Finite-Horizon Consumption and Investment Problem with Loss Aversion to Consumption Changes

Abstract. In this paper, we investigate the optimal consumption and investment problem for an agent exhibiting constant intertemporal loss aversion (ILA) over a finite planning horizon, meaning that the agent’s sensitivity to consumption reductions remains time-invariant and behaviorally consistent across the horizon. The agent incurs a utility penalty whenever consumption decreases, with the cost proportional to the corresponding utility loss and scaled by a time-varying function. Adopting the dual-martingale approach, we derive the agent’s dual problem, which takes the form of a finite-horizon, two-sided singular control problem. The dual value function is shown to satisfy a two-dimensional parabolic Hamilton–Jacobi–Bellman (HJB) equation with gradient constraints, giving rise to a parabolic double obstacle problem with a time-dependent lower obstacle. Through careful partial differential equation (PDE) analysis, we establish both the regularity and monotonicity of the resulting free boundaries over time. These analytical results enable us to prove a duality theorem and explicitly characterize the optimal strategies. Our model explains several well-documented stylized facts in household consumption behavior, including excess smoothness, excess sensitivity, the disappearance of these effects under large shocks, and asymmetric consumption responses to positive and negative wealth shocks. We find that, under constant ILA, the agent adjusts consumption more frequently and becomes less risk-averse as the horizon approaches.