Risk measures on the space of infinite sequences

Abstract

Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner, Coherent Measures of Risk. Risk Management: Value at Risk and Beyond, 1998) and insurance risk measures (Wang, Insur Math Econ 21:173–183, 1997). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In Heyde et al. (Working Paper, Columbia University, 2007), the concept of natural risk statistic is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space ${\mathbb R^n}$ . One drawback of these kind of risk measures is their dependence on the space dimension n. In order to circumvent this issue, we propose a way to define a family {ρ _n }_n=1,2,... of natural risk statistics whose members are defined on ${\mathbb{R}^n}$ and related in an appropriate way. This construction requires the generalization of natural risk statistics to the space of infinite sequences l ^∞.