Extension of coherent lower previsions to unbounded random variables

Abstract

This chapter focuses on the extension of coherent lower previsions from the set of bounded random variables to a larger set. One important shortcoming of the existing theory of coherent lower previsions is that it only deals with random variables that are bounded, whereas in engineering, applications involving unbounded random variables abound. An intuitive, ad hoc way of dealing with an unbounded random variable is to approximate it by a sequence of bounded ones, and to use limit arguments in order to extend notions defined in the context of the bounded random variables to their unbounded counterparts, in the hope that the eventual result will not depend on the exact form of the approximation. The chapter defines an extension of a coherent lower prevision to a linear space of previsible, not necessarily bounded random variables, and highlights that this extension still has properties similar to those of coherent lower previsions. Thus, previsibility coincides with the existing notion of D-integrability when the coherent lower previsions are linear, and an extended lower prevision can be written as the lower envelope of the extensions of the dominating linear previsions of the original.