Mutual aid insurance with a three-state Markov chain

ABSTRACT

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.

KEYWORDS:

• Control
• mutual aid insurance
• three-state process
• inhomogeneous Markov chain

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is the interest rate of U.S. Treasury bonds with a 3-month maturity from 2009 to 2020, which is obtained from the Fred database.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant numbers 12371475, 12171360, 11871052 and 11771329], Key Project of the National Social Science Foundation of China [grant number 21AZD071], and Natural Science Foundation of Tianjin City [grant number 20JCYBJC01160].