Explicit Solution to a government debt reduction problem: a stochastic control approach

We analyze the problem of optimal reduction of the debt-to-GDP ratio in a stochastic control setting. The debt-to-GDP dynamics are modeled through a stochastic differential equation in which fiscal policy simultaneously affects both debt accumulation and GDP growth. A key feature of the framework is the introduction of a cost functional that captures the disutility of fiscal surpluses and the perceived benefit of fiscal deficits, thus incorporating the macroeconomic trade-off between tighten and expansionary policies. By applying the Hamilton-Jacobi-Bellman approach, we provide explicit solutions in the case of linear GDP response to the fiscal policies. We rigorously analyze threshold-type fiscal strategies in the case of linear impact of the fiscal policy and provide closed-form solutions for the associated value function in relevant regimes. A sensitivity analysis is conducted by varying key model parameters, confirming the robustness of our theoretical findings. The application to debt reduction highlights how fiscal costs and benefits influence optimal interventions, offering valuable insights into sustainable public debt management under uncertainty.