  The paper studies a system of Hamilton-Jacobi equations, arising from a
stochastic optimal debt management problem in an infinite time horizon with
exponential discount, modeled as a noncooperative interaction between a
borrower and a pool of risk-neutral lenders. In this model, the borrower is a
sovereign state that can decide how much to devaluate its currency and which
fraction of its income should be used to repay the debt. Moreover, the borrower
has the possibility of going bankrupt at a random time and must declare
bankruptcy if the debt reaches a threshold x*. When bankruptcy occurs, the
lenders only recover a fraction of their capital. To offset the possible loss
of part of their investment, the lenders buy bonds at a discounted price which
is not given a priori. This leads to a nonstandard optimal control problem. We
establish an existence result of solutions to this system and in turn recover
optimal feedback payment strategy $u*(x)$ and currency devaluation $v*(x)$. In
addition, the behavior of $(u*,v*)$ near $0$ and x* is studied.
