Research On Some Optimal Mean-Variance Investment And Reinsurance Problems

This paper considers optimal investment and reinsurance problems under the meanvariance criterion,which involves pre-commitment strategy,open-loop equilibrium strategy and closed-loop equilibrium strategy.First,we consider optimal mean-variance reinsurance and investment problem with constraints in a non-Markovian regime-switching model.The strategies are constrained in the non-negative cone and all coefficients in the model except the interest rate are stochastic processes adapted the filtration generated by a Markov chain.With the help of a backward stochastic differential equation driven by the Markov chain,we obtain the optimal strategy and optimal cost explicitly under the non-Markovian regime-switching model.The cases with one risky asset and Markov regime-switching model are considered as special cases.Second,we consider a general mean-variance reinsurance,new business and investment problem,where the claim processes of original and new businesses are modeled by two different risk processes and the safety loadings of reinsurance and new business are different.The retention level of the insurer is constrained in[0,1]and the controls of new business and risky investment are required to be non-negative.This model relaxes the limitations of those in existing research.By using the projection onto the convex set controls valued in,we obtain an open-loop equilibrium reinsurance-new business-investment strategy explicitly.We also show that the obtained equilibrium strategy is the optimal one among all deterministic strategies in the sense that it yields the smallest mean-variance cost.In the case where original and new businesses are the same,the equilibrium strategy is given in closed-form and its sensitivities to safety loadings are shown by numerical examples.At last,by comparing with the case where acquiring new business is prohibited,we show that allowing writing new policies indeed improves the performance of the insurer's risk management.Finally,we study an optimal reinsurance problem under the ?-maximin meanvariance criterion proposed in Li et al.(2016).We generalize Li et al.(2016)by considering a full range of ambiguity preferences and allowing general form of reinsurance contracts.We find equilibrium reinsurance strategies are of the excess-of-loss form for ambiguityaverse preferences.But for ambiguity-loving preferences,excess-of-loss reinsurance may not be optimal,and equilibrium strategy may not be unique.An ambiguity-averse insurer retains less risk if she is more ambiguous to the reference measure,but an ambiguity-loving insurer not necessarily retains more risk if she has a high level of ambiguity.Our finding suggests that a highly-ambiguity-loving preference may only exhibit when the ambiguity level is very low,and hence,consistent with the empirical studies showed that decision makers can be ambiguity loving if they consider themselves more knowledgeable or competent than the other players.