Second-order Approximations Of Risk Measures And Concentrations, And Characterizations Of Risk Aversions

Regular variation (RV) has become one of the key notions which appears in a natural way in applied probability, statistics, risk management, and other fields. There are a variety of concepts extending RV, among which are the ex-tended regular variation (ERV), second-order regular variation (2RV) and second-order extended regular variation (2ERV). Here,2RV and2ERV are termed as the second-order conditions. The quantification of diversification benefits due to risk aggregation has received more attention in the recent literature. Because risk managers become more and more concerned with the tail area of risks due to the excessive prudence of regulatory framework, there is an urgent need to establish second-order approximations of some risk measures and risk concentrations for extreme tails. The second-order condition provides a platform to do such a study.Risk aversion is a crucial concept in the economics. Risk aversion is the atti-tude that induces people to avoid uncertainty, to be protected from unpredictable events, and to buy some financial products. Various notions of risk aversion have been introduced in the literature, for example, the weak risk aversion, the strong risk aversion, the monotone risk aversion, the left-monotone risk aversion, and the right-monotone risk aversion. There are some literatures on characterizing these notions of risk aversion in the framework of the rank-dependent expected utility (RDEU) model for bounded random variables. It is interesting to identify sufficient and/or necessary conditions on the RDEU model such that a decision maker exhibits some notion of risk aversion.The purposes of this thesis are to study the second-order approximations of some risk measures and risk concentrations under the second-order condition, and to characterize the concepts of risk aversion in the RDEU model. The main contributions of this thesis are as follows. 1. Based on the classical Drees-type inequalities for ERV,2RV and2ERV func-tions, we establish new Drees-type inequalities with arbitrary auxiliary func-tions. This kind of inequalities has potential applications. The connections between2RV and2ERV are investigated carefully.2. Degen et al.(2010) derived second-order approximations of the risk con-centration based on the Value-at-Risk (VaR) for iid loss variables with a common survival function possessing the properties of2RV and of asymp-totic smoothness. We remove the assumption of the asymptotic smoothness, and reestablish the second-order expansions of risk concentration based on VaR and conditional tail expectation (CTE).3. We establish the second-order approximations of Haezendonck-Goovaerts risk measure under the2RV condition upon the tail quantile function U(t), and also reprove the main results in Tang and Yang (2012), concerning the first-order approximations of the Haezendonck-Goovaerts risk measure under the ERV condition upon U(t).4. In the framework of the RDEU model, we characterize the left-monotone and the right-monotone risk aversions for unbounded random variables, and remove the gap in the proof of the main result in Ryan (2006) concerning the characterization of the left-monotone risk aversion for bounded random vari-ables. We also extend the characterizations of the strong risk aversion and the monotone risk aversion obtained respectively by Chew et al.(1987) and Chateauneuf et al.(2005) from bounded to unbounded random variables.5. Cheung (2010) characterized the comonotonicity of random variables in terms of the convex order, provided the underlying probability space is atomless. We remove the assumption that the probability space is atom-less, and give a new and simple proof for such a characterization.