High-level dependence in time series models

We present several notions of high-level dependence for stochastic processes, which have appeared in the literature. We calculate such measures for discrete and continuous-time models, where we concentrate on time series with heavy-tailed marginals, where extremes are likely to occur in clusters. Such models include linear models and solutions to random recurrence equations; in particular, discrete and continuous-time moving average and (G)ARCH processes. To illustrate our results we present a small simulation study.     

Limit laws for upper and lower extremes from stationary mixing sequences

The aim of this paper is to examine the weak limiting behavior of upper and lower extremes from stationary sequences satisfying dependence conditions similar to D and D′ introduced by Leadbetter (Z. Wahrsch. Verw. Gebiete28 (1974), 289–303). By establishing the convergence in distribution of an associated sequence of point processes, the joint limiting distribution of any collection of upper and lower extremes can be determined. Sufficient and, in some cases, necessary conditions for the asymptotic independence of the upper and lower extremes are also given.     

<Emphasis Type="Italic">k</Emphasis>th-order Markov extremal models for assessing heatwave risks

Heatwaves are defined as a set of hot days and nights that cause a marked short-term increase in mortality. Obtaining accurate estimates of the probability of an event lasting many days is important. Previous studies of temporal dependence of extremes have assumed either a first-order Markov model or a particularly strong form of extremal dependence, known as asymptotic dependence. Neither of these assumptions is appropriate for the heatwaves that we observe for our data. A first-order Markov assumption does not capture whether the previous temperature values have been increasing or decreasing and asymptotic dependence does not allow for asymptotic independence, a broad class of extremal dependence exhibited by many processes including all non-trivial Gaussian processes. This paper provides a kth-order Markov model framework that can encompass both asymptotic dependence and asymptotic independence structures. It uses a conditional approach developed for multivariate extremes coupled with copula methods for time series. We provide novel methods for the selection of the order of the Markov process that are based upon only the structure of the extreme events. Under this new framework, the observed daily maximum temperatures at Orleans, in central France, are found to be well modelled by an asymptotically independent third-order extremal Markov model. We estimate extremal quantities, such as the probability of a heatwave event lasting as long as the devastating European 2003 heatwave event. Critically our method enables the first reliable assessment of the sensitivity of such estimates to the choice of the order of the Markov process.     

Bivariate extreme value distributions based on polynomial dependence functions

In this paper, we are concerned with bivariate differentiable models for joint extremes for dependent data sets. This question is often raised in hydrology and economics when the risk driven by two (or more) factors has to be quantified. Here we give a full characterization of polynomial models by means of their dependence function and dependence measure. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

Numerical invariant tori of symplectic integrators for integrable Hamiltonian systems

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.     

Optimal portfolio selection for general provisioning and terminal wealth problems

In Dhaene et al. (2005), multiperiod portfolio selection problems are discussed, using an analytical approach to find optimal constant mix investment strategies in a provisioning or a savings context. In this paper we extend some of these results, investigating some specific, real-life situations. The problems that we consider in the first section of this paper are general in the sense that they allow for liabilities that can be both positive or negative, as opposed to Dhaene et al. (2005), where all liabilities have to be of the same sign. Secondly, we generalize portfolio selection problems to the case where a minimal return requirement is imposed. We derive an intuitive formula that can be used in provisioning and terminal wealth problems as a constraint on the admissible investment portfolios, in order to guarantee a minimal annualized return. We apply our results to optimal portfolio selection.     

非光滑半定规划的一阶最优性条件

首次考虑了非光滑半定规化问题．运用与非线性规划类似的技巧,把现存的理论扩展到约束是结构稀疏矩阵的情况,给出了其一阶最优性条件。考虑了严格互补条件不成立的情形．在约束矩阵为对角阵条件下,所用的正则条件与传统非线性优化意义下的是一致的．     

Bayesian uncertainty management in temporal dependence of extremes

Both marginal and dependence features must be described when modelling the extremes of a stationary time series. There are standard approaches to marginal modelling, but long- and short-range dependence of extremes may both appear. In applications, an assumption of long-range independence often seems reasonable, but short-range dependence, i.e., the clustering of extremes, needs attention. The extremal index 0 < ?? ≤ 1 is a natural limiting measure of clustering, but for wide classes of dependent processes, including all stationary Gaussian processes, it cannot distinguish dependent processes from independent processes with ?? = 1. Eastoe and Tawn (Biometrika 99, 43–55 2012) exploit methods from multivariate extremes to treat the subasymptotic extremal dependence structure of stationary time series, covering both 0 < ?? < 1 and ?? = 1, through the introduction of a threshold-based extremal index. Inference for their dependence models uses an inefficient stepwise procedure that has various weaknesses and has no reliable assessment of uncertainty. We overcome these issues using a Bayesian semiparametric approach. Simulations and the analysis of a UK daily river flow time series show that the new approach provides improved efficiency for estimating properties of functionals of clusters.