Extremal dependence measure and extremogram: the regularly varying case

The dependence of large values in a stochastic process is an important topic in risk, insurance and finance. The idea of risk contagion is based on the idea of large value dependence. The Gaussian copula notoriously fails to capture this phenomenon. Two notions in a process or vector context which summarize extremal dependence in a function comparable to a correlation function are the extremal dependence measure (EDM) and the extremogram. We review these ideas and compare the two tools and end with a central limit theorem for a natural estimator of the EDM which allows drawing confidence bands comparable to those provided by Bartlett’s formula in a classical context of sample correlation functions.     

Concurrency and Commitment: Network Scheduling and Its Consequences for Diffusion 1

Network ties are thought to be concurrent—one can “have” many friends at once, for instance—but their concrete enactment is largely serial and episodic, guided by priorities that steer a person from one encounter to the next. Further, dyadic encounters require that two people be simultaneously available to interact, creating the need for coordinated scheduling. Here I study the consequences of scheduling for network diffusion, using a computer simulation that interposes a scheduling process between a pre-existing network and instances of contagion. The pace and extent of diffusion are shown to depend upon the interaction of network topology, contagion rule (on first-contact versus at some threshold), and whether actors try to remedy past scheduling imperfections. Scheduling turns central actors into diffusion bottlenecks, but can also trigger early adoption by giving actors false readings on the status of their network alters. The implications of scheduling extend beyond diffusion, to other outcomes such as decision-making, as well as to network evolution.     

DERIVATION OF TWO WELL‐BEHAVED THEORETICAL CONTAGION INDICES AND THEIR SAMPLING PROPERTIES AND APPLICATION FOR ASSESSING FOREST LANDSCAPE DIVERSITY

Abstract Studies of spatial patterns of landscapes are useful to quantify human impact, predict wildlife effects, or describe variability of landscape features. A common approach to identify and quantify landscape structure is with a landscape scale model known as a contagion index. A contagion index quantifies two distinct components of landscape diversity: composition and configuration. Some landscape ecologists promote the use of relative contagion indices. It is demonstrated that relativized contagion indices are mathematically untenable. Two new theoretical contagion indices, Γ₁ and Γ₂ , are derived using a mean value approach (i.e., statistical expected value) instead of entropy. Behavior of Γ₁ and Γ₂ was investigated with simulated random, uniform, and aggregated landscapes. They are shown to be well‐behaved and sensitive to composition and configuration. Distributional properties of and are derived. They are shown to be asymptotically unbiased, consistent, and asymptotically normally distributed. Variance formulas for and are developed using the delta method. The new index models are used to examine landscape diversity on three physiographic provinces in Alabama by analyzing the pattern and changes in forest cover types over the recent past. In comparing and , use of in analysis of variance gave a more conservative test of contagion.     

On spatial contagion and multivariate GARCH models

We propose a method for defining and measuring spatial contagion between two financial markets via conditional copulas. Some theoretical results on monotonicity and asymptotic properties of Gaussian copulas with respect to conditioning are presented. Next, we combine the spatial contagion approach with time series models. We investigate which model from a large family of multivariate GARCH is the best tool for modelling spatial contagion. In an empirical study, we show that among models designed for general fit, a two‐step model fitting procedure reduces the ability to describe the contagion effect. This is a feature of copula‐GARCH models. Copyright © 2013 John Wiley & Sons, Ltd.     

Measuring the subprime crisis contagion: Evidence of change point analysis of copula functions

In this paper, we first determine the existence of structural changes in the dependence between time series of equity index returns of two markets using the change point testing method. The method is based on Archimedean copula functions, which are able to comprehensively describe dependence characteristics of random variables. The degree of financial contagion between markets is subsequently estimated using the tail dependence coefficient of copula functions before and after the change point. We empirically test our method by investigating financial contagion during the subprime crisis between the US S&P 500 index and five Asian markets, namely China, Japan, Korea, Hong Kong and Taiwan. Our results show that a statistically significant change point exists in the dependence between the US market and all Asian stock markets except Taiwan. The upper tail dependence is larger after the time of change, implying the existence of contagion during the banking crisis between the US and the Asian economies. The degree of financial contagion is also estimated and found to be consistent with market events and media reports during that period.     

A Multilevel Event History Model of Social Diffusion: Medical Innovation Revisited

This article presents a multilevel event history model of social diffusion and applies it to Coleman, Katz, and Menzel's (1966 Coleman , J. S. , Katz , E. , &; Menzel , H. ( 1966 ). Medical Innovation: A Diffusion Study . Indianapolis , IN : Bobbs-Merrill . [Google Scholar]) data on the adoption of tetracycline by physicians. The simplest form of a multilevel model allows a random intercept. In the present application of this simple model to the Medical Innovation data, structured for an event history analysis, the physicians are nested in city and time. Random intercepts capture effects of contextual conditions that are shared by event history cases with the same city–time status. The intercepts also reflect any baseline internal contagion effects, that is, the proportion of physicians in the city–time network who have adopted the drug at time t ? 1. Here, I show that Van den Bulte and Lilien's (2001 Van den Bulte , C. &; Lilien , G. L. ( 2001 ). Medical innovation revisited: social contagion versus marketing effort . American Journal of Sociology , 106 , 1409 – 1435 .[Crossref], [Web of Science ®] , [Google Scholar]) finding of an important contextual effect of drug firms' marketing effort is misleading. I also show that the social network in which physicians are situated significantly contributes to their adoptions, controlling for baseline internal contagion effects and individual-level characteristics of physicians, which have been emphasized in investigations of these data.     

Ein lokal überlinear konvergentes Verfahren zur Bestimmung von Rückkehrpunkten implizit definierter Raumkurven

Summary A numerical technique is described for computing turning points of a space curveL implicitly defined by a nonlinear system ofn equations inn+1 variables. The basic idea is a local parametrization ofL where the parameter that gives the next iterate is determined by applying one step of the well-known method for minimizing a real function using cubic Hermite interpolation with two nodes. The method is shown to convergeQ-super-linearly and withR-order of at least two. A numerical example concerning the analysis of nonlinear resistive circuits shows the algorithm to work effectively on real life problems.

     

The geometry of a quasilinear system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables

The geometry of a system of two partial differential equations containing the first and second partial derivatives of two functions in two independent variables is studied by using the Cartan method of invariant forms and the group-theoretic method of extensions and enclosings due to G. F. Laptev (for finite groups) and A. M. Vasil’ev (for infinite groups). Systems of quasilinear equations with the first and second partial derivatives of two functions u and v in two independent variables x and y are classified.     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

金融风险在股票市场的传染效应及联动行为分析

近年来金融危机频发并表现出了易传染性,引起了众多学者的高度关注。以动态条件相关模型研究美欧股市与中、日、韩股市间的时变相关性,并结合内生多重结构突变模型划分危机传染阶段,选用溢出指数模型分析股市间的风险溢出特性;随后,定义股市间相互影响的联动模式并构建不同传染阶段的加权有向网络图分析股市间的联动行为。研究表明:美欧股市对中日韩股市有明显的传染效应,被传染的速度和持续时间均不相同;金融传染和风险溢出展现出一定的不一致性,危机期间日股的风险溢出效应强于美股;传染效应在联动网络中表现为联动模式的高聚类性和高联动性,相比欧债危机,次贷危机时期股市间展现出更强的联动行为;日股与美欧股市在两次危机中均表现出最强的联动性,其所受影响也最大。     

On prime representing polynomials

A heuristic method is presented to determine the number of primesp ≤x, represented by an irreducible polynomialf(n), without non-trivial fixed factor (f(y)<∈Z[y]; n∈Z. The method is applied to two specific polynomials and the results are compared with those of the heuristic approach of Hardy and Littlewood.     

Fourth-order finite difference scheme for a system of quasilinear elliptic equations

The system of two quasilinear elliptic equations is approximated by the method of lines, which has the truncation error O(h²) at points neighboring the boundary and O(h⁴) at the most interior points. It is proved that the global error of the method is O(h⁴) at all mesh points. The two-point boundary value problem for the system of ordinary differential equations that arises from the method of lines is solved by the O(h⁴) convergent finite difference scheme, suitable to the equations of the form u_xx = f(x, u) without the first derivative u_x. The system of algebraic equations obtained by the full discretization is solved by Gauss elimination method for three diagonal matrices combined with the method of iterations. A numerical example is presented.     

An approximation of incompressible miscible displacement in porous media by mixed finite elements and symmetric finite volume element method of characteristics

A nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium. A sequential implicit time‐stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated by a mixed finite element method and the concentration is approximated by a combination of a modified symmetric finite volume element method and the method of characteristics. Optimal order convergence in H¹ and in L² are proved for full discrete schemes. Finally, some numerical experiments are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Two‐level Newton iterative method for the 2D/3D steady Navier‐Stokes equations

A combination method of the Newton iteration and two‐level finite element algorithm is applied for solving numerically the steady Navier‐Stokes equations under the strong uniqueness condition. This algorithm is motivated by applying the m Newton iterations for solving the Navier‐Stokes problem on a coarse grid and computing the Stokes problem on a fine grid. Then, the uniform stability and convergence with respect to ν of the two‐level Newton iterative solution are analyzed for the large m and small H and h << H. Finally, some numerical tests are made to demonstrate the effectiveness of the method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Joint availability of systems modelled by finite semi–markov processes

Consider a repairable system at the time instants t and t + x, where t, x ≥0. The joint availability of the system at these time instants is defined as the probability of the system being functional in both t and t + x. A set of integral equations is derived for the joint availability of a system modelled by a finite semi–Markov process. The result is applied to the semi–Markov model of a two–unit system with sequential preventive maintenance. The method used for the numerical solution of the resulting system of integral equations is a two–point trapezoidal rule. The system of implementation is the matrix computation package MATLAB on the Apple Macintosh SE/30. The numerical results obtained by this method are shown to be in good agreement with those from simulation.     

The load transfer between two rigid spherical inclusions in an elastic medium

The paper is concerned with the load transfer problem between two rigid spherical inclusions in an elastic matrix. A reflection-type formula is developed which is accurate up to and including terms ofO(a/R)⁴, wherea is the maximum radius of the inclusions, andR is the centre-to-centre distance between the two inclusions. Asymptotic results are derived at near touching showing a weak logarithmic singularity in the load transfer. The results are verified by a direct numerical calculation using a boundary collocation method. The numerical method uses Kelvin's general solution as the basis functions for the approximate solution and is highly accurate and efficient, even at near touching.     

Difference of two sets and estimation of clarke generalized Jacobian via quasidifferential

The notion of difference for two convex compact sets inR ⁿ , proposed by Rubinovet al, is generalized toR ^m×n . A formula of the difference for the two sets, which are convex hulls of a finite number of points, is developed. In the light of this difference, the relation between Clarke generalized Jacobian and quasidifferential, in the sense of Demyanov and Rubinov, for a nonsnooth function, is established. Based on the relation, the method of estimating Clarke generalized Jacobian via quasidifferential for a certain class of functions, is presented.     

An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

For a differential equationdx/dt=f(t, x) withf _t (t, x),f _x (t, x) computable, the author presents a new one-step method of high-order accuracy. A rule of controlling the mesh size is given and the method is compared with the Runge-Kutta method in two numerical examples.Dedicated to Professor Dr. Dr. h. c. L. Collatz for his 60^th birthday     

Analysis of the Lorenz Equations for Large r

In the limit of large r, the Lorenz equations become “almost” conservative. In this limit, one can use the method of averaging (or some equivalent) to obtain a set of two autonomous differential equations for two slowly varying amplitude functions B and D. A stable fixed point of these equations represents the stable periodic solution which is observed at large r. There is an invariant line B = D on which the method breaks down and the averaged equations are no longer valid. In this paper we show how to extend the validity of the analysis by Poincaré mapping B and D across this line. This extended analysis provides (in principl ) a complete recipe for constructing approximate solutions, and shows how a strange invariant set can occur in connection with an essentially analytically constructed two-dimensional mapping.