离散时间无状态不确定过程的对策论上期望
Game-Theoretic Upper Expectations for Discrete-Time Finite-State Uncertain Processes
摘要：
博弈论上期望是联合（全局）概率模型，它用上鞅数学描述了不确定过程的行为;资本过程对应于可用的投注策略。与（更常见的）测度论期望泛函相比，它们不受可测性或精度等限制性假设的约束，但成功地保留甚至推广了它们的许多基本性质。我们专注于一个离散的时间设置，当地的状态空间是有限的，在这个特定的背景下，建立在现有的工作Shafer和Vovk的主要开发商的框架博弈论的上限期望。在第一部分中，我们研究谢弗和Vovk的本地上期望的特性，并显示它是如何与瓦利的行为概念的连贯性。第二部分包括在一个更全球性的水平上，在那里几个替代的定义，以及广泛的属性的研究博弈论上的期望派生，例如迭代上的期望，与当地模型的兼容性，一致性的法律，.然而，我们的主要贡献，涉及这些运营商的连续性行为。我们证明了连续性与非增加序列的所谓的低切和连续性与非增加序列的有限函数。此外，我们表明，博弈论上的期望值是唯一确定的域上的有限函数的限制下有界的值，并另外表明，对于任何这样的限制，限制序列可以构造在这样一种方式，博弈论上的期望值是连续的关于这个特定的序列。
英文摘要：
Game-theoretic upper expectations are joint (global) probability models that mathematically describe the behaviour of uncertain processes in terms of supermartingales; capital processes corresponding to available betting strategies. Compared to (the more common) measure-theoretic expectation functionals, they are not bounded to restrictive assumptions such as measurability or precision, yet succeed in preserving, or even generalising many of their fundamental properties. We focus on a discrete-time setting where local state spaces are finite and, in this specific context, build on the existing work of Shafer and Vovk; the main developers of the framework of game-theoretic upper expectations. In a first part, we study Shafer and Vovk's characterisation of a local upper expectation and show how it is related to Walley's behavioural notion of coherence. The second part consists in a study of game-theoretic upper expectations on a more global level, where several alternative definitions, as well as a broad range of properties are derived, e.g. the law of iterated upper expectations, compatibility with local models, coherence properties,... Our main contribution, however, concerns the continuity behaviour of these operators. We prove continuity with respect to non-increasing sequences of so-called lower cuts and continuity with respect to non-increasing sequences of finitary functions. We moreover show that the game-theoretic upper expectation is uniquely determined by its values on the domain of bounded below limits of finitary functions, and additionally show that, for any such limit, the limiting sequence can be constructed in such a way that the game-theoretic upper expectation is continuous with respect to this particular sequence.
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