On Nonlinear Expectations and Markov Chains under Model Uncertainty

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.