Optimal Consumption with Reference to Past Spending Maximum

TL;DR

This paper develops a model for optimal consumption that considers past maximum spending, deriving explicit strategies and thresholds using advanced mathematical techniques, with implications for financial decision-making.

Contribution

It introduces a novel approach to optimal consumption with a path-dependent reference, providing explicit solutions and thresholds using dual transforms and smooth-fit principles.

Findings

Explicit closed-form solutions for consumption and investment strategies.

Identification of thresholds in wealth affecting consumption decisions.

Numerical examples illustrating financial implications.

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.