An optimal allocation of risk and insurance problems (Q2839624)

The thesis considers a model of a reinsurance market with \(n\) insurance companies each holding a portfolio of insurance contracts. In this reinsurance market the companies can conclude treaties which redistribute the commitments that the companies had in the initial situation. The methods of convex analysis are used to characterize optimal allocation with respect to the subgradients of convex lower semicontinuous nonfinite functions defined on products of Banach spaces and to give appropriate existence and uniqueness results in the framework.NEWLINENEWLINE The core of the study is a generalization of the results of two earlier papers which allows to include more abstract function spaces like Orlitz spaces and Orlitz hearts. The thesis is divided into three parts.NEWLINENEWLINE The first part collects the terminology and standard results of convex analysis (Chapter 1) and makes a short review of the history of the robust representation of risk measures (Chapter 2).NEWLINENEWLINE Part II fully describes optimal allocations as solutions of the convex minimization problem \(\inf_y\{f(x-y)+g(y)\}\) in a general environment of products of Banach spaces, dealing with their existence and uniqueness (Chapter 3). Then, the results developed in Chapter 3 are used to give a new methodology to derive optimal allocations with respect to the expected risk on Orlitz hearts (Chapter 4). Chapter 5, the last chapter of Part II, draws a connection between optimal allocation with respect to law invariant multivariate risk measure on \(L_d^p(P)\) and the concepts of \(\mu\)-comonotonicity, maximal correlation risk measures and worst-case scenarios.NEWLINENEWLINE Part III deals with the variation \(\inf_y\{f(x-y+g(y))\}\) of the optimal allocation problem studied in Part II and also gives interpretations as insurance games as well as models extensions and versions.NEWLINENEWLINE The thesis is an interesting view on optimal risk allocation with applications to insurance.