Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems
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| Format: | Preprint |
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2025
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| _version_ | 1866912644470931456 |
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| author | Calvia, Alessandro Cannerozzi, Federico Ferrari, Giorgio |
| author_facet | Calvia, Alessandro Cannerozzi, Federico Ferrari, Giorgio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_11158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems Calvia, Alessandro Cannerozzi, Federico Ferrari, Giorgio Optimization and Control Probability 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. |
| title | Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems |
| topic | Optimization and Control Probability |
| url | https://arxiv.org/abs/2510.11158 |