Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

Surfaces of Constant Mean Curvature Bounded by Two Planar Curves

In this paper we study constant mean curvature compact surfaces with two Jordan curves in parallel planes as boundary and we investigate the point at which the surface inherits the symmetries of its boundary.     

Parametric estimation of discretely sampled Gamma-OU processes

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.     

图像稳定技术中的像运动描述及其处理分析

随着图像系统的广泛应用,对图像的稳定要求也越来越高。图像的运动和不稳定是常见的,往往是由于载体的运动引起的,为了达到图像稳定的目的,就需要对其运动进行描述,然后采用相应的算法进行补偿。给出了运动对图像清晰度的影响及像偏移的估算,并对数字电子图像稳定系统进行了简单阐述。     

具有扩散的Lotka-Volterra系统的持续性

利用最大值原理结合上、下解的方法，讨论了一类具有扩散的竞争－捕食的Lotka-Volterra系统静态解的存在性与持续生存。     

An admissible minimax estimator of a bounded scale-parameter in a subclass of the exponential family under scale-invariant squared-error loss

A subclass of the scale-parameter exponential family is considered and for the rth power of the scale parameter, which is lower bounded, an admissible minimax estimator under scale-invariant squared-error loss is presented. Also, an admissible minimax estimator of a lower-bounded parameter in the family of transformed chi-square distributions is given. These estimators are the pointwise limits of a sequence of Bayes estimators. Some examples are given.     

Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps

We consider Markov processes built from pasting together pieces of strong Markov processes which are killed at a position dependent rate and connected via a transition kernel. We give necessary and sufficient conditions for local absolute continuity of probability laws for such processes on a suitable path space and derive an explicit formula for the corresponding likelihood ratio process. The main tool is the consideration of the process between successive jumps – what we call ‘elementary experiments’ – and criteria for absolute continuity of laws of the process there. We apply our results to systems of branching diffusions with interactions and immigrations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Statistical Inference with Fractional Brownian Motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.     

Estimation in the continuous time mover‐stayer model with an application to bond ratings migration

The usual tool for modelling bond ratings migration is a discrete, time‐homogeneous Markov chain. Such model assumes that all bonds are homogeneous with respect to their movement behaviour among rating categories and that the movement behaviour does not change over time. However, among recognized sources of heterogeneity in ratings migration is age of a bond (time elapsed since issuance). It has been observed that young bonds have a lower propensity to change ratings, and thus to default, than more seasoned bonds. The aim of this paper is to introduce a continuous, time‐non‐homogeneous model for bond ratings migration, which also incorporates a simple form of population heterogeneity. The specific form of heterogeneity postulated by the proposed model appears to be suitable for modelling the effect of age of a bond on its propensity to change ratings. This model, called a mover–stayer model, is an extension of a Markov chain. This paper derives the maximum likelihood estimators for the parameters of a continuous time mover–stayer model based on a sample of independent continuously monitored histories of the process, and develops the likelihood ratio statistic for discriminating between the Markov chain and the mover–stayer model. The methods are illustrated using a sample of rating histories of young corporate issuers. For these issuers the default probabilities predicted by the Markov chain and mover–stayer models are different. In particular for 1–4 years old bonds the mover–stayer model estimates substantially lower default probabilities from rating C than a Markov chain. Copyright © 2004 John Wiley & Sons, Ltd.     

四阶非线性边值问题解的存在性与上下解方法

该文讨论四阶常微分方程边值问题u^(4)(t)=f(t,u,u″), t∈［0,1］,u(0)=u(1)=u″(0)=u″(1)=0解的存在性, 其中f(t,u,v):［0,1］×R×R→R为Carathéodory函数. 在不限制f关于u,v的增长阶, 不假定f关于u,v的单调性的一般情形下, 用上下解方法获得了解的存在性结果,并讨论了单调迭代求解的有效性.