Optimal Consumption-Investment for General Utility with a Drawdown Constraint over a Finite-Time Horizon

TL;DR

This paper develops a model for optimal consumption and investment over a finite horizon considering loss aversion and drawdown constraints, using advanced stochastic control techniques to derive explicit strategies.

Contribution

It introduces a dual transformation approach to solve a complex nonlinear HJB variational inequality with drawdown constraints, providing explicit feedback strategies.

Findings

Explicit characterization of optimal strategies via dual formulation

Existence of solutions to the obstacle problem under standard constraints

Derivation of time-dependent free boundaries for the control problem

Abstract

We study an optimal investment and consumption problem over a finite-time horizon, in which an individual invests in a risk-free asset and a risky asset, and evaluate utility using a general utility function that exhibits loss aversion with respect to the historical maximum of consumption. Motivated by behavioral finance and habit formation theory, we model the agent's preference for maintaining a standard of living by imposing constraints on declines from the peak consumption level. To solve the resulting Hamilton-Jacobi-Bellman (HJB) variational inequality, which is fully nonlinear, we apply a dual transformation, transforming the original problem into a linear singular control problem with a constraint. By differentiating the value function further, we reduce the constrained linear singular control problem to a linear obstacle problem. We prove the existence of a solution to the obstacle problem under standard constraints. It allows us to characterize the optimal consumption and investment strategies through piecewise analytical feedback forms derived from the dual formulation. Our analysis contributes to the literature on habit formation, drawdown constraints, and stochastic control by explicitly characterizing the time-dependent free boundaries and the associated optimal feedback strategies.