On the Optimal Management of Public Debt: a Singular Stochastic Control Problem

Giorgio Ferrari
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Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University

No 709, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

Abstract: Consider the problem of a government that wants to reduce the debt-to-GDP (gross domestic product) ratio of a country. The government aims at choosing a debt reduction policy which minimizes the total expected cost of having debt, plus the total expected cost of interventions on the debt ratio. We model this problem as a singular stochastic control problem over an inï¬ nite time horizon. In a general not necessarily Markovian framework, we ï¬ rst show by probabilistic arguments that the optimal debt reduction policy can be expressed in terms of the optimal stopping rule of an auxiliary optimal stopping problem. We then exploit such a link to characterize the optimal control in a two-dimensional Markovian setting in which the state variables are the level of the debt-to-GDP ratio and the current inflation rate of the country. The latter follows uncontrolled Ornstein–Uhlenbeck dynamics and affects the growth rate of the debt ratio. We show that it is optimal for the government to adopt a policy that keeps the debt-to-GDP ratio under an inflation- dependent ceiling. This curve is given in terms of the solution of a nonlinear integral equation arising in the study of a fully two-dimensional optimal stopping problem.

Keywords: singular stochastic control; optimal stopping; free boundary; nonlinear integral equation; debt-to-GDP ratio; inflation rate; debt ceiling (search for similar items in EconPapers)
Pages: 40
Date: 2025-06-24
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