Premium Pricing With Non-Linear Measure

In this paper, we used some nonlinear methods to measure risk and try to solve the problems in the field of premium principles. One useful method is Choquet integral-also be called Choquet Expectation[1], which has been widely used in calculated the premium of risk. Another useful method is g-expectation, which was found by Pengl997[40], based on backward stochastic differential equation (BSDE). Recently many distinguished work have been done on this topic in many fields, especially in financial mathematics, econometric, stochastic control and partial differential equations. Chen2005[46) shows us Choquet expectation and g-expectation is equal in some special field, which builds a useful links between the two non-linear expectations.There are four Chapters in this article: Chapter 1 and 2 focus on the premium principle with Choquet integral. In the condition of "Preserves stop-loss ordering" and "Additive for comonotonic risks", the premium principle can be represented and decomposed by Choquet integral. We show them in an insurance marketing model. In Chapter 3 we discuss the nonadditive measure, which is the basic conception of Choquet integral. We define the essentialmeasure of a nonadditive measure and get many imported results. Chapter 4 turns to the g-expectation under the condition of comonotonic additive and also get two mainly approaches.In Chapter 1, the section 1 can be read as the introduction, in which we conclude eight properties of Premium principles from many related articles. Compared with other authors improving the Choquet pricing from Comonotonic, we focus on the property named as Preserves stop-loss ordering(SL), which is widely used in insurance contracts. One main result is theorem1.1: the insurance premium can be represented as Choquet Expectation if it is "Preserves stop-loss ordering" and "Additive for comonotonic risks". This Choquet expectation is very similar with the Wang(1996)[8] integral of X with respect to distorted probability, and they are equal if X is Bernoulli(u).In Chapter 2, lemma 2.1 build the links between the Choquet expectation andthe mathematical expectation of X. The Choquet pricing of insurance contract is the upper envelopes of the possible expectations and the Choquet pricing of surrender is the lower envelope. Theorem 2.1 shows that in a continuous probability space, a Choquet expectation can be represented as a sum of expectation EpX and safety loading ^m^based on the set M). That is an approach of the result by Waegenaere et. (2003)[7]. In section 2, we consider an insurance marketing model and the key conception named as "Ambiguity", which was first used to explain the Ellsberg paradox . In this model, P is the approximate of the objective probability and M is the set of possible probability. The scale of the M shows the knowledge of the risk: if we have more information about the risk, M. will be smaller, that means less "Ambiguilty" and lower safety loading. In this view, theorem. 2.1 shows that risk and Ambiguilty are different, one is objective that we can not effect it, another is subjective that we can change. Example 2.3 give an obvious explain about it and an important conclusion can be draw: as the progress of insurance, the "risk" cannot be reduced but "Ambiguity "can. The reason for insurance is Ambiguity Aversion. Theorem 2.1 also can explain the reason for surrender and the fee of surrender.In chapter 3, we discuses the non additive measure-the basic of the Choquet expectation [36], In section 1, the outer and inner measure of nonadditive measure fi is extended with many useful properties. In section 2, theorem3.2 and theorem3.3 get two interesting results: an non additive measure will be measure after inner and outer measure calculations. Thus essentialmeasure (/i*)?, (/x*)* is founded. We give two examples to illuminate that. In section 3, theorem 3.4 represent the P and M. with essentialmeasure of /i,that is only one use of the essencialmeasure.In chapter 4, many mathematic equations are used to describe some properties of g-expectation with Comonotonic Additivity[45]. Two main results are: Theorem 4.1 shows the generaror g is positive homogeneous and theorem 4.2 shows Jensen's inequality for g-expectation holds for any increasing convex function.