The Expansion Of Haezendonck Risk Measure

Risk will be met unavoidably in the economic system,so the importance of risk management is outstanding day by day.The first step of risk management is to carry on quantization of the risk.There are a lot of ways to quantize to the risk.Haezendonck quantize to the risk by his own way, known as Haezendonck risk measure.But when the risk becomes vary great, the lose or the income will become a little larger, and the function vision should not be straight line, the shape should change corresponding to the actual case. So we expand the Haezendonck risk measure further. Then, we use the knowledge of probability and statistics and the skill of mathematics analysis to get some properties of the expanded Haezendonck risk measure, which has certain directive significance to actual life.This text divides six sections.Section one, introduction.Section two, the risk and risk management.Introduced the risk briefly, through the recognition of risk, we realize the importance of risk management, hence we strengthen to take precautions against the risk.Section three, the risk measure.Introduced a main content in risk management further—the risk measure, which draws this paper.Section four, the Haezendonck risk measure.The Haezendonck risk measure is a kind of quantization of form to the risk, therefore, we realize that for this kind of risk measure, which provides a function, we can define a kind of risk measure.Section five, the expanded Haezendonck risk measure.We are inspired, having expanded the Haezendonck risk measure. We attempt to let it be used more. Then we have also received some properties of the expandedrisk measure. Every theorem is sketched as follow:Definition 1. Let S be a set of axioms for risk measures, and a, 0 < a < 1, be a level. A risk measure ir[ ■ ] = X(s,a)[ ' ] == ^al ? ] is called (S,a)-consistent if ?r[ ? ] is a rule that assigns a value to each risk X satisfying the axioms S and such that n[X] > F^l(a), where F^1(a)is the ath quantile of the risk X, and is defined, as usual, by F^ia) = inf{x : F(x) > a}.Theorem 1. Let X be a risk variable , and let ip be a non-negative, strictly increasing, and continuous funtion on [0,+00) with i^x(a), ira[X,x] > x.Definition 2. Let X be a risk variable , let ip be a non-negative, strictly increasing, and continuous funtion with and v(+°°) = +°°) and let 0 < a < 1 be arbitrarily fixed.We callttq[*]= inf *?[*,*]-oo 0;B3:Translation invariance: ira[X + a] = na[X] + a for any a.Collary 1. Let ttq[X] be the expanded Haezendonck risk measure of a risk variable X, na[X, x] is the unique solution of the equation 1—a = E[ip(Also let 0 satisfying the restrictions in Definition 2 and F(-) be a distribution, and let 0 < a < 1 be arbitrarily given.Then we have Cl:Monotonicity: If X {x is convex (concave).Theorem 4. Let le* 7rL[-^?a;3'* = 1>2, be the solutions of (2) with...