Optimal investment problem between two insurers with value-added service

Service has become an important factor that affects insurance holders’ purchase behaviors, competition and even the survival of insurers. This paper introduces value-added service into the optimal investment problem between two competing insurers, one provides value-added service while the other does not. The surplus processes of the two insurers are assumed to follow classical Cramér-Lundberg (C-L) model. Both of the two insurers 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 insurer wants to maximize the expectation of the difference between her terminal wealth and that of her competitor, and to minimize the variance of the difference between her terminal wealth and that of her competitor. By solving the corresponding extended Hamilton-Jacobi-Bellman (HJB) equations, we derive the equilibrium service level, investment strategies and the corresponding equilibrium value functions. In addition, some special cases of our model are provided. Finally, the economic implications of our findings are illustrated. It is interesting to find that for the insurer with value-added service, the equilibrium value function in the case of providing value-added service is larger than that without value-added service under some given assumptions.