The Optimal Investment-reinsurance Problem Under Uncertain Circumstances

Because of their extensive scope of business coverage,insurance companies are exposed to the outbreak of numerous high-risk events.For example,the sudden outbreak of COVID-19 pandemic in 2019 and the flood in Henan in 2021,both events increase the short-term claims to insurance companies.As a result,the surplus process of insurance companies faces high uncertainty.On the other hand,insurance companies need to allocate the surplus funds to stocks and other financial markets to manage wealth growth.To control the risk in the financial market,insurance companies often adopt conservative investment-reinsurance strategies.In four steps,this analysis examines the optimal investment-reinsurance strategy from two aspects,which are the different risk preferences and competition among insurance companies,under the uncertainty of stock market models.In the first part,we assume that the insurance companies follow exponential utility and they are uncertain on the market stock model and surplus process model.Under this assumption,we introduce stock option trading to hedge the financial market risk.The optimal investment and reinsurance strategies in both cases of complete and incomplete markets are analyzed and deduced.We find a suboptimal strategy can be adopted if an insurance company ignores certain factors,which will cause corresponding utility losses.We specifically examine four suboptimal strategies to calculate the corresponding utility loss and strictly prove that the loss value is positive.Because the reference model and alternative model in the market are difficult to judge statistically,for which reason there exists error probability.We calculate the error probability formula for both cases.In the second part,we use the homotopy analysis method to study the optimal investment and reinsurance problem under uncertain circumstances with general utility functions.By constructing the extended value function and Taylor expansion,the recursive relationship between the front and rear terms of Taylor series is derived.We prove that for any initial guess solution,the Taylor series of the extended value function corresponding to a special parameter can converge to the value function of the original problem.The effectiveness of the homotopy analysis method is verified by comparing its solution with the analytical solution for two special utility functions.To verify the universality of the application of the homotopy analysis method,we demonstrate to solve for the mixed power function utility function which cannot obtain the analytical solutions.In the third part,we consider the case of two insurance companies competing with each other in the market.Here,we introduce the mean variance criterion to consider not only the expectation,but also the variance of wealth value.The optimization goal is the weighted average wealth of both insurance companies.Under the uncertainty of the model,we first establish the alternative model of stock price and surplus process.Utilizing the work of Bjork（2010）,we establish the extended HJB equation corresponding to the optimization value function.The optimal investment and reinsurance strategy at equilibrium is solved with a numerical analysis example.In the last part,we study the optimal investment and reinsurance strategy under uncertain circumstances,when the number of insurance companies competing in the market tends to infinity.The question is analyzed in two steps.Firstly,considering the case of limited insurance companies in the market,the optimization goal is the weighted average of the wealth of insurance company and its competitors.According to the principle of dynamic programming,we derive the HJB equation corresponding to the value function of a single insurance company and solve for the equilibrium strategy.Next,let the number of insurance companies extend to infinity and the parameters of the market model are assumed to be random variables.We analyze the optimal investment and reinsurance strategies of representative insurance companies using the mean field principle.The optimal strategies corresponding to special parameters are used to explain the economic impact of equilibrium strategies.