Optimal Investment–Consumption–Insurance Problem of a Family with Stochastic Income under the Exponential O-U Model

Yang Wang (), Jianwei Lin, Dandan Chen and Jizhou Zhang
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Yang Wang: School of Finance and Business, Shanghai Normal University, Shanghai 200234, China
Jianwei Lin: Fujian Key Laboratory of Financial Information Processing, Putian University, Putian 351100, China
Dandan Chen: Mathematics and Science College, Shanghai Normal University, Shanghai 200234, China
Jizhou Zhang: School of Finance and Business, Shanghai Normal University, Shanghai 200234, China

Mathematics, 2023, vol. 11, issue 19, 1-19

Abstract: A household consumption and optimal portfolio problem pertinent to life insurance (LI) in a continuous time setting is examined. The family receives a random income before the parents’ retirement date. The price of the risky asset is driven by the exponential Ornstein–Uhlenbeck (O-U) process, which can better reflect the state of the financial market. If the parents pass away prior to their retirement time, the children do not have any work income and LI can be purchased to hedge the loss of wealth due to the parents’ accidental death. Meanwhile, utility functions (UFs) of the parents and children are individually taken into account in relation to the uncertain lifetime. The purpose of the family is to appropriately maximize the weighted average of the corresponding utilities of the parents and children. The optimal strategies of the problem are achieved using a dynamic programming approach to solve the associated Hamilton–Jacobi–Bellman (HJB) equation by employing the convex dual theory and Legendre transform (LT). Finally, we aim to examine how variations in the weight of the parents’ UF and the coefficient of risk aversion affect the optimal policies.

Keywords: life insurance (LI); optimal investment and consumption (OIC); exponential O-U progress; HJB equation; Legendre transform (LT) (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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