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Stationary Mean-Field singular control of an Ornstein-Uhlenbeck process

Federico Cannerozzi

TL;DR

We study a stationary mean-field control problem with singular controls for a mean-reverting Ornstein-Uhlenbeck process under an ergodic, mean-field dependent cost. A novel potential stationary mean-field game is introduced, yielding a bijection between MFC optimal controls and equilibria of the MFG. Existence of an equilibrium is shown, and the MFG is solved explicitly via a two-barrier, Dynkin-game structure; this provides a concrete optimal reflection policy for the original MFC. The approach combines a constrained optimization on the stationary mean with a Lagrange multiplier, a fixed-point analysis, and regularity results, complemented by numerical illustrations of parameter sensitivity.

Abstract

Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game explicitly, thereby providing a solution to the original stationary mean-field control problem.

Stationary Mean-Field singular control of an Ornstein-Uhlenbeck process

TL;DR

We study a stationary mean-field control problem with singular controls for a mean-reverting Ornstein-Uhlenbeck process under an ergodic, mean-field dependent cost. A novel potential stationary mean-field game is introduced, yielding a bijection between MFC optimal controls and equilibria of the MFG. Existence of an equilibrium is shown, and the MFG is solved explicitly via a two-barrier, Dynkin-game structure; this provides a concrete optimal reflection policy for the original MFC. The approach combines a constrained optimization on the stationary mean with a Lagrange multiplier, a fixed-point analysis, and regularity results, complemented by numerical illustrations of parameter sensitivity.

Abstract

Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game explicitly, thereby providing a solution to the original stationary mean-field control problem.
Paper Structure (8 sections, 18 theorems, 112 equations, 3 figures)

This paper contains 8 sections, 18 theorems, 112 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. From mean-field control solutions to mean-field game solutions
  4. Solution to the ergodic singular control problem
  5. Solution to the constrained optimization problem
  6. Proving the necessary condition
  7. From mean-field game solutions to mean-field control solutions
  8. Numerical illustrations

Key Result

Theorem 1

Let Assumption ass:phi_rho hold true. Suppose that $\xi^\star \in {\mathcal{B}} _{mfc}$ is optimal for the stationary MFC problem eq:cost:ergodic, and let $p^\star_\infty \in {\mathcal{P}} (\mathbb{R})$ denote its stationary distribution. Set $\theta^\star= \int_{\mathbb{R}} x \, p^\star_\infty(dx)$

Figures (3)

  • Figure 1: Sensitivity with respect to $\delta$. $\delta$ varies from $0.1$ to $5$. On the left panel, we show the optimal reflection boundaries $a_\pm(\theta^\star,\lambda^\star)$, whereas on the right panel we show the parameter $\theta^\star$ at equilibrium. Here, $\sigma=1$, $\rho=1.5$, $\varphi= 1$, $\psi = 0.5$, $\bar{x} = \bar{\theta} = 1$, $K_+ = K_- = 1$.
  • Figure 2: Sensitivity with respect to $\sigma$. $\sigma$ varies from $0.1$ to $4$. On the left panel, we show the optimal reflection boundaries $a_\pm(\theta^\star,\lambda^\star)$, whereas on the right panel we show the parameter $\theta^\star$ at equilibrium. Here, $\delta=1$, $\rho=1.5$, $\varphi= 1$, $\psi = 0.5$, $\bar{x} = \bar{\theta} = 1$, $K_+ = K_- = 1$.
  • Figure 3: Sensitivity with respect to $\varphi$. $\varphi$ varies from $0.1$ to $\rho = 5$. On the left panel, we show the optimal reflection boundaries $a_\pm(\theta^\star,\lambda^\star)$, whereas on the right panel we show the parameter $\theta^\star$ at equilibrium. Here, $\delta = \sigma=1$, $\rho=5$, $\psi = 0.5$, $\bar{x} = 1=\bar{\theta} = 1$, $K_+ = K_- = 1$.

Theorems & Definitions (40)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Lemma 4
  • ...and 30 more