Distortion risk measures in random environments: construction and axiomatic characterization
The risk of a financial position shines through by means of the fluctuation of its market price. The factors affecting the price of a financial position include not only market internal factors, but also other various market external factors. The latter can be understood as sorts of environments to which financial positions have to expose. Motivated by this observation, this paper aims to design a novel axiomatic approach to risk measures in random environments. We construct a new distortion-type risk measure, which can appropriately evaluate the risk of financial positions in the presence of environments. After having studied its fundamental properties, we also axiomatically characterize it by proposing a novel set of axioms. Furthermore, its coherence and dual representation are investigated. The new class of risk measures in random environments is rich enough, for example, it not only can recover some known risk measures such as the common weighted value at risk and range value at risk, but also can induce other new specific risk measures such as risk measures in the presence of background risk. Examples are given to illustrate the new framework of risk measures. This paper gives some theoretical results about risk measures in random environments, helping one to have an insight look at the potential impact of environments on risk measures of positions.