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Optimal Consumption with Reference to Past Spending Maximum

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Listed:
  • Shuoqing Deng
  • Xun Li
  • Huyen Pham
  • Xiang Yu

Abstract

This paper studies the infinite-horizon optimal consumption 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 the 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 & Huyen Pham & Xiang Yu, 2020. "Optimal Consumption with Reference to Past Spending Maximum," Papers 2006.07223, arXiv.org, revised Mar 2022.
    • Handle: RePEc:arx:papers:2006.07223
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. 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.
    3. Chonghu Guan & Jiacheng Fan & Zuo Quan Xu, 2023. "Optimal dividend payout with path-dependent drawdown constraint," Papers 2312.01668, arXiv.org.
    4. Chonghu Guan & Zuo Quan Xu, 2023. "Optimal ratcheting of dividend payout under Brownian motion surplus," Papers 2308.15048, arXiv.org.
    5. Lijun Bo & Huafu Liao & Xiang Yu, 2020. "Optimal Tracking Portfolio with A Ratcheting Capital Benchmark," Papers 2006.13661, arXiv.org, revised Apr 2021.
    6. 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.
    7. Bahman Angoshtari & Erhan Bayraktar & Virginia R. Young, 2021. "Optimal Investment and Consumption under a Habit-Formation Constraint," Papers 2102.03414, arXiv.org, revised Nov 2021.
    8. 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, June.
    9. Lijun Bo & Yijie Huang & Xiang Yu, 2023. "An extended Merton problem with relaxed benchmark tracking," Papers 2304.10802, arXiv.org, revised Mar 2024.

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