Two shock and wear systems under repair standing a finite number of shocks

A shock and wear system standing a finite number of shocks and subject to two types of repairs is considered. The failure of the system can be due to wear or to a fatal shock. Associated to these failures there are two repair types: normal and severe. Repairs are as good as new. The shocks arrive following a Markovian arrival process, and the lifetime of the system follows a continuous phase-type distribution. The repair times follow different continuous phase-type distributions, depending on the type of failure. Under these assumptions, two systems are studied, depending on the finite number of shocks that the system can stand before a fatal failure that can be random or fixed. In the first case, the number of shocks is governed by a discrete phase-type distribution. After a finite (random or fixed) number of non-fatal shocks the system is repaired (severe repair). The repair due to wear is a normal repair. For these systems, general Markov models are constructed and the following elements are studied: the stationary probability vector; the transient rate of occurrence of failures; the renewal process associated to the repairs, including the distribution of the period between replacements and the number of non-fatal shocks in this period. Special cases of the model with random number of shocks are presented. An application illustrating the numerical calculations is given. The systems are studied in such a way that several particular cases can be deduced from the general ones straightaway. We apply the matrix-analytic methods for studying these models showing their versatility.     

Reliability analysis for devices subject to competing failure processes based on chance theory

This paper studies the reliability for devices subject to independent competing failure processes of degradation and shocks in an uncertain random environment. The continuous degradation is governed by an uncertain process, and external shocks arrive according to an uncertain random renewal reward process, in which the inter-arrival times of shocks and the shock sizes are assumed to be random variables and uncertain variables, respectively. The device reliability is defined as the chance measure that the uncertain degradation signals do not exceed a soft failure threshold L, and the uncertain random shocks do not cause the device failure. The device reliability is obtained by employing chance theory under four different shock patterns. Finally, a case study on a gas insulated transmission line is carried out to show the implementation of the proposed model.     

A diffusion model for exchange rates I: Theoretical introduction

The rates of exchange between r continuously traded currencies are modelled by a diffusion process. The motivation for this kind of model is in terms of long term trends in the exchange rates combined with random fluctuations in these rates which occur as a result of continous evaluation of information by the dealers in the currency market together with a continuous revision of expectations about the short-term course of the exchange rates of the currencies they are dealing in. The formulation as a diffusion process permits considerable theoretical development. On a qualitative level, stability of exchange rates is investigated using Hashiminsky's criterion and the components of an unstable regime quite well identified. Government intervention in exchange markets may be studied at a simple level by using local times. The properties of local times suggest a certain amount of insight into how optimal intervention performs. An important feature of the diffusion formulation is the derivation of explicit formulae for the numerical calculation of quantities of interest in terms of the model. This is illustrated by partial differential equations for local times and an equation obtained from the Feynman-Kac formula which permits interpretation in terms of forward exchange rates. Numerical techniques permit explicit calculation from such equations.     

基于GARCH方法研究非效率投资中金融冲击的影响机理

代理理论认为,在动荡和不确定环境下,管理者会做出偏离企业价值最大化的非效率投资行为。为探究金融冲击这个带有不确定色彩的因素,是否会恶化企业非效率投资行为,论文先是使用一个数学模型来说明金融冲击与非效率投资的可能关系,而后以GARCH方法的条件异方差来度量金融冲击的潜变量股市冲击、货币冲击和汇率冲击。最后,实证检验了这三个金融冲击潜变量和非效率投资的关系。研究发现:货币冲击和汇率冲击会显著提升非效率投资,而股市冲击能够显著降低非效率投资。进一步研究表明,货币冲击和汇率冲击会通过显著增加过度投资来加大非效率投资行为,而股市冲击则通过显著减少投资不足来达到降低非效率投资行为。     

Transport‐equilibrium schemes for computing nonclassical shocks. Scalar conservation laws

This paper presents a new numerical strategy for computing the nonclassical weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of concave–convex and convex–concave flux functions. In such situations the so‐called nonclassical shocks, violating the classical Oleinik entropy criterion and selected by a prescribed kinetic relation, naturally arise in the resolution of the Riemann problem. Enforcing the kinetic relation from a numerical point of view is known to be a crucial but challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic relation at the discrete level. The proposed strategy is based on the use of a numerical flux function and random numbers. We prove that the resulting scheme enjoys important consistency properties. Numerous numerical evidences illustrate the validity of our approach. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

On fail‐safe systems under random shocks

In practical situations, systems often suffer shocks from external stressing environments, stressing the system at random. These random shocks may have non‐ignorable effects on the system's reliability. In this paper, we provide sufficient (and necessary) conditions on components' lifetimes and their surviving probabilities from random shocks for comparing the lifetimes of two fail‐safe systems by means of the usual stochastic, hazard rate, and likelihood ratio orderings. Numerical examples are presented to highlight these theoretical results as well.     

Regenerative structure of Markov chains simulated via common random numbers

A standard strategy in simulation, for comparing two stochastic systems, is to use a common sequence of random numbers to drive both systems. Since regenerative output analysis of the steady-state of a system requires that the process be regenerative, it is of interest to derive conditions under which the method of common random numbers yields a regenerative process. It is shown here that if the stochastic systems are positive recurrent Markov chains with countable state space, then the coupled system is necessarily regenerative; in fact, we allow couplings more general than those induced by common random numbers. An example is given which shows that the regenerative property can fail to hold in general state space, even if the individual systems are regenerative.     

Reliability modeling for mutually dependent competing failure processes due to degradation and random shocks

Many systems are subject to two mutually dependent competing risks namely degradation and random shocks, and they can fail due to two competing modes of failure, soft and hard failure. Soft failure occurs when the total degradation performance, including continuous degradation and sudden degradation increments caused by random shocks, exceeds a certain critical threshold level. Hard failure occurs when the magnitude of any shock (extreme shock model) or the accumulated damage of shocks (cumulative shock model) is beyond some strength threshold level, which is affected by the temporal degradation performance. From viewpoints of Stress-Strength models and Cumulative damage/shock model, a realistic reliability model is developed in this article for mutually dependent competing failure processes due to degradation and random shocks. Finally, a numerical example of Micro-Electro-Mechanical System (MEMS) is conducted to illustrate the implementation and effectiveness of the proposed model.     

Skewness Term-Structure Tests

In this paper, we conduct skewness term-structure tests to check whether the temporal structure of risk-neutral skewness is consistent with rational expectations. Because risk-neutral skewness is substantially mean reverting, skewness shocks should decay quickly and risk-neutral skewness of more distant option should display the rationally expected smoothing behaviour. Using an equilibrium asset and option-pricing model in a production economy under jump diffusion with stochastic jump intensity, we derive this elasticity analytically. In an empirical application of the model using more than 20 years of data on S&P500 index options, we find that this elasticity turns out to be different than suggested under rational expectations – smaller on the short end (underreaction) and larger on the long end (overreaction) of the ‘skewness curve’.     

有关数学期望计算的一个典型错误

对条件期望与无条件期望混淆不分,是计算二维随机变量函数数学期望时常犯的典型错误之一,结合实例分析,指出合理利用随机变量函数数学期望的定义或全概率公式,可避免此类错误的产生.     

Treasury Management Model with Foreign Exchange Exposure

In this paper we formulate a model for foreign exchange exposure management and (international) cash management taking into consideration random fluctuations of exchange rates. A vector error correction model (VECM) is used to predict the random behaviour of the forward as well as spot rates connecting dollar and sterling. A two-stage stochastic programming (TWOSP) decision model is formulated using these random parameter values. This model computes currency hedging strategies, which provide rolling decisions of how much forward contracts should be bought and how much should be liquidated.The model decisions are investigated through ex post simulation and backtesting in which value at risk (VaR) for alternative decisions are computed. The investigation (a) shows that there is a considerable improvement to “spot only” strategy, (b) provides insight into how these decisions are made and (c) also validates the performance of this model.     

带有相依重尾冲击的Poisson噪音过程尾的一致渐近性质

本文考虑了带有某种相依重尾冲击的Poisson噪音过程尾的一致渐近性质.当冲击是二元上尾渐近独立的非负随机变量具有长尾和控制变化尾分布且噪音函数具有正的上下界时,得到了过程尾概率的一致渐近公式.进而,当冲击具有连续的一致变化尾分布时,去除了噪音函数具有正的下界的限制.对于噪音函数不一定具有正的上界的情形,当冲击具有两两负象限相依结构时,也得到了一致渐近性结果.     

Some applications of free Fisher information on frame theory

We introduce the notion of operator-valued free Fisher information with respect to a positive map of a random variable in an operator-valued noncommutative probability space and point out its close relations to the modular frames arising from conditional expectations. Then we can apply this notion on the study of frame theory, especially on the disjointness problem of modular frames arising from conditional expectations.     

Novel biogeography-based optimization algorithm with hybrid migration and global-best Gaussian mutation

The Biogeography-Based Optimization algorithm and its variants have been used widely for optimization problems. To get better performance, a novel Biogeography-Based Optimization algorithm with Hybrid migration and global-best Gaussian mutation is proposed in this paper. Firstly, a linearly dynamic random heuristic crossover strategy and an exponentially dynamic random differential mutation one are presented to form a hybrid migration operator, and the former is used to get stronger local search ability and the latter strengthen the global search ability. Secondly, a new global-best Gaussian mutation operator is put forward to balance exploration and exploitation better. Finally, a random opposition learning strategy is merged to avoid getting stuck in local optima. The experiments on the classical benchmark functions and the complexity functions from CEC-2013 and CEC-2017 test sets, and the Wilcoxon, Bonferroni-Holm and Friedman statistical tests are used to evaluate our algorithm. The results show that our algorithm obtains better performance and faster running speed compared with quite a few state-of-the-art competitive algorithms. In addition, experimental results on Minimum Spanning Tree and K-means clustering optimization show that our algorithm can cope with these two problems better than the comparison algorithms.     

Optimal replacement policy based on maximum repair time for a random shock and wear model

We study a \(\delta \) shock and wear model in which the system can fail due to the frequency of the shocks caused by external conditions, or aging and accumulated wear caused by intrinsic factors. The external shocks occur according to a Bernoulli process, i.e., the inter-arrival times between two consecutive shocks follow a geometric distribution. Once the system fails, it can be repaired immediately. If the system is not repairable in a pre-specific time D, it can be replaced by a new one to avoid the unnecessary expanses on repair. On the other hand, the system can also be replaced whenever its number of repairs exceeds N. Given that infinite operating and repair times are not commonly encountered in practical situations, both of these two random variables are supposed to obey general discrete distribution with finite support. Replacing the finite support renewal distributions with appropriate phase-type (PH) distributions and using the closure property associated with PH distribution, we formulate the maximum repair time replacement policy and obtain analytically the long-run average cost rate. Meanwhile, the optimal replacement policy is also numerically determined by implementing a two-dimensional-search process.     

A note on replacement policy for a system subject to non-homogeneous pure birth shocks

A system is subject to shocks that arrive according to a non-homogeneous pure birth process. As shocks occur, the system has two types of failures. Type-I failure (minor failure) is removed by a general repair, whereas type-II failure (catastrophic failure) is removed by an unplanned replacement. The occurrence of the failure type is based on some random mechanism which depends on the number of shocks occurred since the last replacement. Under an age replacement policy, a planned (or scheduled) replacement happens whenever an operating system reaches age T. The aim of this note is to derive the expected cost functions and characterize the structure of the optimal replacement policy for such a general setting. We show that many previous models are special cases of our general model. A numerical example is presented to show the application of the algorithm and several useful insights.     

Modeling Dependencies in Operational Risk with Hybrid Bayesian Networks

This paper addresses the problem of quantifying and modeling financial institutions’ operational risk in accordance with the Advanced Measurement Approach put forth in the Basel II Accord. We argue that standard approaches focusing on modeling stochastic dependencies are not sufficient to adequately assess operational risk. In addition to stochastic dependencies, causal topological dependencies between the risk classes are typically encountered. These dependencies arise when risk units have common information- and/or work-flows and when failure of upstream processes imply risk for downstream processes. In this paper, we present a modeling strategy that explicitly captures both topological and stochastic dependencies between risk classes. We represent the operational-risk taxonomy in the framework of a hybrid Bayesian network (BN) and provide an intuitively compelling approach for handling causal relationships and external influences. We demonstrate the use of hybrid BNs as a tool for mapping causal dependencies between frequencies and severities of risk events and for modeling common shocks. Monte-Carlo simulations illustrate that the impact of topological dependencies on triggering overall system breakdowns can be substantial.     

A random review replacement model for a system subject to compound poisson shocks

We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply     

Multivariate Hierarchical Copulas with Shocks

A transformation to obtain new multivariate hierarchical copulas, starting with an arbitrary copula, is introduced. In addition to the hierarchical structure, the presented construction principle explicitly supports singular components. These may be interpreted as the effect of local or global shocks to the underlying random variables. A large spectrum of dependence patterns can be achieved by the presented transformation, which seems promising for practical applications. Moreover, copulas arising from this construction are similarly admissible with respect to analytical tractability and sampling routines as the original copula. Finally, several well-known families of copulas may be interpreted as special cases.