In this paper 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 death time  of an insured. By using the expansion of 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  market model and  time of death, we address the risk-minimisation problem for a large class of equity-linked mortality and/or mortality contracts. 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  . This constitutes our first main original contribution, while the second main contribution lies in describing, as explicit as possible, the optimal strategy when hedging mortalityrisks using the optional martingale representation .
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 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 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 .
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In this paper we consider
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that contains
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risk-minimisation problem
risk-minimization problem \`a la F\"ollmer-Sondermann
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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
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
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theorem, when (local ) martingales of
results, which state that any local martingale in
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are
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can be decomposed into several and precise orthogonal local martingales
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main original contribution, while the second main contribution lies in describing, as explicit as possible, the optimal strategy when hedging mortalityrisks using the optional martingale representation
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