Finite horizon portfolio selection with a negative wealth constraint

This paper studies an optimal consumption and portfolio strategy with a negative wealth constraint for a finitely-lived economic agent. That is, we allow the individual agent to borrow partially against her/his future labor income during a finite time horizon. We derive the associated Hamilton–Jacobi–Bellman equation and use the Mellin transform to obtain the integral equation representation satisfied by the free boundary. Moreover, we derive an analytic representation for the optimal consumption, wealth, and portfolio, and provide some numerical implications for the optimal strategies.