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

Alessandro Calvia, Federico Cannerozzi and Giorgio Ferrari
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Alessandro Calvia: Center for Mathematical Economics, Bielefeld University
Federico Cannerozzi: Center for Mathematical Economics, Bielefeld University
Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University

No 754, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

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 longterm 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.

Keywords: ergodic singular stochastic control; Dynkin games; free boundaries; Skorokhod reflection; variational inequality; optimal inventory management (search for similar items in EconPapers)
Pages: 37
Date: 2025-10-21
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