Optimal Consumption with Reference to Past Spending Maximum

Shuoqing Deng, Xun Li, Huyên Pham, Xiang Yu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)217-266
Number of pages50
JournalFinance and Stochastics
Issue number2
Publication statusPublished - Apr 2022


  • Consumption running maximum
  • Exponential utility
  • Path-dependent reference
  • Piecewise feedback control
  • Verification theorem

ASJC Scopus subject areas

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


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