  We discuss a class of debt management problems in a stochastic environment
model. We propose a model for the debt-to-GDP (Gross Domestic Product) ratio
where the government interventions via fiscal policies affect the public debt
and the GDP growth rate at the same time. We allow for stochastic interest rate
and possible correlation with the GDP growth rate through the dependence of
both the processes (interest rate and GDP growth rate) on a stochastic factor
which may represent any relevant macroeconomic variable, such as the state of
economy. We tackle the problem of a government whose goal is to determine the
fiscal policy in order to minimize a general functional cost. We prove that the
value function is a viscosity solution to the Hamilton-Jacobi-Bellman equation
and provide a Verification Theorem based on classical solutions. We investigate
the form of the candidate optimal fiscal policy in many cases of interest,
providing interesting policy insights. Finally, we discuss two applications to
the debt reduction problem and debt smoothing, providing explicit expressions
of the value function and the optimal policy in some special cases.
