An equilibrium model of debt and bankruptcy^∗

Department of Mathematics, Penn State University, University Park, PA 16802, USA.
bressan@math.psu.edu; khaivycy@gmail.com

Received: 22 July 2015
Revised: 7 June 2016
Accepted: 7 June 2016

Abstract

We study optimal strategies for a borrower, who services a debt in an infinite time horizon, taking into account the risk of possible bankruptcy. In a first model, the interest rate as well as the instantaneous bankruptcy risk are given, increasing functions of the total amount of debt. In a second model only the bankruptcy risk is given, while the interest rate is determined from a Nash equilibrium, in a game between the borrower and a pool of risk-neutral lenders. This yields a non-standard optimal control problem for the borrower, where the dynamics involves all future values of the control. For this model, optimal repayment strategies are constructed, in open-loop form. In addition, for optimal strategies in feedback form, our analysis shows that the value function should satisfy a new kind of nonlinear, degenerate elliptic equation.