## 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.

Original language | English (US) |
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Pages (from-to) | 953-982 |

Number of pages | 30 |

Journal | ESAIM - Control, Optimisation and Calculus of Variations |

Volume | 22 |

Issue number | 4 |

DOIs | |

State | Published - Oct 1 2016 |

## All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Control and Optimization
- Computational Mathematics