Optimal Risk Allocation Under Quasi-convex Risk Measures

In the research of theory and practice of finance,insurance and reinsurance,and related fields, one of the core work is to measure the risk accurately and to optimize the allocation of risk. More and more experts and scholars have given extensive attention and in-depth study to the optimal risk allocations in the realm of various risk measures. In recent years, the research and application of the optimal allocation of financial risks under coherent risk measure and convex risk measure, have obtained a series of results. Taking into account the fact that the quasi-convex risk measure can better describe the liquidity risk in the real financial market, this paper will extend the problem of optimal risk allocation w.r.t convex risk measure to the quasi-convex case on the basis of the previous studies, and some meaningful conclusions are obtained.Firstly, this paper studies the problem of optimal risk allocations for single-asset and multi-assets w.r.t. convex risk measure:derives judgment method that for single asset Pareto optimal allocation and minimizing the total risk allocation are equivalent; studies the problem of optimal risk allocations of risk vectors, derives that pareto optimal allocation equivalents to minimizing the total weighted risk allocation in multi-assets case, provides the the connections between optimal solution and subdifferential, and comonotonic characteristics of optimal allocations under law-invariant case.Secondly, the problem of optimal allocation for risk positions w.r.t. quasi-convex risk measure is established and studied.we proof that the solution of the problem of optimal risk allocations is weakly Pareto optimal in single-asset case, and only ensure the weakly Pareto; proposes the definition of multivariate quasi-convex risk measure, provides the solution of the problem of optimal risk allocations and its contacts with the quasi-convex subdifferential in multi-assets case.Finally, this paper studies the problem of reinsurance policy under quasi-convex risk measures and the analytical solution of the problem is given.The results obtained in this paper are new, which can be regarded as a direct generalization of the related results of risk allocation under the convex risk measure.