A stochastic control approach to public debt management

Abstract

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.

Proofs of secondary results

Proof of Theorem 4.1

We adapt the proof of [18, Chapter 4.3] to our framework. Proposition 3.5 ensures the continuity of v. Let $({\bar{x}},{\bar{z}})\in (0,+\infty )\times {\mathbb {R}}$ and take a test function $\varphi \in {\mathcal {C}}^{2,2}((0,+\infty )\times {\mathbb {R}})$ such that

$$\begin{aligned} 0 = (v-\varphi )({\bar{x}},{\bar{z}}) = \max _{(x,z)\in (0,+\infty )\times {\mathbb {R}}}(v-\varphi )(x,z) \;. \end{aligned}$$

Let us observe that $v\le \varphi$ by construction. Since v is continuous, there exists a sequence $\{(x_n,z_n)\}_{n\ge 1}$ such that

$$\begin{aligned} (x_n,z_n) \rightarrow ({\bar{x}},{\bar{z}}) \qquad \text {and } \qquad v(x_n,z_n) \rightarrow v({\bar{x}},{\bar{z}}) \qquad \text {as } n\rightarrow +\infty \;. \end{aligned}$$

Correspondingly, we must have that

$$\begin{aligned} \gamma _n \doteq v(x_n,z_n) - \varphi (x_n,z_n) \rightarrow 0 \qquad \text {as } n\rightarrow +\infty \;. \end{aligned}$$

Now let us consider a control ${\bar{u}}_t={\bar{u}}$ $\forall t>0$, for some arbitrary constant ${\bar{u}}\in [-U_1,U_2]$. Moreover, let introduce a sequence of stopping times $\{\tau _n\}_{n\ge 0}$ as follows:

$$\begin{aligned} \tau _n = \inf \left\{ s\ge 0 \mid \max \left\{ \left| X^{{\bar{u}},x_n}_s-x_n\right| ,\left| Z^{z_n}_s-z_n\right| \right\} > \epsilon \right\} \wedge h_n \qquad n\ge 1\;, \end{aligned}$$

for some fixed $\epsilon >0$ and $\{h_n\}_{n\ge 1}$ such that

$$\begin{aligned} h_n\rightarrow 0 \;, \qquad \frac{\gamma _n}{h_n} \rightarrow 0 \qquad \text {as } n\rightarrow +\infty \;. \end{aligned}$$

Here $\{Z^{z_n}_t\}_{t\ge 0}$ denotes the solution of the SDE (2.1) with initial condition $Z^{z_n}_0=z_n$. By the dynamic programming principle (see e.g. [18, Theorem 3.3.1]) for any $n\ge 1$ we have that

$$\begin{aligned} v(x_n,z_n) \le {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t} f(X^{{\bar{u}},x_n}_t,Z^{z_n}_t,{\bar{u}})\,dt + e^{-\lambda \tau _n}v(X^{{\bar{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;, \end{aligned}$$

hence

$$\begin{aligned} \varphi (x_n,z_n) + \gamma _n \le {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t} f(X^{{\bar{u}},x_n}_t,Z^{z_n}_t,{\bar{u}})\,dt + e^{-\lambda \tau _n}\varphi (X^{{\bar{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;. \end{aligned}$$

Applying Itô’s formula we get that

$$\begin{aligned} e^{-\lambda \tau _n}\varphi (X^{{\bar{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n})= & {} \varphi (x_n,z_n) + \int _0^{\tau _n} e^{-\lambda t}[{\mathcal {L}}^{{\bar{u}}} \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t) \nonumber \\&- \lambda \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t)] \,dt + M_{\tau _n} \;, \end{aligned}$$

(A.1)

where

$$\begin{aligned} M_t= & {} \int _0^t e^{-\lambda s}\frac{\partial {\varphi }}{\partial {x}}(X^{{\bar{u}},x_n}_s,Z^{z_n}_s)\sigma X^{{\bar{u}},x_n}_s \,dW_s \\&+ \int _0^t e^{-\lambda s}\frac{\partial {\varphi }}{\partial {z}}(X^{{\bar{u}},x_n}_s,Z^{z_n}_s)\sigma _Z(Z^{z_n}_s) \,dW^Z_s \;. \end{aligned}$$

Clearly $\{M_t\}_{t\ge 0}$ is a local martingale (having $\{\tau _n\}_{n\ge 0}$ as localizing sequence of stopping times) because the integrand functions are continuous and hence bounded on the compact sets. Taking expectations in Eq. (A.1) and using the previous inequality yields

$$\begin{aligned} \begin{aligned}&\gamma _n+\varphi (x_n,z_n) \\&\quad = \gamma _n - {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t}[{\mathcal {L}}^{{\bar{u}}} \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t) - \lambda \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t)]\,dt \biggr ]\\&\quad \quad + {\mathbb {E}}\biggl [ e^{-\lambda \tau _n}\varphi (X^{{\bar{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \\&\quad \le {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t}f(X^{{\bar{u}},x_n}_t,Z^{z_n}_t,{\bar{u}})\,dt \biggr ] + {\mathbb {E}}\biggl [ e^{-\lambda \tau _n}\varphi (X^{{\bar{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;, \end{aligned} \end{aligned}$$

that is, dividing by $h_n$ (using that $\tau _n\le h_n$),

$$\begin{aligned} \frac{\gamma _n}{h_n} \le {\mathbb {E}}\biggl [ \frac{1}{h_n}\int _0^{\tau _n}e^{-\lambda t}[{\mathcal {L}}^{{\bar{u}}} \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t) + f(X^{{\bar{u}},x_n}_t,Z^{z_n}_t,{\bar{u}}) - \lambda \varphi (X^{{\bar{u}},x_n}_t,Z^{z_n}_t)]\,dt \biggr ] \;. \end{aligned}$$

Letting $n\rightarrow +\infty$ we have that $X^{{\bar{u}},x_n}_t\rightarrow X^{{\bar{u}},{\bar{x}}}_t$ and $Z^{z_n}_t\rightarrow Z^{{\bar{z}}}_t$, $\forall t \ge 0$ ${\mathbb {P}}-a.s.$ and the right-hand side converges to ${\mathcal {L}}^{{\bar{u}}} \varphi ({\bar{x}},{\bar{z}}) + f({\bar{x}},{\bar{z}},{\bar{u}})-\lambda \varphi ({\bar{x}},{\bar{z}})$ by the mean value theorem for integrals. Hence

$$\begin{aligned} {\mathcal {L}}^{{\bar{u}}} \varphi ({\bar{x}},{\bar{z}}) + f({\bar{x}},{\bar{z}},{\bar{u}}) - \lambda \varphi ({\bar{x}},{\bar{z}}) \ge 0 \;. \end{aligned}$$

Since ${\bar{u}}$ is arbitrary, taking the infimum we obtain that v is a viscosity subsolution of Eq. (4.2) (see Eq. (4.3)).

Now we prove that v is a viscosity supersolution. To this end, we take a test function $\varphi \in {\mathcal {C}}^{2,2}((0,+\infty )\times {\mathbb {R}})$ such that

$$\begin{aligned} 0 = (v-\varphi )({\bar{x}},{\bar{z}}) = \min _{(x,z)\in (0,+\infty )\times {\mathbb {R}}}(v-\varphi )(x,z) \;. \end{aligned}$$

By definition of the value function, we can find a strategy $\{{\hat{u}}_t\}_{t\ge 0}\in {\mathcal {U}}$ such that

$$\begin{aligned} v(x_n,z_n) + h_n^2 \ge {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t} f(X^{{\hat{u}},x_n}_t,Z^{z_n}_t,{\hat{u}}_t)\,dt + e^{-\lambda \tau _n}v(X^{{\hat{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;, \end{aligned}$$

and hence

$$\begin{aligned} \varphi (x_n,z_n) + \gamma _n + h_n^2 \ge {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t} f(X^{{\hat{u}},x_n}_t,Z^{z_n}_t,{\hat{u}}_t)\,dt + e^{-\lambda \tau _n}v(X^{{\hat{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;. \end{aligned}$$

Using Eq. (A.1) and $v\ge \varphi$ we obtain that

$$\begin{aligned}&\gamma _n + h_n^2 + {\mathbb {E}}\biggl [ e^{-\lambda \tau _n}\varphi (X^{{\hat{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] - {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t}[{\mathcal {L}}^{{\hat{u}}} \varphi (X^{{\hat{u}},x_n}_t,Z^{z_n}_t) - \lambda \varphi (X^{{\hat{u}},x_n}_t,Z^{z_n}_t)]\,dt \biggr ] \\&\quad \ge {\mathbb {E}}\biggl [ \int _0^{\tau _n} e^{-\lambda t}f(X^{{\hat{u}},x_n}_t,Z^{z_n}_t,{\hat{u}}_t)\,dt \biggr ] + {\mathbb {E}}\biggl [ e^{-\lambda \tau _n}\varphi (X^{{\hat{u}},x_n}_{\tau _n},Z^{z_n}_{\tau _n}) \biggr ] \;, \end{aligned}$$

and dividing by $h_n$ we get

$$\begin{aligned} \begin{aligned} \frac{\gamma _n}{h_n} + h_n&\ge {\mathbb {E}}\biggl [ \frac{1}{h_n}\int _0^{\tau _n}e^{-\lambda t}[{\mathcal {L}}^{{\hat{u}}} \varphi (X^{{\hat{u}},x_n}_t,Z^{z_n}_t) + f(X^{{\hat{u}},x_n}_t,Z^{z_n}_t,{\hat{u}}_t) - \lambda \varphi (X^{{\hat{u}},x_n}_t,Z^{z_n}_t)]\,dt \biggr ] \\&\ge {\mathbb {E}}\biggl [ \frac{1}{h_n}\int _0^{\tau _n}\inf _{u\in [-U_1,U_2]}\{{\mathcal {L}}^{u} \varphi (X^{u,x_n}_t,Z^{z_n}_t) + f(X^{u,x_n}_t,Z^{z_n}_t,u) - \lambda \varphi (X^{u,x_n}_t,Z^{z_n}_t) \}\,dt \biggr ] \;. \end{aligned} \end{aligned}$$

Observing that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{\tau _n}{h_n} = \min \left\{ \lim _{n\rightarrow +\infty }\frac{\inf \left\{ s\ge 0 \mid \max \left\{ \left| X^{{\bar{u}},x_n}_s-x_n\right| ,\left| Z^{z_n}_s-z_n\right| \right\} \ge \epsilon \right\} }{h_n}, 1 \right\} = 1 \;, \end{aligned}$$

using the mean value theorem for integrals again, we finally get the inequality

$$\begin{aligned} \inf _{u\in [-U_1,U_2]}\{ {\mathcal {L}}^{u} \varphi ({\bar{x}},{\bar{z}}) + f({\bar{x}},{\bar{z}},u) - \lambda \varphi ({\bar{x}},{\bar{z}}) \} \le 0 \;, \end{aligned}$$

and hence v is a viscosity supersolution of Eq. (4.2) (see Eq. (4.4)).