Mortality Risk Minimisation and Optional Martingale Representation Theorem for Enlarged Filtration

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 mortality risks using the optional martingale representation.