scholarly journals On the extension property of dilatation monotone risk measures

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Massoomeh Rahsepar ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Natural Extension ◽  
Random Variables ◽  
Extension Property ◽  
Fatou Property ◽  
Quasi Convexity

Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] \rho\colon\mathcal{X}\to(-\infty,\infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ⁢ ( L 1 , L ) \sigma(L^{1},\mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] \overline{\rho}\colon L^{1}\to(-\infty,\infty] . Moreover, ρ ¯ \overline{\rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ \overline{\rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.

scholarly journals The strong Fatou property of risk measures

2018 ◽  
Vol 6 (1) ◽  
pp. 183-196 ◽  
Author(s):  
Shengzhong Chen ◽  
Niushan Gao ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Lower Semicontinuous ◽  
Rearrangement Invariant Space ◽  
Invariant Space ◽  
Fatou Property ◽  
Rearrangement Invariant Spaces ◽  
Dual Representations ◽  
Wide Range ◽  
Invariant Functional ◽  
Invariant Spaces

AbstractIn this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property,whichwas introduced in [19], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on a rearrangement invariant space X with the strong Fatou property is (X, L1) lower semicontinuous and that the converse is true on a wide range of rearrangement invariant spaces. We also study inf-convolutions of law-invariant or surplus-invariant risk measures that preserve the (strong) Fatou property.

On risk measuring in the variance-gamma model

2018 ◽  
Vol 35 (1-2) ◽  
pp. 23-33 ◽  
Author(s):  
Roman V. Ivanov
Keyword(s):  
Value At Risk ◽  
Hypergeometric Functions ◽  
Risk Measures ◽  
Random Variables ◽  
Gamma Model ◽  
Variance Gamma ◽  
Stochastic Volatilities ◽  
Investment Returns ◽  
Variance Gamma Model ◽  
Analytical Expressions

AbstractIn this paper, we discuss the problem of calculating the primary risk measures in the variance-gamma model. A portfolio of investments in a one-period setting is considered. It is supposed that the investment returns are dependent on each other. In terms of the variance-gamma model, we assume that there are relations in both groups of the normal random variables and the gamma stochastic volatilities. The value at risk, the expected shortfall and the entropic monetary risk measures are discussed. The obtained analytical expressions are based on values of hypergeometric functions.

Error Oblate

2010 ◽  
Vol 40-41 ◽  
pp. 941-947
Author(s):  
Yang Sheng You ◽  
Yang Hua Gao ◽  
Yan Ying Chen ◽  
Xin Jie Deng
Keyword(s):  
Data Processing ◽  
Normal Distribution ◽  
Natural Extension ◽  
Random Variables ◽  
Error Ellipse ◽  
Theoretical Significance ◽  
Concept Theory

Error Ellipse can’t correctly depict the distribution of random variables in non-normal distribution, so this paper presents the concept, theory and methods of Error Oblate. Error Oblate is the natural extension of Error Ellipse. It is widely applicable and carries great theoretical significance in data processing.

scholarly journals Robustness regions for measures of risk aggregation

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Silvana M. Pesenti ◽  
Pietro Millossovich ◽  
Andreas Tsanakas
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Random Variables ◽  
Distribution Functions ◽  
Aggregation Function ◽  
Convex Risk Measures ◽  
Linear Growth Condition ◽  
Risk Aggregation ◽  
Definition Of

AbstractOne of risk measures’ key purposes is to consistently rank and distinguish between different risk profiles. From a practical perspective, a risk measure should also be robust, that is, insensitive to small perturbations in input assumptions. It is known in the literature [14, 39], that strong assumptions on the risk measure’s ability to distinguish between risks may lead to a lack of robustness. We address the trade-off between robustness and consistent risk ranking by specifying the regions in the space of distribution functions, where law-invariant convex risk measures are indeed robust. Examples include the set of random variables with bounded second moment and those that are less volatile (in convex order) than random variables in a given uniformly integrable set. Typically, a risk measure is evaluated on the output of an aggregation function defined on a set of random input vectors. Extending the definition of robustness to this setting, we find that law-invariant convex risk measures are robust for any aggregation function that satisfies a linear growth condition in the tail, provided that the set of possible marginals is uniformly integrable. Thus, we obtain that all law-invariant convex risk measures possess the aggregation-robustness property introduced by [26] and further studied by [40]. This is in contrast to the widely-used, non-convex, risk measure Value-at-Risk, whose robustness in a risk aggregation context requires restricting the possible dependence structures of the input vectors.

scholarly journals Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

2018 ◽  
Vol 22 (2) ◽  
pp. 395-415 ◽  
Author(s):  
Niushan Gao ◽  
Denny Leung ◽  
Cosimo Munari ◽  
Foivos Xanthos
Keyword(s):  
Risk Measures ◽  
Orlicz Spaces ◽  
Fatou Property ◽  
Law Invariant Risk Measures

An extension of a lemma of Wald

1966 ◽  
Vol 3 (01) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y 1, y 2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

Series expansions for convolutions of Pareto distributions

2015 ◽  
Vol 32 (1) ◽  
Author(s):  
Quang Huy Nguyen ◽  
Christian Y. Robert
Keyword(s):  
Infinite Series ◽  
Asymptotic Expansions ◽  
Value At Risk ◽  
Risk Measures ◽  
Random Variables ◽  
Series Expansions ◽  
Sums Of Random Variables ◽  
Scale Parameters ◽  
Pareto Distributions ◽  
Higher Order Terms

AbstractAsymptotic expansions for the tails of sums of random variables with regularly varying tails are mainly derived in the case of identically distributed random variables or in the case of random variables with the same tail index. Moreover, the higher-order terms are often given under the condition of existence of a moment of the distribution. In this paper, we obtain infinite series expansions for convolutions of Pareto distributions with non-integer tail indices. The Pareto random variables may have different tail indices and different scale parameters. We naturally find the same constants for the first terms as given in the previous asymptotic expansions in the case of identically distributed random variables, but we are now able to give the next additional terms. Since our series expansion is not asymptotic, it may be also used to compute the values of quantiles of the distribution of the sum as well as other risk measures such as the Tail Value at Risk. Examples of values are provided for the sum of at least five Pareto random variables and are compared to those determined via previous asymptotic expansions or via simulations.

An extension of a lemma of Wald

1966 ◽  
Vol 3 (1) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y1, y2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

scholarly journals Distortion risk measures for sums of random variables

2004 ◽  
Vol 26 (4) ◽  
pp. 631-641
Author(s):  
Grzegorz Darkiewicz ◽  
Jan Dhaene ◽  
Marc Goovaerts
Keyword(s):  
Risk Measures ◽  
Random Variables ◽  
Sums Of Random Variables ◽  
Distortion Risk Measures
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