Dini–Lipschitz criterion: Difference between revisions

Revision as of 09:43, 21 August 2013

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Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) In financial mathematics, a distortion risk measure is a type of risk measure which is related to the cumulative distribution function of the return of a financial portfolio.

Mathematical definition

The function $ρ_{g} : L^{p} \to ℝ$ associated with the distortion function $g : [0, 1] \to [0, 1]$ is a distortion risk measure if for any random variable of gains $X \in L^{p}$ (where $L^{p}$ is the L^p space) then

ρ_{g} (X) = - \int_{0}^{1} F_{- X}^{- 1} (p) d \tilde{g} (p) = \int_{- \infty}^{0} \tilde{g} (F_{- X} (x)) d x - \int_{0}^{\infty} g (1 - F_{- X} (x)) d x

where $F_{- X}$ is the cumulative distribution function for $- X$ and $\tilde{g}$ is the dual distortion function $\tilde{g} (u) = 1 - g (1 - u)$ .^[1]

If $X \leq 0$ almost surely then $ρ_{g}$ is given by the Choquet integral, i.e. $ρ_{g} (X) = - \int_{0}^{\infty} g (1 - F_{- X} (x)) d x .$ ^[1]^[2] Equivalently, $ρ_{g} (X) = 𝔼^{ℚ} [- X]$ ^[2] such that $ℚ$ is the probability measure generated by $g$ , i.e. for any $A \in ℱ$ the sigma-algebra then $ℚ (A) = g (ℙ (A))$ .^[3]

Properties

In addition to the properties of general risk measures, distortion risk measures also have:

Law invariant: If the distribution of $X$ and $Y$ are the same then $ρ_{g} (X) = ρ_{g} (Y)$ .
Monotone with respect to first order stochastic dominance.
1. If $g$ is a concave distortion function, then $ρ_{g}$ is monotone with respect to second order stochastic dominance.
$g$ is a concave distortion function if and only if $ρ_{g}$ is a coherent risk measure.^[1]^[2]

Examples

Value at risk is a distortion risk measure with associated distortion function $g (x) = {\begin{cases} 0 & if 0 \leq x < 1 - α \\ 1 & if 1 - α \leq x \leq 1 \end{cases} .$ ^[2]^[3]
Conditional value at risk is a distortion risk measure with associated distortion function $g (x) = {\begin{cases} \frac{x}{1 - α} & if 0 \leq x < 1 - α \\ 1 & if 1 - α \leq x \leq 1 \end{cases} .$ ^[2]^[3]
The negative expectation is a distortion risk measure with associated distortion function $g (x) = x$ .^[1]

References

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[PortfolioOpt-1] 1.0 ^1.1 ^1.2 ^1.3 Template:Cite doi

[Wirch-2] 2.0 ^2.1 ^2.2 ^2.3 ^2.4 Template:Cite web

[PropertiesDRM-3] 3.0 ^3.1 ^3.2 Template:Cite doi

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+{{context|date=May 2012}}
+In [[financial mathematics]], a '''distortion risk measure''' is a type of [[risk measure]] which is related to the [[cumulative distribution function]] of the [[rate of return|return]] of a [[financial portfolio]].
+== Mathematical definition ==
+The function <math>\rho_g: L^p \to \mathbb{R}</math> associated with the [[distortion function]] <math>g: [0,1] \to [0,1]</math> is a ''distortion risk measure'' if for any [[random variable]] of gains <math>X \in L^p</math> (where <math>L^p</math> is the [[Lp space|L<sup>p</sup> space]]) then
+: <math>\rho_g(X) = -\int_0^1 F_{-X}^{-1}(p) d\tilde{g}(p) = \int_{-\infty}^0 \tilde{g}(F_{-X}(x))dx - \int_0^{\infty} g(1 - F_{-X}(x)) dx</math>
+where <math>F_{-X}</math> is the cumulative distribution function for <math>-X</math> and <math>\tilde{g}</math> is the dual distortion function <math>\tilde{g}(u) = 1 - g(1-u)</math>.<ref name="PortfolioOpt"/>
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+If <math>X \leq 0</math> [[almost surely]] then <math>\rho_g</math> is given by the [[Choquet integral]], i.e. <math>\rho_g(X) = -\int_0^{\infty} g(1 - F_{-X}(x)) dx.</math><ref name="PortfolioOpt">{{cite doi|10.1007/978-0-387-77439-8_25}}</ref><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|format=pdf|accessdate=March 10, 2012}}</ref>  Equivalently, <math>\rho_g(X) = \mathbb{E}^{\mathbb{Q}}[-X]</math><ref name="Wirch"/> such that <math>\mathbb{Q}</math> is the [[probability measure]] generated by <math>g</math>, i.e. for any <math>A \in \mathcal{F}</math> the [[sigma-algebra]] then <math>\mathbb{Q}(A) = g(\mathbb{P}(A))</math>.<ref name="PropertiesDRM">{{cite doi|10.1007/s11009-008-9089-z}}</ref>
+=== Properties ===
+In addition to the properties of general risk measures, distortion risk measures also have:
+# ''Law invariant'': If the distribution of <math>X</math> and <math>Y</math> are the same then <math>\rho_g(X) = \rho_g(Y)</math>.
+# ''Monotone'' with respect to first order [[stochastic dominance]].
+## If <math>g</math> is a [[concave function|concave]] distortion function, then <math>\rho_g</math> is monotone with respect to second order stochastic dominance.
+# <math>g</math> is a [[concave function|concave]] distortion function if and only if <math>\rho_g</math> is a [[coherent risk measure]].<ref name="PortfolioOpt"/><ref name="Wirch"/>
+== Examples ==
+* [[Value at risk]] is a distortion risk measure with associated distortion function <math>g(x) = \begin{cases}0 & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.</math><ref name="Wirch"/><ref name="PropertiesDRM"/>
+* [[Conditional value at risk]] is a distortion risk measure with associated distortion function <math>g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x < 1-\alpha\\ 1 & \text{if }1-\alpha \leq x \leq 1\end{cases}.</math><ref name="Wirch"/><ref name="PropertiesDRM"/>
+* The negative [[expected value|expectation]] is a distortion risk measure with associated distortion function <math>g(x) = x</math>.<ref name="PortfolioOpt"/>
+== See also ==
+* [[Risk measure]]
+* [[Coherent risk measure]]
+* [[Deviation risk measure]]
+* [[Spectral risk measure]]
+== References ==
+{{reflist}}
+*{{cite journal|last=Wu|first=Xianyi|coauthors=Xian Zhou|title=A new characterization of distortion premiums via countable additivity for comonotonic risks|journal=Insurance: Mathematics and Economics|date=April 7, 2006|volume=38|issue=2|pages=324–334|url=http://www.sciencedirect.com/science/article/pii/S016766870500123X|accessdate=March 14, 2012}}
+[[Category:Mathematical finance]]
+[[Category:Financial risk]]

Dini–Lipschitz criterion: Difference between revisions

Revision as of 09:43, 21 August 2013

Contents

Mathematical definition

Properties

Examples

See also

References

Navigation menu

Dini–Lipschitz criterion: Difference between revisions

Revision as of 09:43, 21 August 2013

Mathematical definition

Properties

Examples

See also

References

Navigation menu

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