Choquet Integral Representation Of Premium Functional And Related Problems Research

Expected utility theory is one of the classical theory of economic and financial research field.However it does not expiain all the problems in this field. Especially Allais paradox and Ellsberg Paradox proposed made people more clearly understand the nature defect of the classical linear expected utility theory. Therefore, many e-conomists began to seek an non-linear expected utility theory to describe the economic and financial issues. It is worth noting that the French mathematician Choquet in 1953, first proposed the capacity theory and given a nonlinear operator by: Which is known as the Choquet integral. Choquet integral is an non-additive measure integration, with widely applications in the pricing of insurance, financial and economic fields.Many economists and mathematicians investigated the distorted probability mea-sure Choquet integration in probability space and achieved a lot of properties of dis-torted probability measure. For this reason, it is interesting to determine whether there is a distorted capacity issue in the capacity space. In this paper we will discuss the problems in capacity space as follows:(1) When the premium functional H satisfies law invariance, additivity of comono-tonic, normativity, monotonicity, whether there exists a distortion function g, by using the distorted capacity of Choquet integral to describe the premium functional?(2) The properties of the premiums functional H or distorted function g need to de-termined in order to keep the distorted capacity go v satisfing the inverse 2-alternating? The same time for a special class of capacity space, whether the distorted capacity go V have the 2-alternating or inverse 2-alternating?The problem mentioned above with their corresponding results discussed in three chapters in this paper.The first chapter introduces the research background and status, research contents and preliminaries;Some literature reviews about the classic premium pricing axiom of Choquet inte-gral in the probability space are summarized in the second chapter.Chapter three is the main result, we proved that if the premium functional H satis-fies law invariance, additivity of comonotonic, normativity, monotonicity in the capac-ity space, then there exists a pseudo-distortion function g, so that we can use pseudo- distorted capacity of Choquet integral to explain the premium functional (Theorem 3.1 ). As a necessary and sufficient condition of the premium functional it can be expressed as a distorted premium functional is that the premium functional satisfy law invariance (Theorem 3.2). While we also have a further research of relevant properties, and ob-tain that the distorted capacity gov satisfying inverse 2-alternating. At the same time we find that for a special class of capacity space, the distorted capacity goV satisfies the 2-alternating and inverse 2-alternating. Finally, we show that the distorted premium functional H[X] with right continuous function g of non-negative random variable X can be expressed as a weighted average of the quantiles of random variable X.