Quadratic reflected BSDEs and related obstacle problems for PDEs

Abstract

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem.     

Ordering Comparison of Logarithmic Series Random Variables with their Mixtures

The problem of comparing some known distributions in various types of stochastic orderings has been of interest to many authors. In particular, several authors have been recently concerned with the comparison of Poisson, binomial, and negative binomial distributions with their respective mixtures. Incidentally, these distributions are among the four well-known distributions of the family of generalized power series distributions (GPSD's). The remaining distribution is the logarithmic series distribution. In this paper, we shall be concerned with comparing this remaining distribution of the class GPSD with its mixture in terms of various types of stochastic orderings such as the simple stochastic, likelihood ratio, uniformly more variable, convex, hazard rate and expectation orderings. Derivation of the results in this case prove to be computationally trickier than the other three. The special case when the means of the two distributions are the same is also discussed. Finally, an illustrative explicit example is provided.     

Barnard Convex Sets

In calculating significance levels for statistical non inferiority tests, the critical regions that satisfy the Barnard convexity condition have a central role. According to a theorem proved by Röhmel and Mansmann (1999 Röhmel , J. , Mansmann , U. ( 1999 ). Unconditional nonasymptotic one sided tests for independent binomial proportions when the interest lies in showing noninferiority and or superiority . Biometr. J. 2 : 149 – 170 .[Crossref], [Web of Science ®] , [Google Scholar]), when the critical regions satisfy this condition, the significance level for non inferiority tests can be calculated much more efficiently. In this study, the sets that fulfil the Barnard convexity condition are called Barnard convex sets, and because of their relevance, we studied their properties independently of the context from which the sets originated. Among several results, we found that Barnard convex sets are a convex geometry and that each Barnard convex set has a unique basis. Also, we provide an algorithm for calculating the Barnard convex hull of any set. Finally, we present some applications of the concept of the Barnard convex hull of a set for non inferiority tests.     

凸度量空间中广义压缩的Ishikawa型迭代格式的收敛性

本文研究了凸度量空间中广义压缩的Ｉｓｈｉｋａｗａ型迭代格式的收剑性，推广了文［３］的主要结果。     

Simultaneous equivariant estimation of the parameters of linear models

Consider a family of distributions which is invariant under a group of transformations. In this paper, we define an optimality criterion with respect to an arbitrary convex loss function and we prove a characterization theorem for an equivariant estimator to be optimal. Then we consider a linear model Y=Xβ+ε, in which ε has a multivariate distribution with mean vector zero and has a density belonging to a scale family with scale parameter σ. Also we assume that the underlying family of distributions is invariant with respect to a certain group of transformations. First, we find the class of all equivariant estimators of regression parameters and the powers of σ. By using the characterization theorem we discuss the simultaneous equivariant estimation of the parameters of the linear model.     

A nonlinear optimization methodology for VLSI fixed-outline floorplanning

Floorplanning is a critical step in the physical design of VLSI circuits. The floorplanning optimization problem can be formulated as a global optimization problem minimizing wire length, with the area of each rectangular module fixed while the module’s height and width are allowed to vary subject to aspect ratio constraints. While classical floorplanning seeks to simultaneously minimize the wire length and the area of the floorplan without being constrained by a fixed outline for the floorplan, state-of-the-art technologies such as System-On-Chip require the solution of fixed-outline floorplanning. Fixing the outline of the floorplan makes the problem significantly more difficult. In this paper, we propose a two-stage nonlinear-optimization-based methodology specifically designed to perform fixed-outline floorplanning by minimizing wire length while simultaneously enforcing aspect ratio constraints on soft modules and handling a zero deadspace situation. In the first stage, a convex optimization globally minimizes an approximate measure of wire length. Using the solution of the first stage as a starting point, the second stage minimizes the wire length by sizing the modules subject to the prescribed aspect ratios, and ensuring no overlap. Computational results on standard benchmarks demonstrate that the model is competitive with other floorplanning approaches in the literature. Research supported by a Postgraduate Award, Discovery Grants 312125 and 15296, and RTI Grant 314668 from the Natural Sciences and Engineering Research Council of Canada, by an Ontario Graduate Scholarship, and by a MITACS grant.     

A counterexample to a conjecture on random shadows

We construct a counterexample to a conjecture in stereology. It is well known that two convex solid bodies which cast congruent orthographic shadows on any common plane must necessarily be congruent to each other. This characterization fails if the orientations of the shadows are allowed to differ. Such counterexamples as have been constructed for the latter case fail to have the same distribution of random shadows projected onto planes with random orientation. Therefore P. McCullagh has conjectured that two bodies whose random shadows have the same distribution are necessarily congruent. We construct a counterexample to this conjecture. Some implications are drawn for the development of a general theory of stereological reconstruction based upon random projected shadows or images.     

Selection of the Transformation Variable in the Laplace Transform Method of Estimation

The work of this paper is based on the innovative approach of Feigin et al. (1983), who estimate parameters of lifetime distributions by equating empirical and theoretical Laplace transforms. We show that the optimal choice of the transform variable depends critically upon the number of sampling times, the way they are spaced, and how the empirical transform is formed. Two new approaches for choosing the transform variable, viz. using cross-validation or constrained optimisation, are introduced and shown to have potential for wide-ranging use.     

关于具有常数带宽的凸闭曲线的一个附注

本文举例说明具有常数带宽的凸闭曲线不一定是圆。