In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markovian stochastic optimization problems. Similar to existing notions of distance for general stochastic processes, our distance is based on transportation metrics. As opposed to the extant literature, the proposed distance is problem specific, i.e., dependent on the data of the problem whose objective value we want to bound. As a result, we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature. We derive several properties of the proposed semi-metric and demonstrate its use in a stylized numerical example.

中文翻译：

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线性马尔可夫随机优化问题的稳定性结果

在本文中，我们提出了一种马尔可夫过程的半度量，它允许限制线性马尔可夫随机优化问题的最优值。与一般随机过程的现有距离概念类似，我们的距离基于运输指标。与现有文献相反，建议的距离是特定于问题的，即取决于我们想要限制其目标值的问题的数据。因此，我们能够考虑约束和目标函数中的随机性问题，从而放宽现有文献中的假设。我们推导出了所提出的半度量的几个属性，并在一个程式化的数值示例中演示了它的使用。