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Optimal consumption with reference to past spending maximum

Author

Listed:
  • Shuoqing Deng

    (University of Michigan)

  • Xun Li

    (The Hong Kong Polytechnic University)

  • Huyên Pham

    (Université de Paris and CREST-ENSAE)

  • Xiang Yu

    (The Hong Kong Polytechnic University)

Abstract

This paper studies the infinite-horizon optimal consumption problem with a path-dependent reference under exponential utility. The performance is measured by the difference between the nonnegative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process, and hence the value function depends on two state variables. The Hamilton–Jacobi–Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that a classical solution to the HJB equation and feedback optimal investment and consumption strategies can be obtained in closed form in each region. A complete proof of the verification theorem is provided, and numerical examples are presented to illustrate some financial implications.

Suggested Citation

  • Shuoqing Deng & Xun Li & Huyên Pham & Xiang Yu, 2022. "Optimal consumption with reference to past spending maximum," Finance and Stochastics, Springer, vol. 26(2), pages 217-266, April.
    • Handle: RePEc:spr:finsto:v:26:y:2022:i:2:d:10.1007_s00780-022-00475-w
    DOI: 10.1007/s00780-022-00475-w
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Zongxia Liang & Xiaodong Luo & Fengyi Yuan, 2023. "Consumption-investment decisions with endogenous reference point and drawdown constraint," Mathematics and Financial Economics, Springer, volume 17, number 6, December.
    2. Lijun Bo & Shihua Wang & Xiang Yu, 2022. "A mean field game approach to equilibrium consumption under external habit formation," Papers 2206.13341, arXiv.org, revised Mar 2024.
    3. Geonwoo Kim & Junkee Jeon, 2024. "Optimal Consumption and Investment with Income Adjustment and Borrowing Constraints," Mathematics, MDPI, vol. 12(22), pages 1-14, November.
    4. Chonghu Guan & Jiacheng Fan & Zuo Quan Xu, 2023. "Optimal dividend payout with path-dependent drawdown constraint," Papers 2312.01668, arXiv.org.
    5. Lijun Bo & Yijie Huang & Xiang Yu, 2023. "An extended Merton problem with relaxed benchmark tracking," Papers 2304.10802, arXiv.org, revised Jul 2024.
    6. Minglian Lin & Indranil SenGupta, 2023. "Analysis of optimal portfolio on finite and small-time horizons for a stochastic volatility model with multiple correlated assets," Papers 2302.06778, arXiv.org, revised Dec 2023.
    7. Li, Xun & Yu, Xiang & Zhang, Qinyi, 2023. "Optimal consumption and life insurance under shortfall aversion and a drawdown constraint," Insurance: Mathematics and Economics, Elsevier, vol. 108(C), pages 25-45.
    8. Lijun Bo & Yijie Huang & Kaixin Yan & Xiang Yu, 2024. "Optimal consumption under relaxed benchmark tracking and consumption drawdown constraint," Papers 2410.16611, arXiv.org.
    9. Chonghu Guan & Zuo Quan Xu, 2023. "Optimal ratcheting of dividend payout under Brownian motion surplus," Papers 2308.15048, arXiv.org, revised Jul 2024.

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    More about this item

    Keywords

    Exponential utility; Consumption running maximum; Path-dependent reference; Piecewise feedback control; Verification theorem;
    All these keywords.

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G41 - Financial Economics - - Behavioral Finance - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making in Financial Markets
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory

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