Mass distributions of two-dimensional extreme-value copulas and related results

Working with Markov kernels (conditional distributions) and right-hand derivatives D ⁺ A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) C _A distribute mass. Underlining the usefulness of working directly with D ⁺ A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of C _A concentrates its mass on the graphs of some increasing homeomorphisms f _t , we determine which EVC assigns maximum mass to the union of the graphs of \(f_{t_{1}},\ldots ,f_{t_{N}}\), derive the absolutely continuous component of an arbitrary EVC C _A and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs C _A which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel \(K_{C_{A}}(x,\cdot )\) of C _A have full support [0,1] for every x∈[0,1].     

Spatially homogeneous copulas

We consider spatially homogeneous copulas, i.e. copulas whose corresponding measure is invariant under a special transformations of \([0,1]^2\), and we study their main properties with a view to possible use in stochastic models. Specifically, we express any spatially homogeneous copula in terms of a probability measure on [0, 1) via the Markov kernel representation. Moreover, we prove some symmetry properties and demonstrate how spatially homogeneous copulas can be used in order to construct copulas with surprisingly singular properties. Finally, a generalization of spatially homogeneous copulas to the so-called (m, n)-spatially homogeneous copulas is studied and a characterization of this new family of copulas in terms of the Markov \(*\)-product is established.

     

A remark about the interpolation of spaces of continuous, vector-valued functions

If A is a sectorial operator on a Banach space X, then the space C([0,1];(X,D(A))_θ,∞) is a subspace of the interpolation space (C([0,1];X),C([0,1];D(A)))_θ,∞. The inclusion is strict in general.     

A note on approximation properties of q-Durrmeyer operators

In this paper, the approximation properties of q-Durrmeyer operators D_n,q(f;x) for f∈C[0,1] are discussed. The exact class of continuous functions satisfying approximation process lim_n→∞D_n,q(f;x)=f(x) is determined. The results of the paper provide an elaboration of the previously-known ones on operators D_n,q.     

Smoothness of solutions of almost hypoelliptic equations

A two-dimensional linear differential operator P(D) = P(D ₁, D ₂) is called almost hypoelliptic if all derivatives D ^α P of the characteristic polynomial P(ζ) = P(ζ ₁, ζ ₂) are estimated by P(ζ). Assuming that {Ω _κ = (x ₁, x ₂) ∈ E ² : |x ₁| < κ, x ₂ ∈ R ¹}, the paper proves that if the width κ of the strip Ω _κ exceeds some C = C(P) > 0, then all solutions {u} of the almost hypoelliptic equation P(D)u = 0 in a Sobolev space are infinitely smooth functions with respect to x ₁.     

Intervals of 1-Lipschitz aggregation operators,quasi-copulas,and copulas with given affine section

Best lower and upper bounds for 1-Lipschitz aggregation operators with a given affine section are given. These are used to determine best bounds for quasi-copulas and copulas with a given affine section. However, in general there is no greatest copula with a given non-decreasing affine section. These results are used to study (quasi-)copulas with arbitrary affine sections. A significant part of this work was done during a visit of the second author at the Johannes Kepler University, Linz (Austria). The second author was supported by the grant VEGA 1/3012/06 and by the Science and Technology Assistance Agency (Contract No. APVT-20-003204). Both authors would like to thank the anonymous referee whose comments (including the two copulas C ₁ and C ₂ given in the Conclusion) not only solved the originally stated open problem in the negative, but also allowed them to formulate two more interesting open problems.     

Products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ, M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We consider linear operators induced by products of these operators on weighted Bergman spaces on D. The boundedness is established by using Carleson-type measures.     

Semi-analytical Formula for Pricing Bilateral Counterparty Risk of CDS with Correlated Credit Risks

Based on the framework of [7], we discuss pricing bilateral counterparty risk of CDS, where each individual default intensity is modeled by a shifted CIR process with jump (JCIR++), and the correlation between the default times is modeled by a copula function. We present a semi-analytical formula for pricing bilateral counterparty risk of CDS, which is more convenient to compute through calculating multiple numerical integration or using Monte-Carlo simulation without simulating default times. Moreover, we obtain simpler formulae under FGM copulas, Bernstein copulas and C^A,B copulas, which can be applied for speeding up the computation and reducing the pricing error. Numerical results under FGM copulas and C^A,B copulas show that our method performs better both in computation speed and accuracy.     

Mathematical modeling of real-world images

The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR ₃, and exploring theHausdorff distance between these point sets, a metric spaceIM _D of functions is defined. The main purpose is to show that the functionsf∈IM _D, defined on the squareD=[0,1]², are appropriate mathematical models of real world images. The properties of the metric spaceIM _D are studied and methods of approximation for the purpose of image compression are presented. The metric spaceIM _D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM _D may be digitized and represented by a pixel functionp _n, withn pixels, in such a way that the distance betweenf andp _n is no greater than 2n ^?1/2. It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

基于倾斜-t Copula的半参数马尔可夫链

在不指定时间序列结构的情况下,我们的分布模型是基于多变量离散时间的相应马尔可夫族和相关变量一维的边际分布.这样的模型可以同时处理时间序列之间的相互依赖和每个时间序列沿时间方向的依赖.具体的参数copula被指定为倾斜-t. 倾斜-t Copla能够处理不对称,偏斜和粗尾的数据分布.三个股票指数日均收益的实证研究表明,倾斜-t copula的马尔可夫模型要比以下模型更好:倾斜正态Copula马可夫, t-copula马可夫, 倾斜-t copula但无马尔可夫特性.     

Sampling distribution of Gini's index of diversity

In the case of a multinomial distribution Π₁(1-Π₁)+Π₂(1-Π₂)+…+ Π_k(1-Π_k) is at times referred to as Gini's index of diversity. In this paper, we present the distributional properties of the statistic

, based on samples of size n for a homogeneous multinomial distribution. For 2≤n≤ 12 and 2≤k≤12, we give a short table of the pdf, cdf, and the moments of D. For large values of n, we mention some results for the asymptotic distribution of D for the general multinomial distribution.     

Fixed points of Mizoguchi–Takahashi contraction on a metric space with a graph and applications

In this paper we extend Mizoguchi–Takahashi's fixed point theorem for multi-valued mappings on a metric space endowed with a graph. As an application, we establish a fixed point theorem on an ε  -chainable metric space for mappings satisfying Mizoguchi–Takahashi contractive condition uniformly locally. Also, we establish a result on the convergence of successive approximations for certain operators (not necessarily linear) on a Banach space as another application. Consequently, this result yields the Kelisky–Rivlin theorem on iterates of the Bernstein operators on the space C[0,1]$C[0,1]$ and also enables us study the asymptotic behaviour of iterates of some nonlinear Bernstein type operators on C[0,1]$C[0,1]$.     

A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

Existence and sharp regularity results for linear parabolic non-autonomous integro-differential equations

We study existence, uniqueness and maximal regularity of the strict solutionu∈C ¹([0,T],E) of the integro-differential equation \(u'(t) - A(t)u(t) - \int {_0^1 } B(t,s)u(s)ds = f(t),t \in [0,T],\) with the initial datumu(0)=x, in a Banach spaceE, {itA(itt)}_f∈|0,1| is a family of generators of analytic semigroups whose domainsD _A(t) are not constant int as well as (possibly) not dense inE, whereas {itB(itt)}_0≦11≦T is a family of closed linear operators withD _B(t,s) ?D _A(s) ∨t∈[s, T]. We prove necessary and sufficient conditions for existence of the strict solution and for Hölder continuity of its derivative; well-posedness of the problem with respect to the Hölder norms is also shown.     

Positive solution of singular Dirichlet boundary value problems for second order differential equation system

This paper investigates the existence of positive solutions of singular Dirichlet boundary value problems for second order differential system. A necessary and sufficient condition for the existence of C[0,1]×C[0,1] positive solutions as well as C¹[0,1]×C¹[0,1] positive solutions is given by means of the method of lower and upper solutions and the fixed point theorems. Our nonlinearity fi(t,x₁,x₂) may be singular at x₁=0, x₂=0, t=0 and/or t=1, i=1,2.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

The metric dimension of Cayley digraphs

A vertex x in a digraph D is said to resolve a pair u, v of vertices of D if the distance from u to x does not equal the distance from v to x. A set S of vertices of D is a resolving set for D if every pair of vertices of D is resolved by some vertex of S. The smallest cardinality of a resolving set for D, denoted by dim(D), is called the metric dimension for D. Sharp upper and lower bounds for the metric dimension of the Cayley digraphs Cay(Δ:Γ), where Γ is the group Z_n1⊕Z_n2⊕?⊕Z_nm and Δ is the canonical set of generators, are established. The exact value for the metric dimension of Cay({(0,1),(1,0)}:Z_n⊕Z_m) is found. Moreover, the metric dimension of the Cayley digraph of the dihedral group D_n of order 2n with a minimum set of generators is established. The metric dimension of a (di)graph is formulated as an integer programme. The corresponding linear programming formulation naturally gives rise to a fractional version of the metric dimension of a (di)graph. The fractional dual implies an integer dual for the metric dimension of a (di)graph which is referred to as the metric independence of the (di)graph. The metric independence of a (di)graph is the maximum number of pairs of vertices such that no two pairs are resolved by the same vertex. The metric independence of the n-cube and the Cayley digraph Cay(Δ:D_n), where Δ is a minimum set of generators for D_n, are established.     

Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces

Let ψ be a holomorphic function on the open unit disk D and φ a holomorphic self-map of D. Let C_φ,M_ψ and D denote the composition, multiplication and differentiation operator, respectively. We find an asymptotic expression for the essential norm of products of these operators on weighted Bergman spaces on the unit disk. This paper is a continuation of our recent paper concerning the boundedness of these operators on weighted Bergman spaces.     

Некоторые точные инт егральные оценки про изводных рациональных и алгеб раических функций. Пр иложения

Exact estimates are obtained for integrals of absolute values of derivatives and gradients, for integral moduli of continuity and for major variations of piecewise algebraic functions (in particular, for polynomials, rational functions, splines, etc.). These results are applied to the problems of approximation theory and to the estimates of Laurent and Fourier coefficients. Typical results:

1. IfE is a measurable subset of the circle or of a line in thez-plane andR(z) is a rational function of degree ≦n, ¦R(z)¦≦ (z∈E), then ∝_E ¦R′(z)¦dz¦≦ 2πn; the latter estimate is exact forn=0, 1, ... and everyE with positive measure;
2. Iff(x ₁,x ₂, ...,x _m) is a real valued piecewise algebraic function of order (n, k) on the unit ballD?R ^m (in particular, a real valued rational function of order ≦n), and ¦f¦≦1 onD, then ∝_D¦gradf¦dx≦2π ^m/2n/Π(m/2); herem≧1, n≧0, 1≦k<∞;
3. LetE=Π={z∶¦z¦=1}, and letc _m(R) be the mth Laurent coefficient ofR onΠ,C _m(n)=sup{¦cm(R)¦}, where sup is taken over allR from 1), then 1/7 min {n/¦m¦, 1} ≦C _m(n) ≦ min {n/¦m¦, 1}.