Research On Investment And Reinsurance Problem With A Defaultable Security Under The Smooth Ambiguity Utility

With the continuous improvement of people’s living standards,the domestic insurance industry is developing rapidly.With the increasing business of insurance companies,the risks they faced are also increasing.In order to ensure the sustainable operation and development of the companies,insurance companies invest the surplus and obtain income from the financial investment to improve their compensation ability.At the same time,the insurance companies purchase reinsurance from the reinsurance companies to avoid the risk of their own claims.In recent years,insurers are actively investing in defaultable bonds.Defaultable corporate bonds are less risky than stock investments,and more profitable than risk-free bonds.Therefore,it is of great significance to study the investment portfolio of insurance companies with defaultable bonds.In addition,the parameters in the financial model are difficult to estimate with precision,therefore,it is very necessary to consider the impact of model uncertainty on insurance companies’ investment and reinsurance strategies.Some literatures use robust optimal strategies for solving the worst-case scenario to deal with model uncertainty,but this will cause the strategy to be too conservative and result in significant utility loss,so this paper considers the robust strategies under the average case.This paper considers the optimal investment and reinsurance problem with a defaultable security for an insurer in an environment with parameter uncertainties.Suppose that the insurer is uncertain about the specific information of the claim,specifically,the insurance claim is exponentially distributed,the rate parameter in the exponential distribution and the arrival intensity of the claims are uncertain,respectively.The insurer is allowed to invest in a risk-free bond,a stock and a defaultable bond,where the stock’s price process satisfies the Heston’s SV model.Moreover,the insurer is allowed to purchase proportional reinsurance and aims to maximize the smooth ambiguity utility of the terminal wealth of the insurer.By applying the approach of stochastic control,according to the theory of stochastic control,the corresponding HJB equations are established for the pre-default case and post-default case based on the principle of dynamic programming and the time-consistent investment and reinsurance strategies of the insurer were derived,respectively.Finally,a sensitivity analysis of the strategies is provided through numerical simulation,the effects of model parameters on the equilibrium investment and reinsurance strategies is discussed.