Sublinear Expectations and Risk Measures

Abstract

A sublinear expectation is also called the upper expectation or the upper prevision, and this notion is used in situations when the probability models have uncertainty. In this chapter, we present the basic notion of sublinear expectations and the corresponding sublinear expectation spaces. We give the representation theorem of a sublinear expectation and the notions of distributions and independence within the framework of sublinear expectations. We also introduce a natural Banach norm of a sublinear expectation in order to get the completion of a sublinear expectation space which is a Banach space. As a fundamentally important example, we introduce the notion of coherent risk measures in finance.