关键词: 投资与风险控制, 非零和博弈, 纳什均衡, Hamilton-Jacobi-Bellman方程

Abstract: Recently, GRASSELLI (2017) proposed a new SV model, called the 4/2 (that is, 1/2+3/2) SV model, which assumes that the instaneous variance is a linear combination of the 1/2 and the 3/2 terms. So, it combines the properties of both Heston’s SV and 3/2 models. In addition, the 4/2 model has some new features that are not contemplated in Heston’s SV model and 3/2 model. In view of the advantages of the 4/2 SV model, in this paper we try to consider the optimal investment and risk control strategies for two competing insurers under relative performance criteria. Specially, we set up a combined financial and insurance market consisting of one risk-free asset, one risky asset, and two dependent risk process representing the liabilities per unit (or per policy), for k=1,2. We apply the standard Cramér-Lundberg diffusion approximation model for the risk process R_k, and assume that each insurer can directly control her liability exposure, or alternatively, each insurer can decide the total amount of liabilities measured by units (or the number of policies) L_k times R_k. One can easily see that such an assumption is equivalent to allowing the insurer to purchase proportional reinsurance to manage her risk exposure from underwriting. Under the framework of Nash equilibrium theory, the relative performance concerns of insurers is used to describe their game behaviors,a non-zero-sum game model is constructed which maximizes the expected exponential utility of his terminal surplus relative to that of his competitor. Applying the techniques of stochastic dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equations for both insurers are obtained, and the Nash equilibrium strategies of both insurers are established by solving HJB equations. Moreover, two special cases are discussed. Finally, some numerical examples are conducted to illustrate the effects of several model parameters on the Nash equilibrium strategies and draw some economic interpretations from these results.
Numerical examples demonstrate that the relative performance concerns of the insurer increase the retained insurance policies and the amount invested in the risky asset, which implies that the competition would lead the insurers to be much more risk-seeking.
Several possible extensions of our work deserve further investigation. The first extension is to apply other utility functions in establishing objective functions in the game framework. Under such a formulation, explicit expressions for Nash equilibrium strategies might be difficult to derive. However, we could apply suitable numerical approximation methods when solving the system of HJB equations. Furthermore, this paper only considers games with perfect revelation, or perfect observation. It is well known that such assumption is too stringent, and the partial observation assumption is more realistic. In such a case, it would be interesting to investigate the game with asymmetric information. We leave these suggestions for future research.

Key words: investment and risk control, non-zero-sum game, Nash equilibrium, Hamilton-Jacobi-Bellman equation