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arXiv · 2608.15243

Ergodic Stochastic Optimal Control Problems

Chenglin Ma·Huaizhong Zhao
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Abstract

In this article, we introduce a novel approach to solving the ergodic stochastic optimal control problem whose dynamics is driven by a controlled stochastic differential equation. The coefficients of this stochastic differential equation are non-autonomous but periodic in time. We first prove that the infinite horizon average stochastic optimal control problem is ergodic, i.e., the value function equals a constant $\rho$ which is independent of the initial conditions. Based on the result of ergodicity, we construct an auxiliary function $w(t,x)$ that is well-defined and periodic in time. We can prove that the pair $(w,\rho)$ satisfies the dynamic programming principle and is a viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Finally, we apply our results to the study of ergodic backward stochastic differential equations. Our method is even new in the homogeneous case.

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Chenglin Ma, Huaizhong Zhao. 2026-08-15. Ergodic Stochastic Optimal Control Problems. https://arxiv.org/abs/2608.15243

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