Theory Of Expected Utility And Ordering Of Risks

Expected utility theory is one of effective approaches to choices under uncertainties. Firstly, in the present paper, the researches of the utility theory carried out for nearly three centuries are reviewed generally. The basic concepts like risk aversion, certainty equivalent, risk premium are explained based on the theory of expected utility in the context of economics. Meanwhile, a feasible condition for a contract is analysized and used to demonstrate an unusual application of the theory to actuarial practice. Secondly, we give a full description of the ordering of risks under the expected utility hypothesis in the context of actuarial sciences. Mainly, we discuss the definition of stop-loss orders and their corresponditig characterization. A useful method for judging a stop-loss order, ie. a crossing-type condition for their cumulative distribution functions is also introduced. Finally, the Frechet space and Frechet bounds are summarily introduced and a detailed discussion of the mutually exclusive risks, a kind of extreme correlated risks, is carried out. Especially, by means of the first degree stop-loss order, we show that the mutually exclusive risk portfolio in a Frechet space leads to the smallest stop-loss reinsurance premium.This paper is mainly a survey of some new results in recent research on risk theory. Considering little attention paid to the field in our country, the aim of this paper is to provide theoritical basis for the research in the field. Besides, we correct the drawbacks in some current references and first present the defining properties of the infinite degree stop-loss order. Two simple characterizations of the stochastic order are also obtained.