A note on convex risk statistic

By weakening the comonotonic subadditivity axiom, we give the definition of the comonotonic convex risk statistic. Motivated by Ahmed et al. (2008) [1], we establish the representation results for the comonotonic convex risk statistics and the law-invariance convex risk statistics by using the convex analysis.     

Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures

In this paper, we extend the concept of tail subadditivity (Belles-Sampera et al., 2014a; Belles-Sampera et al., 2014b) for distortion risk measures and give sufficient and necessary conditions for a distortion risk measure to be tail subadditive. We also introduce the generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure. To further illustrate the applications of the tail subadditivity, we propose multivariate tail distortion (MTD) risk measures and generalize the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016). The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, we discuss the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks and explore the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations.     

An overview of representation theorems for static risk measures

In this paper, we give an overview of representation theorems for various static risk measures: coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity, law-invariant coherent or convex risk measures, risk measures with comonotonic subadditivity or convexity and respecting stochastic orders. This work was supported by National Natural Science Foundation of China (Grant No. 10571167), National Basic Research Program of China (973 Program) (Grant No. 2007CB814902), and Science Fund for Creative Research Groups (Grant No. 10721101)     

CreditRisk+模型下商业银行经济资本配置研究

对金融资产风险的度量与经济资本的分配应该体现分散化效应,传统的V aR方式不能保证分散化效应的次可加性.本文讨论了基于T a ilV aR这一新的风险度量与经济资本分配标准,并在违约率均值不变情况下,对C red itR isk+模型下的商业银行经济资本分配进行了实证分析.     

非线性期望和非线性估价的次可加性

证明了在适当的条件下F-期望的次可加性蕴涵条件F-期望的次可加性,F-估价的次可加性蕴涵Ft-相容非线性估价的次可加性.     

When does the subadditivity theorem for multiplier ideals hold?

Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.

     

Risk measures with comonotonic subadditivity or convexity on product spaces

In this paper, by an axiomatic approach, we propose the concepts of comonotonic subadditivity and comonotonic convex risk measures for portfolios, which are extensions of the ones introduced by Song and Yan(2006)Representation results for these new introduced risk measures for portfolios are given in terms of Choquet integralsLinks of these newly introduced risk measures to multi-period comonotonic risk measures are representedFinally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.     

Subadditivity of The Entropy and its Relation to Brascamp–Lieb Type Inequalities

We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on ${\mathbb {R}^n}We prove a general duality result showing that a Brascamp–Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This opens a new approach to the proof of Brascamp–Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on \mathbb Rⁿ{\mathbb {R}^n}, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp–Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard’s inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr?dinger operators. Though we focus mainly on the case of random variables in \mathbb Rⁿ{\mathbb {R}^n} in this paper, we discuss extensions to other settings as well.     

A Counterexample for Subadditivity of Multiplier Ideals on Toric Varieties

We construct a 3-dimensional complete intersection toric variety on which the subadditivity formula doesn't hold, answering negatively a question by Takagi and Watanabe. A combinatorial proof of the subadditivity formula on 2-dimensional normal toric varieties is also provided.     

Risk measures on the space of infinite sequences

Axiomatically based risk measures have been the object of numerous studies and generalizations in recent years. In the literature we find two main schools: coherent risk measures (Artzner, Coherent Measures of Risk. Risk Management: Value at Risk and Beyond, 1998) and insurance risk measures (Wang, Insur Math Econ 21:173–183, 1997). In this note, we set to study yet another extension motivated by a third axiomatically based risk measure that has been recently introduced. In Heyde et al. (Working Paper, Columbia University, 2007), the concept of natural risk statistic is discussed as a data-based risk measure, i.e. as an axiomatic risk measure defined in the space \mathbb Rⁿ{\mathbb R^n} . One drawback of these kind of risk measures is their dependence on the space dimension n. In order to circumvent this issue, we propose a way to define a family {ρ _n } _n=1,2,... of natural risk statistics whose members are defined on \mathbbRⁿ{\mathbb{R}^n} and related in an appropriate way. This construction requires the generalization of natural risk statistics to the space of infinite sequences l ^∞.     

A note on arbitrage,approximate arbitrage and the fundamental theorem of asset pricing

We provide a critical analysis of the proof of the fundamental theorem of asset pricing given in the paper Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing by B. Wong and C.C. Heyde [Stochastics 82 (2010), pp. 189–200] in the context of incomplete Itô-process models. We show that their approach can only work in the known case of a complete financial market model and give an explicit counter example.     

一个概率不等式的推广

通过对于概率性质中次可加性的分析以及和若尔当公式的比较,猜想并证明了由次可加性推广的类似级数展开的一般不等式形式.     

The Search for the Ultimate Baseball Statistic

Abstract

The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century by David Salsburg. New York: W. H. Freeman and Co., 2001, pp. xi + 340, $23.95. Reviewed by Todd E. Bodner

Statisticians of the Centuries by C. C. Heyde and E. Seneta (Eds.). New York: Springer-Verlag, 2001, pp. xii + 500, $45.95. Reviewed by Todd E. Bodner     

Rates of convergence for the laws of large numbers for independent Banach-valued random variables

In this paper, results of Lai, Heyde, and Rohatgi concerning the convergence rates for the laws of large numbers are extended for the case of independent random variables taking values in a separable Banach space.     

A sharp inequality for martingales and its applications

In this paper, the best constant with respect to an inequality for martingales in Hall and Heyde (1980) is obtained. As a consequence, some large deviations with martingale difference and moving average process are established.     

Inference for intermediate Haezendonck–Goovaerts risk measure

Recently Haezendonck–Goovaerts (H–G) risk measure has received much attention in actuarial science. Nonparametric inference has been studied by Ahn and Shyamalkumar (2014) and Peng et al. (2015) when the risk measure is defined at a fixed level. In risk management, the level is usually set to be quite near one by regulators. Therefore, especially when the sample size is not large enough, it is useful to treat the level as a function of the sample size, which diverges to one as the sample size goes to infinity. In this paper, we extend the results in Peng et al. (2015) from a fixed level to an intermediate level. Although the proposed maximum empirical likelihood estimator for the H–G risk measure has a different limit for a fixed level and an intermediate level, the proposed empirical likelihood method indeed gives a unified interval estimation for both cases. A simulation study is conducted to examine the finite sample performance of the proposed method.     

Asymptotic sequential Rademacher complexity of a finite function class

For a finite function class, we describe the large sample limit of the sequential Rademacher complexity in terms of the viscosity solution of a G-heat equation. In the language of Peng’s sublinear expectation theory, the same quantity equals to the expected value of the largest order statistics of a multidimensional G-normal random variable. We illustrate this result by deriving upper and lower bounds for the asymptotic sequential Rademacher complexity.     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

Forward–backward SDEs with random terminal time and applications to pricing special European-type options for a large investor

In this paper, we first discuss the solvability of coupled forward–backward stochastic differential equations (FBSDEs, for short) with random terminal time. We prove the existence and uniqueness of adapted solution to such FBSDEs under some natural assumptions. The method of proof adopted is to construct a contraction mapping related to the solutions of a sequence of backward SDEs. Our monotonicity-type assumptions are different from those in Hu and Peng (1995) [4], Peng and Shi (2000) [11], and so on. As a corollary of our main result, the solvability of FBSDEs with a finite time horizon is discussed. Finally, the existence and uniqueness theorem of the solution to FBSDEs with a finite time horizon is applied to price special European-type options for a large investor.