Mutual aid insurance with a three-state Markov chain

Mutual aid insurance is a collective type of insurance where the policyholders share the potential losses or risks that they may face. In this paper, we establish a mathematical structure for mutual aid insurance through a three-state (good, bad and death) process, which is driven by an inhomogeneous Markov chain. The objective of maximizing an individual's lifetime utility is achieved by addressing a stochastic control problem that involves both mutual aid insurance and life insurance. We obtain the explicit expressions for optimal consumption, investment strategies, and life insurance premiums by employing the corresponding Hamilton-Jacobi-Bellman equation. In the end, we carry out a numerical analysis to show the significance of mutual aid insurance and demonstrate the optimal mutual aid insurance premium.