The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from $$L^2$$ is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

中文翻译：

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连续随机分布下的动态概率约束

该论文研究了动态概率约束（机会约束）的分析特性。潜在的随机分布应该是连续的。在第一部分中，分析了具有依赖于随机过程的过去观察的决策规则的一般多阶段模型。研究了概率函数的（弱序列）（半）连续性或解的存在性等基本属性。事实证明，根据决策规则是嵌入 Lebesgue 空间还是 Sobolev 空间，结果显着不同。在第二部分中，研究了最简单的有意义的两阶段模型，其决策规则来自 $$L^2$$。考虑了更具体的属性，如 Lipschitz 连续性和概率函数的可微性。在高斯情况下，这些属性的显式可验证条件与显式梯度公式一起提供。讨论了这些公式在必要优化条件的背景下的应用，并给出了解决方案的具体识别。