# The Optimal Control of Government Stabilization Funds

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Stabilization Fund Model

**Definition 1.**(Control) Let L and U be two $\left\{{\mathcal{F}}_{t}\right\}$-adapted, non-negative, and non-decreasing stochastic processes from $[0,\infty )\times \mathsf{\Omega}$ to $[0,\infty )$, with sample paths that are left-continuous with right-limits. The pair $(L,U)$ is called a stochastic singular control. By convention, we set ${U}_{0}={L}_{0}=0$.

**Remark**

**1.**

**Problem**

**1.**

**Definition**

**2.**

**Remark**

**2.**

**Proof.**

**Example**

**1.**

## 3. The Value Function and a Verification Theorem

**Proposition**

**1.**

**Proof.**

**Definition**

**3.**

- (i)
- $\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {X}_{t}^{v}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}x+{\int}_{0}^{t}\lambda (\theta -{X}_{s}^{v})ds+{\int}_{0}^{t}\sigma \phantom{\rule{0.166667em}{0ex}}d{W}_{s}-{U}_{t}^{v}+{L}_{t}^{v},\phantom{\rule{1.em}{0ex}}\forall t\in [\phantom{\rule{0.166667em}{0ex}}0,\infty ),P-a.s.,\hfill \end{array}$satisfies the following three conditions:
- (ii)
- ${X}_{t}^{v}\in \overline{\mathcal{C}},\phantom{\rule{1.em}{0ex}}\forall t\in (0,\infty ),\phantom{\rule{0.277778em}{0ex}}P-a.s.;$
- (iii)
- ${\int}_{0}^{\infty}{I}_{\left\{{X}_{t}^{v}\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\mathcal{C}\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}{\mathsf{\Sigma}}_{2}\right\}}d{L}_{t}^{v}=0,\phantom{\rule{1.em}{0ex}}P-a.s.;$
- (iv)
- ${\int}_{0}^{\infty}{I}_{\left\{{X}_{t}^{v}\phantom{\rule{0.166667em}{0ex}}\in \phantom{\rule{0.166667em}{0ex}}\mathcal{C}\phantom{\rule{0.166667em}{0ex}}\cup \phantom{\rule{0.166667em}{0ex}}{\mathsf{\Sigma}}_{1}\right\}}d{U}_{t}^{v}=0,\phantom{\rule{1.em}{0ex}}P-a.s..$

**Theorem**

**1.**

**Proof.**

## 4. The Analytical Solution

**Definition 4.**(Fund band) Let v be a function that satisfies the HJB Equation (12), and ${\mathcal{C}}^{v}$ the corresponding continuation region. If ${\mathcal{C}}^{v}\ne \mathsf{\varnothing}$, the stabilization fund band $[{a}^{v},{b}^{v}]$ is given by

**Theorem**

**2.**

- (i)
- ${\widehat{X}}_{t}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}x+{\int}_{0}^{t}\lambda (\theta -{\widehat{X}}_{s})ds+{\int}_{0}^{t}\sigma d{W}_{s}-{\widehat{U}}_{t}+{\widehat{L}}_{t},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall t\in [\phantom{\rule{0.166667em}{0ex}}0,\infty ),P-a.s.,$
- (ii)
- ${\widehat{X}}_{t}\in [a,b],\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall t\in (\phantom{\rule{0.166667em}{0ex}}0,\infty ),\phantom{\rule{0.277778em}{0ex}}P-a.s.,$
- (iii)
- ${\int}_{0}^{\infty}{I}_{\left\{{\widehat{X}}_{t}\phantom{\rule{0.166667em}{0ex}}>\phantom{\rule{0.277778em}{0ex}}a\right\}}d{\widehat{L}}_{t}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P-a.s.,$
- (iv)
- ${\int}_{0}^{\infty}{I}_{\left\{{\widehat{X}}_{t}\phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.277778em}{0ex}}b\right\}}d{\widehat{U}}_{t}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}P-a.s.$

**Proof.**

**Remark**

**3.**

## 5. Time to Increase or Decrease the Stabilization Fund

**Theorem**

**3.**

**Proof.**

## 6. Analysis of the Solution

#### 6.1. The Optimal Stabilization Fund Management and the Optimal Stabilization Fund Band

#### 6.2. The Stabilization Fund Target $\rho $ and the Optimal Stabilization Fund Band $[a,b]$

**Proposition**

**2.**

**Proof.**

**Example**

**2.**

#### 6.3. Time to Increase or Decrease the Stabilization Fund

#### 6.4. Comparative Statics Analysis

**Remark**

**4.**

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Remark 2

**Proof.**

## Appendix B. Proof of Proposition 1

**Proof.**

## Appendix C.

**Lemma**

**A1.**

**Proof.**

## Appendix D. Proof of Theorem 1

**Proof.**

## Appendix E. Lemma A2 and Its Proof under the Assumption That the Cost Function Is Symmetric and $\theta =\rho $

**Lemma**

**A2.**

**Remark**

**A1.**

**Proof**

**of**

**Remark.**

**Proof**

**of**

**Lemma**

**A2.**

## Appendix F. Proof of Theorem 2

**Proof.**

## Appendix G. Proof of Theorem 3

## Appendix H. Solution of the Problem with the Cost Function h(x) = e ^{-νx} + e^{ηx}

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${\mathit{k}}_{\mathit{L}}=\mathbf{0.4}\phantom{\rule{1.em}{0ex}}$ | ${\mathit{k}}_{\mathit{L}}=\mathbf{0.5}\phantom{\rule{1.em}{0ex}}$ | ${\mathit{k}}_{\mathit{L}}=\mathbf{0.6}\phantom{\rule{1.em}{0ex}}$ | |

a | 1.5350 | 1.4944 | 1.4581 |

b | 2.4302 | 2.4619 | 2.4893 |

$b-a$ | 0.8951 | 0.9674 | 1.0312 |

$\mathbf{\sigma}=\mathbf{0.6}$ | $\mathbf{\sigma}=\mathbf{0.75}$ | $\mathbf{\sigma}=\mathbf{0.9}$ | |

$\phantom{\rule{1.em}{0ex}}a\phantom{\rule{1.em}{0ex}}$ | 1.5596 | 1.49445 | 1.43349 |

$\phantom{\rule{1.em}{0ex}}b\phantom{\rule{1.em}{0ex}}$ | 2.3965 | 2.46192 | 2.52302 |

$\phantom{\rule{1.em}{0ex}}b-a\phantom{\rule{1.em}{0ex}}$ | 0.8369 | 0.96746 | 1.08905 |

$\mathbf{\theta}=\mathbf{0.0}$ | $\mathbf{\theta}=\mathbf{2.0}$ | $\mathbf{\theta}=\mathbf{2.5}$ | |

a | 1.54592 | 1.49445 | 1.48116 |

b | 2.51587 | 2.46192 | 2.44880 |

$b-a$ | 0.96995 | 0.967467 | 0.96764 |

$\mathbf{\lambda}=\mathbf{0.05}$ | $\mathbf{\lambda}=\mathbf{0.10}$ | $\mathbf{\lambda}=\mathbf{0.15}$ | |

$\phantom{\rule{1.em}{0ex}}a\phantom{\rule{1.em}{0ex}}$ | 1.50497 | 1.49995 | 1.48370 |

$\phantom{\rule{1.em}{0ex}}b\phantom{\rule{1.em}{0ex}}$ | 2.46463 | 2.46192 | 2.45911 |

$\phantom{\rule{1.em}{0ex}}b-a\phantom{\rule{1.em}{0ex}}$ | 0.95966 | 0.96746 | 0.97540 |

$\phantom{\rule{1.em}{0ex}}{\mathit{k}}_{\mathit{L}}=\mathbf{0.4}\phantom{\rule{1.em}{0ex}}$ | $\phantom{\rule{1.em}{0ex}}{\mathit{k}}_{\mathit{L}}=\mathbf{0.5}\phantom{\rule{1.em}{0ex}}$ | $\phantom{\rule{1.em}{0ex}}{\mathit{k}}_{\mathit{L}}=\mathbf{0.6}\phantom{\rule{1.em}{0ex}}$ | |

$\tilde{a}$ | 0.9714 | 0.8647 | 0.7758 |

$\tilde{b}$ | 2.7056 | 2.7207 | 2.7331 |

$\tilde{b}-\tilde{a}$ | 1.7341 | 1.8560 | 1.9572 |

$\mathbf{\sigma}=\mathbf{0.6}$ | $\mathbf{\sigma}=\mathbf{0.75}$ | $\mathbf{\sigma}=\mathbf{0.9}$ | |

$\tilde{a}$ | 0.8695 | 0.8647 | 0.8192 |

$\tilde{b}$ | 2.2461 | 2.7207 | 3.0409 |

$\tilde{b}-\tilde{a}$ | 1.3766 | 1.8560 | 2.2216 |

$\mathbf{\theta}=\mathbf{0.0}$ | $\mathbf{\theta}=\mathbf{2.0}$ | $\mathbf{\theta}=\mathbf{2.5}$ | |

$\tilde{a}$ | 1.0724 | 0.8647 | 0.8135 |

$\tilde{b}$ | 3.0503 | 2.7207 | 2.6337 |

$\tilde{b}-\tilde{a}$ | 1.9779 | 1.8560 | 1.8201 |

$\mathbf{\lambda}=\mathbf{0.05}$ | $\mathbf{\lambda}=\mathbf{0.10}$ | $\mathbf{\lambda}=\mathbf{0.15}$ | |

$\tilde{a}$ | 1.0333 | 0.8647 | 0.7189 |

$\tilde{b}$ | 3.1583 | 2.7207 | 2.3286 |

$\tilde{b}-\tilde{a}$ | 2.1250 | 1.8560 | 1.6097 |

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**MDPI and ACS Style**

Cadenillas, A.; Huamán-Aguilar, R. The Optimal Control of Government Stabilization Funds. *Mathematics* **2020**, *8*, 1975.
https://doi.org/10.3390/math8111975

**AMA Style**

Cadenillas A, Huamán-Aguilar R. The Optimal Control of Government Stabilization Funds. *Mathematics*. 2020; 8(11):1975.
https://doi.org/10.3390/math8111975

**Chicago/Turabian Style**

Cadenillas, Abel, and Ricardo Huamán-Aguilar. 2020. "The Optimal Control of Government Stabilization Funds" *Mathematics* 8, no. 11: 1975.
https://doi.org/10.3390/math8111975