Abstract
This paper is devoted to an investment-consumption and life insurance problem with habit formation and non-exponential utility discounting. General utility functions are employed to evaluate non-habitual consumption and bequest. Distinct from Liu et al. (Non-exponential discounting portfolio management with habit formation. 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 seek open-loop control for each of the pre-commitment solutions and the time-consistent solution. Since the model coefficients are allowed to be random, we use the stochastic maximum principle arising from a perturbation argument to solve our problems. For each of the pre-commitment solutions and the time-consistent solution, the analytical expression is obtained via a flow 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 the semi-analytical solution with numerical results based on simulations for the constant relative risk aversion (CRRA) utility with hyperbolic