Optimal individual health insurance strategy combined investment, consumption, income and public insurance

In this paper, we investigate the optimal investment, consumption, and insurance problem for the policyholder who participates in a type of practical hybrid insurance to mitigate the tail risks of disease expenditures, complementing public health insurance. The policyholder invests with jumps, and the consumption process is also incorporated into the personal health insurance model, which is supported by a salary income and initial assets. By employing a stochastic optimal control framework, we explicitly derive both the optimal strategy and its corresponding value function. Moreover, a verification theorem is established, and the characteristics of the optimal strategy are analyzed. Lastly, a numerical example is provided to demonstrate how the model parameters affect both the optimal strategy and the value function.