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.