  We solve an infinite time-horizon bounded-variation stochastic control
problem with regime switching between $N$ states. This is motivated by the
problem of a government that wants to control the country's debt-to-GDP (gross
domestic product) ratio. In our formulation, the debt-to-GDP ratio evolves
stochastically in continuous time, and its drift -- given by the interest rate
on government debt, net of the growth rate of GDP -- is affected by an
exogenous macroeconomic risk process modelled by a continuous-time Markov chain
with $N$ states. The government can act on the public debt by increasing or
decreasing its level, and it aims at minimising a net expected regime-dependent
cost functional. Without relying on a guess-and-verify approach, but performing
a direct probabilistic study, we show that it is optimal to keep the
debt-to-GDP ratio in an interval, whose boundaries depend on the states of the
risk process. These boundaries are given through a zero-sum optimal stopping
game with regime switching with $N$ states and are characterised through a
system of nonlinear algebraic equations with constraints. To the best of our
knowledge, such a result appears here for the first time. Finally, we put in
practice our methodology in a case study of a Markov chain with $N=2$ states;
we provide a thorough analysis and we complement our theoretical results by a
detailed numerical study on the sensitivity of the optimal debt ratio
management policy with respect to the problem's parameters.
