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

Yike Wang, Jingzhen Liu () and Tak Kuen Siu
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Yike Wang: Chongqing Technology and Business University
Jingzhen Liu: Central University of Finance and Economics

Finance and Stochastics, 2024, vol. 28, issue 1, No 5, 214 pages

Abstract: Abstract 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.

Keywords: Investment–consumption–insurance management; Habit formation; Non-exponential discounting; Stochastic maximum principle; Open-loop Nash equilibrium control; 93E20; 91G80; 91B08; 91B42; 49N90; 35A22 (search for similar items in EconPapers)
JEL-codes: C61 C73 D11 G11 (search for similar items in EconPapers)
Date: 2024
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00780-023-00510-4

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