  We consider a government that aims at reducing the debt-to-gross domestic
product (GDP) ratio of a country. The government observes the level of the
debt-to-GDP ratio and an indicator of the state of the economy, but does not
directly observe the development of the underlying macroeconomic conditions.
The government's criterion is to minimize the sum of the total expected costs
of holding debt and of debt's reduction policies. We model this problem as a
singular stochastic control problem under partial observation. The contribution
of the paper is twofold. Firstly, we provide a general formulation of the model
in which the level of debt-to-GDP ratio and the value of the macroeconomic
indicator evolve as a diffusion and a jump-diffusion, respectively, with
coefficients depending on the regimes of the economy. These are described
through a finite-state continuous-time Markov chain. We reduce via filtering
techniques the original problem to an equivalent one with full information (the
so-called separated problem), and we provide a general verification result in
terms of a related optimal stopping problem under full information. Secondly,
we specialize to a case study in which the economy faces only two regimes, and
the macroeconomic indicator has a suitable diffusive dynamics. In this setting
we provide the optimal debt reduction policy. This is given in terms of the
continuous free boundary arising in an auxiliary fully two-dimensional optimal
stopping problem.
