financial assets, and a larger flow ofinformation that contains additional knowledge about a random time. This randomtime can represent many economic and financial settings, such as the defaulttime of a firm for credit risk, and the death time of an insured for lifeinsurance. By using the expansion of filtration, the random time uncertaintyand its entailed risk are fully considered without any mathematicalrestriction. In this context with no model's specification for the random time,the main challenge lies in finding the dynamics and the structures for thevalue processes of defaultable or mortality and/or longevity securities whichare vital for the insurance securitization. To overcome this obstacle, weelaborate our optional martingale representation results, which state that anymartingale in the large filtration stopped at the random time can be decomposedinto precise and unique orthogonal local martingales (i.e. local martingaleswhose product remains a local martingale). This constitutes our first andprobably the principal contribution. Even though the driving motivation forthis representation resides in credit risk theory, our results are applicableto several other financial and economics contexts, such as life insurance andfinancial markets with random horizon. Thanks to this optional representation,we decompose any defaultable or mortality and/or longevity liability into thesum of "non-correlated" risks using a risk basis. This constitutes our secondcontribution.

A martingale representation theorem and valuation of defaultable securities