The Modification And Optimization Of Haezendonck Risk Measure

The Modification and Optimization of Haezendonck Risk MeasureRisk will be met unavoidably in the fields of finance and insurance , so the importance of risk management is outstanding day by day.The first step of risk management is to carry on quantization to the risk.There are a lot of methods to quantize to the risk. Haezendonck quantizes to the risk by his own way , known as Haezendonck risk measure. But we think that when the risk becomes very great , the lose or. the income will become a little larger and the function vision should be a form steeper than the straight line in the tail at this time, so we revise the Haezendonck risk measure further, and propose the modified Haezendonck risk measure. Then , we use the knowledge of probability and satistics and the skill of mathematics ananlyse to prove some properties of the modifided Haezendonck risk measure, which has certain directive significance to actual life.This text divides seven sections:Section one, Foreword.Section two, Risk.Introduced the risk briefly, through the recognization of risk , we realize the importance of risk management, hence we strengthen to take precautions against the risk.Section three, Risk measure.One important concept in risk management was introduced—the risk measure.Section four, Lemmas.Section five, Haezendonck risk measure.The Haezendonck risk measure is a kind of quantization form to the risk, therefore , we can define a kind of risk measure as soon as such a function was given.Section six, Modified Haezendonck risk measure.We are inspired, having revised the Haezendonck risk measure.We attempt to let it optimize even more and correspond to the imagine of reality even more. Then we also prove some properties of the modified risk measure.Every theorem is sketched as follows:Definition 1. Let S be a set of axioms for risk measures, and a, 0 < a < 1, be a level. A risk measure tt[ ■ ] = n(s,a)[ ' ] = ^a[ ? ] IS called (5,a)-consistent if ir[ ■ ] is a rule that assigns a value to each risk X satisfying the axioms S and such that 7r[X] > F^ (a), where F^ (a)is the ath quantile of the risk X, and is defined, as usual, by F^1(a) = inf{x : F(x) > a}.Theorem 1. Let X be a risk variable , and let...