Optional Service Problem between Two Insurance Companies with Investment

Service has become an important factor that affects insurance holders' purchase behaviors, competition between insurance companies, and even the survival of insurance companies. This paper first introduces the service quality into the optimal investment problem between two competing insurance companies, company 1 and 2. Company 1 provides service while company 2 does not, which results in attracting insurance business from company 2 to company 1, partly. Moreover, the service quality affects the cost, the premium and the demand of the insurance. The surplus processes of the two insurance companies are assumed to follow classical Cram´er-Lundberg (C-L) model. Both the two insurance companies are allowed to invest in a risk-free asset and two different risky assets, respectively. Dynamic mean-variance criterion is considered in this paper. Each insurance company wants to maximize the expectation of the difference of wealth between itself and the other one, and to minimize the variance. By applying stochastic control approach, we establish the corresponding extended Hamilton-Jacobi-Bellman (HJB) system of equations. Furthermore, we derive the equilibrium service and investment strategies, and the corresponding equilibrium value function by solving the extended HJB system of equations. In addition, some special cases of our model are provided, which show that our model and results extend some existing ones in the literature. Finally, the economic implications of our findings are illustrated. It is interesting to find that the equilibrium value function of company 1 who provides service is larger than that of company 2, and for company 1, the equilibrium value function with service is larger than that without service, while the situation of company 2 is opposite