A Nonstandard Approach To Stochastic Processes Under Probability Bounding
This paper studies stochastic processes under probability bounding, using nonstandard conditional lower previsions within the framework of internal set theory. Following Nelson's approach to stochastic processes, we introduce elementary processes which are defined over a finite number of time points and that serve to approximate any standard process, including processes over continuous time. We show that every standard process can be represented by an elementary process, and that the shadow of every elementary process constitutes again a standard process. We then move to demonstrate how elementary processes can be used to define imprecise Markov chains both in discrete and continuous time. To demonstrate the benefits and downsides of this approach, we show how to recover some basic results for continuous time Markov chains through analysis of a nonstandard elementary process.