AI 中文总结
本文提出一种求解由非自治周期系数受控随机微分方程驱动的遍历随机最优控制问题的新方法，证明了其遍历性并构造辅助函数，将结果应用于遍历倒向随机微分方程研究，该方法在齐次情形下亦属首次。
AI 中文摘要
本文介绍一种求解遍历随机最优控制问题的新方法，该问题的动力学由受控随机微分方程驱动，此随机微分方程的系数是非自治的，但随时间呈周期性变化。我们首先证明无限时段平均随机最优控制问题是遍历的，即值函数等于与初始条件无关的常数ρ。基于遍历性的结果，我们构造了一个明确定义且随时间周期性变化的辅助函数w(t,x)，可证明对(w,ρ)满足动态规划原理，且是相关哈密尔顿-雅可比-贝尔曼方程的粘性解。最后，我们将所得结果应用于遍历倒向随机微分方程的研究，该方法在齐次情形下也是新的。
英文摘要
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 $ρ$ 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 $(w,ρ)$ 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.
