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Finance and Stochastics
Published in: Finance and Stochastics 1/2024

06-09-2023

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

Authors: Yike Wang, Jingzhen Liu, Tak Kuen Siu

Published in: Finance and Stochastics | Issue 1/2024

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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.
previous article Optimal investment and consumption for financial markets with jumps under transaction costs
next article A càdlàg rough path foundation for robust finance

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Appendix
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Metadata
Title
Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting
Authors
Yike Wang
Jingzhen Liu
Tak Kuen Siu
Publication date
06-09-2023
Publisher
Springer Berlin Heidelberg
Published in
Finance and Stochastics / Issue 1/2024
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-023-00510-4

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