Optimal Strategies for a Nonlinear Premium-Reserve Model in a Competitive Insurance Market

The calculation of a fair premium is always a challenging topic in the real world insurance applications. In this paper, a nonlinear premium-reserve (P-R) model is presented and the premium is derived by minimizing a quadratic performance criterion. The reserve is a stochastic equation, which includes an additive random nonlinear function of the state, premium and not necessarily Gaussian noise which is, however, independently distributed in time, provided only that the mean value and the covariance of the random function is zero and a quadratic function of the state, premium and other parameters, respectively. In this quadratic representation of the covariance function, new parameters are implemented and enriched further the previous linear models, such as the income insurance elasticity of demand, the number of insured and the inflation in addition to the company's reputation. The quadratic utility function concerns the present value of the reserve. Interestingly, for the very first time, the derived optimal premium in a competitive market environment is also depended on the company's reserve among the other parameters. Finally, a numerical application illustrates the main findings of the paper