Random Variables: Linearity and Order

Abstract

The topic of this book requires us to deal with quantities, such as future payments or prices, whose value is not known in advance. To this effect, we first need to develop the basic mathematics to model uncertainty. This chapter is devoted to introducing the notion of a sample space, corresponding to the set of possible outcomes of a situation of uncertainty, and of a random variable, a quantity that is contingent on those outcomes. We equip the set of random variables with the structure of an ordered vector space. This structure, in particular, allows us to treat the notion of convexity which pervades much of mathematical finance. Although the main topic of our book is mathematical finance, much of the theory of random variables originated with the study of games of chance, a fact that is reflected in most of the examples with which we illustrate the basic theory.