Optimal Consumption with Reference to Past Spending Maximum

Abstract

This paper studies the infinite horizon optimal consumption with a path-dependent reference under the exponential utility. The performance is measured by the difference between the non-negative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional. 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 the classical solution to the HJB equation and the feedback optimal investment and consumption strategies can be obtained in the closed form in each region. The complete proof of the verification theorem is provided and numerical examples are presented to illustrate some financial implications.