Optimal control of debt-to-GDP ratio in an N-state regime switching economy

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 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 we completely characterise them as solutions to 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.