Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting

This paper is devoted to an investment–consumption and life insurance problem with habit formation and non-exponential discounting. General utility functions are employed to evaluate non-habitual consumption and bequest. Distinct from Liu et al. in (Math. Control Relat. Fields 10:761–783, 2020) for consumption habit and feedback control, we assume that past consumption and bequest amounts have an interaction in formulating their endogenous reference levels, and we seek open-loop controls for both the pre-commitment solution and the time-consistent solution. Since the model coefficients are allowed to be random, we use the stochastic maximum principle to solve our problems. For both the pre-commitment and the time-consistent solution, an analytical expression is obtained via a system of forward-backward stochastic differential equations. Additionally, when the model coefficients are Markovian, we show that our problem for open-loop control can also be reduced to solving a Hamilton–Jacobi–Bellman equation, and then we introduce a transformation method for solving the equation. In particular, we provide a semi-analytical solution with numerical results based on simulations for the constant relative risk aversion (CRRA) utility with hyperbolic discounting.