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Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems

Alessandro Calvia, Federico Cannerozzi, Giorgio Ferrari

TL;DR

The paper advances ergodic singular stochastic control by establishing a novel link to optimal stopping via an auxiliary Dynkin game in a multi-dimensional setting. It derives general verification theorems that express optimal control through Skorokhod reflection on $Y$-dependent free boundaries, with the pseudo-potential $(V,\lambda)$ built from the Dynkin game value $U$. The authors fully solve two genuinely two-dimensional inventory problems under partial and full information, exploiting filtering and hypoellipticity to obtain regularity and optimality results. The approach yields a complete characterization of optimal policies and long-run values, demonstrating that ergodic singular control can be effectively solved through dimension-reducing Dynkin game analyses and suitable coordinate transformations. This work broadens the applicability of stopping-time methods to ergodic, multi-dimensional control problems with practical implications for inventory management under uncertainty.

Abstract

In ergodic singular stochastic control problems, a decision-maker can instantaneously adjust the evolution of a state variable using a control of bounded variation, with the goal of minimizing a long-term average cost functional. The cost of control is proportional to the magnitude of adjustments. This paper characterizes the optimal policy and the value in a class of multi-dimensional ergodic singular stochastic control problems. These problems involve a linearly controlled one-dimensional stochastic differential equation, whose coefficients, along with the cost functional to be optimized, depend on a multi-dimensional uncontrolled process Y. We first provide general verification theorems providing an optimal control in terms of a Skorokhod reflection at Y-dependent free boundaries, which emerge from the analysis of an auxiliary Dynkin game. We then fully solve two two-dimensional optimal inventory management problems. To the best of our knowledge, this is the first paper to establish a connection between multi-dimensional ergodic singular stochastic control and optimal stopping, and to exploit this connection to achieve a complete solution in a genuinely two-dimensional setting.

Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems

TL;DR

The paper advances ergodic singular stochastic control by establishing a novel link to optimal stopping via an auxiliary Dynkin game in a multi-dimensional setting. It derives general verification theorems that express optimal control through Skorokhod reflection on -dependent free boundaries, with the pseudo-potential built from the Dynkin game value . The authors fully solve two genuinely two-dimensional inventory problems under partial and full information, exploiting filtering and hypoellipticity to obtain regularity and optimality results. The approach yields a complete characterization of optimal policies and long-run values, demonstrating that ergodic singular control can be effectively solved through dimension-reducing Dynkin game analyses and suitable coordinate transformations. This work broadens the applicability of stopping-time methods to ergodic, multi-dimensional control problems with practical implications for inventory management under uncertainty.

Abstract

In ergodic singular stochastic control problems, a decision-maker can instantaneously adjust the evolution of a state variable using a control of bounded variation, with the goal of minimizing a long-term average cost functional. The cost of control is proportional to the magnitude of adjustments. This paper characterizes the optimal policy and the value in a class of multi-dimensional ergodic singular stochastic control problems. These problems involve a linearly controlled one-dimensional stochastic differential equation, whose coefficients, along with the cost functional to be optimized, depend on a multi-dimensional uncontrolled process Y. We first provide general verification theorems providing an optimal control in terms of a Skorokhod reflection at Y-dependent free boundaries, which emerge from the analysis of an auxiliary Dynkin game. We then fully solve two two-dimensional optimal inventory management problems. To the best of our knowledge, this is the first paper to establish a connection between multi-dimensional ergodic singular stochastic control and optimal stopping, and to exploit this connection to achieve a complete solution in a genuinely two-dimensional setting.

Paper Structure

This paper contains 15 sections, 20 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The Problem
  4. A preliminary verification theorem
  5. The connection to a Dynkin game
  6. Remarks on Theorems \ref{['thm:HJB_sol']} and \ref{['thm:HJB_sol_weak']}
  7. Applications to inventory models
  8. Inventory control with partially observable mean-reversion level
  9. Derivation of the separated problem
  10. The auxiliary Dynkin game for the separated problem
  11. Regularity refinements: a new coordinate system for the value function $U$
  12. Solution of the ergodic singular control problem
  13. Inventory control with observable mean-reversion level
  14. The associated Dynkin game
  15. Solution to the ergodic singular control problem

Key Result

Theorem 2.1

Recall eq:HJB:waiting_exert. Let $(V,\lambda)$ be a solution to eq:HJB such that $V \in {\mathrm C} ^2({\mathcal{I}} )$, $\vert V(x,y) \vert \leq \kappa (1 + \vert x \vert )$ for some $\kappa > 0$ and $(\lambda(Y_t))_{t \in [0,T]}$ is $d\mathbb{P} \otimes dt$ integrable, for any $T > 0$. Then, Moreover, suppose that there exists an admissible control $\xi^\star \in {\mathcal{B}}$ such that $(X_t^

Theorems & Definitions (46)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.2
  • proof
  • Remark 2.4
  • Theorem 2.3
  • ...and 36 more