Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same ...Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same monotonicity as f, everywhere except, perhaps, the small intervals.展开更多
Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is c...Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.展开更多
With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to ...With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f1, f2, …, fn are comonotone, we also obtain the Hölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.展开更多
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）. ...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 integrals. Links of these newly introduced risk measures to multi-period comonotonic risk measures are represented. Finally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.展开更多
This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respecti...This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.展开更多
It is a very important issue for us to explore the effects of the marriage to life. In recent years, many scholars have proved that the marriage can lengthen life-span from different angles. With the development of th...It is a very important issue for us to explore the effects of the marriage to life. In recent years, many scholars have proved that the marriage can lengthen life-span from different angles. With the development of theory of dependence random variables, we discuss the effects of the marriage to life and provide a mathematical basis of the idea that the harmonious marriage can lengthen life-span in this paper. Meanwhile, we analyze the impact of the marriage on life on the basis of the net single premium.展开更多
In this note we establish some appropriate conditions for stochastic equality of two random vari- ables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate ...In this note we establish some appropriate conditions for stochastic equality of two random vari- ables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered.展开更多
For the class of(partially specified)internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models.This allows us to derive meaningf...For the class of(partially specified)internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models.This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor.The proof of our main comparison result is not standard.It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers.An application to real market data shows considerable improvement over the standard method.展开更多
文摘Suppose that a continuous 2re-periodic function f on the real axis changes its monotonicity at points Yi In this paper, for each n _ N, a trigonometric polynomial Pn of order cn is found such that： Pn has the same monotonicity as f, everywhere except, perhaps, the small intervals.
文摘Let a function f E C[-1, 1], changes its monotonisity at the finite collection Y ：= {y1,……, ys} of s points Yi ∈ （-1, 1）. For each n 〉 N（Y）, we construct an algebraic polynomial Pn, of degree 〈 n, which is comonotone with f, that is changes its monotonisity at the same points yi as f, and ｜f（x） - Pn（x）｜ ≤ c（s）ω2 （f1 √1-x^2/n）,x∈ [-1,1] where N（Y） is a constant depending only on Y, c（s） is a constant depending only on s and ω2 （f, t） is the second modulus of smoothness of f.
基金supported by the National Natural Science Foundation of China(no.51374199).
文摘With the development of fuzzy measure theory, the integral inequalities based on Sugeno integral are extensively investigated. We concern on the inequalities of Choquuet integral. The main purpose of this paper is to prove the H?lder inequality for any arbitrary fuzzy measure-based Choquet integral whenever any two of these integrated functions f, g and h are comonotone, and there are three weights. Then we prove Minkowski inequality and Lyapunov inequality for Choquet integral. Moreover, when any two of these integrated functions f1, f2, …, fn are comonotone, we also obtain the Hölder inequality, Minkowski inequality and Lyapunov inequality hold for Choquet integral.
基金Supported by the National Natural Science Foundation of China(11371284)the Natural Science Foundation of Henan Province(14B110037)
文摘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 integrals. Links of these newly introduced risk measures to multi-period comonotonic risk measures are represented. Finally, applications of the newly introduced comonotonic coherent risk measures to capital allocations are provided.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10571167).
文摘This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.
基金Supported by the Natural Science Foundation of Fujian(S0650038)Technologic innovation Foundation of Xiamen University(K70001)
文摘It is a very important issue for us to explore the effects of the marriage to life. In recent years, many scholars have proved that the marriage can lengthen life-span from different angles. With the development of theory of dependence random variables, we discuss the effects of the marriage to life and provide a mathematical basis of the idea that the harmonious marriage can lengthen life-span in this paper. Meanwhile, we analyze the impact of the marriage on life on the basis of the net single premium.
基金supported by the National Natural Science Foundation of China(11571198,11701319)
文摘In this note we establish some appropriate conditions for stochastic equality of two random vari- ables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered.
文摘For the class of(partially specified)internal risk factor models we establish strongly simplified supermodular ordering results in comparison to the case of general risk factor models.This allows us to derive meaningful and improved risk bounds for the joint portfolio in risk factor models with dependence information given by constrained specification sets for the copulas of the risk components and the systemic risk factor.The proof of our main comparison result is not standard.It is based on grid copula approximation of upper products of copulas and on the theory of mass transfers.An application to real market data shows considerable improvement over the standard method.
