Utility maximization of household with information learning and health shocks

This paper discusses an optimal investment–consumption-insurance problem for a wage earner incorporating information learning and health shocks. The information learning mechanism is designed based on the historical investment performance. Critical illness insurance and life insurance can be purchased to hedge against health risk and mortality risk, respectively. Moreover, the wage earner allocates her wealth among consumption and three financial assets continuously over time to maximize the expected discounted utilities. Using the dynamic programming principle coupled with the Hamilton–Jacobi-Bellman (HJB) equations, we obtain analytical expressions for optimal strategies and the respective value functions under various health states. Finally, numerical examples are provided to illustrate the impact of health shocks and information learning mechanism on optimal strategies.