A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion-Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle's causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this 'weak adapted topology' is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

中文翻译：

---

所有适应的拓扑都是平等的

许多研究人员已经在随机过程定律集上引入了拓扑结构。这些作者的一个统一目标是加强通常的弱拓扑，以便充分捕捉随机过程的时间结构。Aldous 定义了基于预测过程的弱收敛的扩展弱拓扑。在经济学文献中，Hellwig 引入了信息拓扑来研究均衡问题的稳定性。Bion-Nadal 和 Talay 介绍了扩散过程定律之间的 Wasserstein 距离版本。Pflug 和 Pichler 考虑嵌套距离（和弱嵌套拓扑）以获得随机多级规划问题的连续性。这些距离可以看作是拉萨尔因果传输问题的对称化，但也有更自然的方法可以从因果传输中推导出拓扑。我们的主要结果是所有这些看似独立的方法都在有限的离散时间内定义了相同的拓扑。此外，我们表明这种“弱适应拓扑”的特征是最粗糙的拓扑，它保证了连续有界奖励函数的最优停止问题的连续性。