This paper considers a market model with two levels of information. The public information generated by the financial assets, and a larger flow of information containing additional knowledge about a death time(random time /horizon) of an insured. By expanding the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. In this context , which catches real features such as correlation between the market model and the time of death, we address the risk-minimization problem \`a la F\"ollmer-Sondermann for a large class of equity-linked mortality contracts. The challenge in this setting, when no modelspecification for these securities nor for the death timeis 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 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 . This constitutes our first principal novel contribution. Thanks to this optional representation, we succeed to decompose the risk in some popular mortality and/or longevity securities into the sum of orthogonal risks using a risk basis. One of the components of this basis is a new martingale, in the large filtration, that possesses nice features. Hence, the dynamics of mortality and longevity securities used in the securitization is described without mortality specification, and this constitutes our second novel contribution. Our third main contribution resides in finding explicitly the risk-minimization strategy as well as the corresponding undiversified risk for a largest class of mortality/longevity linked liabilities with or without the mortality securitization .
We consider a market model where there are two levels of information. The public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm for credit risk, and the death time of an insured for life insurance. By using the expansion of filtration, the random time uncertainty and its entailed risk are fully considered without any mathematical restriction. In this context with no model's specification for the random time, the main challenge lies in finding the dynamics and the structures for the value processes of defaultable or mortality and / or longevity securities which are vital for the insurance securitization. To overcome this obstacle, we elaborate our optional martingale representation results, which state that any martingale in the large filtration stopped at the random time can be decomposed into precise and unique orthogonal local martingales (i.e. local martingales whose product remains a local martingale). This constitutes our first and probably the principal contribution. Even though the driving motivation for this representation resides in credit risk theory, our results are applicable to several other financial and economics contexts, such as life insurance and financial markets with random horizon. Thanks to this optional representation, we decompose any defaultable or mortality and/or longevity liability into the sum of "non-correlated" risks using a risk basis. This constitutes our second contribution .
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