Nearly Comonotone Approximation of Periodic Functions

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.

Some General Inequalities for Choquet Integral

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.

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 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.

The representations of two types of functionals on L^∞(Ω,F) and L^∞(Ω,F,P)

This article gives the representations of two types of real functionals on L∞(Ω,F)or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.

Remarks on Equality of Two Distributions under Some Partial Orders

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.

Upper risk bounds in internal factor models with constrained specification sets

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.