Optimal Investment And Reinsurance Problem For The Insurance Group With Default Risk

With the continuous improvement of people's living standards,the demand for insurance is also increasing,and the domestic insurance market is developing rapidly.With the increasing business of insurance companies,the risks they face are also increasing,and the traditional premium income has been difficult to maintain the company's sustainable operation.As a result,insurance companies reduce risks by buying reinsurance from reinsurers,and increase returns by investing in capital markets.With the continuous expansion of the size of insurance companies,the business of large insurance groups has also expanded from a single insurance business to the simultaneous operation of insurance and reinsurance businesses.It is also more common for insurance groups to have both insurance companies and reinsurance companies.This will not only reduce the risk exposure of the insurance company,but also ensure that the interests of the group will not be affected by excessive reinsurance premiums.Therefore,it is more practical to study the optimal investment and reinsurance problem of insurance company and reinsurance company from the perspective of the group.Therefore,this paper considers the joint interests of the insurance company and the reinsurance company,studying an optimal investment and reinsurance problem of the insurance group with default risk.Assuming that the claim process is described by a Brownian motion with drift,the insurer can purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset,a risky asset and a defaultable bond.Moreover,the reinsurer is allowed to invest in a risk-free asset and a stock.The interest of the insurance group is represented by the exponential utility product of the wealth of the two companies and the weighted sum of the wealth of the two companies,and this paper aims to maximize the expected exponential utility of the insurance group's terminal wealth.By using the dynamic programming approach,the corresponding Hamilton-Jacobi-Bellman(HJB)equations are established for the pre-default case and post-default case,respectively.In both cases,closed-form expressions for the optimal strategies and the corresponding value functions are derived,and the verification theorem is given according to the stochastic control theory.Finally,numerical examples are given to illustrate the effects of model parameters on the optimal investment and reinsurance strategies,and the corresponding economic explanations are given.