A stochastic model of optimal debt management and bankruptcy

A problem of optimal debt management is modeled as a noncooperative interaction between a borrower and a pool of lenders, in an infinite time horizon with exponential discount. The yearly income of the borrower is governed by a stochastic process. When the debt-to-income ratio x(t) reaches a given size x ∗ , bankruptcy instantly occurs. The interest rate charged by the risk-neutral lenders is precisely determined in order to compensate for this possible loss of their investment. For a given bankruptcy threshold x ∗ , existence and properties of optimal feedback strategies for the borrower are studied, in a stochastic framework as well as in a limit deterministic setting. The paper also analyzes how the expected total cost to the borrower changes, depending on different values of x ∗ .