We consider three equilibrium concepts proposed in the literature for time-inconsistent stopping problems, including mild equilibria (introduced in Huang and Nguyen-Huu (2018)), weak equilibria (introduced in Christensen and Lindensjö (2018)), and strong equilibria (introduced in Bayraktar et al. (2021)). The discount function is assumed to be log subadditive and the underlying process is one-dimensional diffusion. We first provide necessary and sufficient conditions for the characterization of weak equilibria. The smooth-fit condition is obtained as a by-product. Next, based on the characterization of weak equilibria, we show that an optimal mild equilibrium is also weak. Then we provide conditions under which a weak equilibrium is strong. We further show that an optimal mild equilibrium is also strong under a certain condition. Finally, we provide several examples including one showing a weak equilibrium may not be strong, and another one showing a strong equilibrium may not be optimal mild.

中文翻译：

---

一维扩散过程的时间不一致停止平衡

我们考虑文献中针对时间不一致停止问题提出的三个均衡概念，包括温和均衡（Huang 和 Nguyen-Huu (2018) 中引入）、弱均衡（Christensen 和 Lindensjö (2018) 中引入）和强均衡（Huang 和 Nguyen-Huu (2018) 中引入） Bayraktar 等人 (2021)）。假设折扣函数是对数次加法，并且底层过程是一维扩散。我们首先为弱平衡的表征提供充分必要条件。平滑拟合条件是作为副产品获得的。接下来，基于弱平衡的表征，我们表明最佳温和平衡也是弱的。然后我们提供弱均衡变为强均衡的条件。我们进一步表明，在特定条件下，最佳温和平衡也是很强的。最后，