The aim of this work is to give an overview on nonlinear expectation and to relate them to other concepts that describe model uncertainty or imprecision in a probabilistic framework. We discuss imprecise versions of stochastic processes with a particular interest in imprecise Markov chains. First, we focus on basic properties and representations of nonlinear expectations with additional structural assumptions such as translation invariance or convexity. In a second step, we discuss how stochastic processes under nonlinear expectations can be constructed via primal and dual representations. We illustrate the concepts by means of imprecise Markov chains with a countable state space, and show how families of Markov chains give rise to imprecise versions of Markov chains. We discuss dual representations and differential equations related to the latter.

中文翻译：

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模型不确定性下的非线性期望和马尔可夫链

这项工作的目的是概述非线性期望，并将它们与在概率框架中描述模型不确定性或不精确性的其他概念相关联。我们讨论随机过程的不精确版本，对不精确的马尔可夫链特别感兴趣。首先，我们关注非线性期望的基本属性和表示，以及附加的结构假设，例如平移不变性或凸性。在第二步中，我们讨论如何通过原始和对偶表示构建非线性期望下的随机过程。我们通过具有可数状态空间的不精确马尔可夫链来说明这些概念，并展示马尔可夫链族如何产生不精确的马尔可夫链版本。我们讨论与后者相关的对偶表示和微分方程。