Optimal Consumption with Loss Aversion and Reference to Past Spending Maximum

Xun Li, Xiang Yu, Qinyi Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review


This paper studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the nonnegative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the utility with respect to consumption, allowing us to focus on an auxiliary HJB variational inequality on the strength of concavification principle and dynamic programming arguments. By applying the dual-transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in piecewise closed form, and some thresholds of the wealth variable are obtained. The optimal consumption and investment control can be derived in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.

Original languageEnglish
Pages (from-to)121-160
Number of pages40
JournalSIAM Journal on Financial Mathematics
Issue number1
Publication statusPublished - Mar 2024


  • concave envelope
  • loss aversion
  • optimal relative consumption
  • path-dependent reference
  • piecewise feedback control

ASJC Scopus subject areas

  • Numerical Analysis
  • Finance
  • Applied Mathematics


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