A note on the first passage time of diffusions with holding and jumping boundary

In this note we are concerned with the first passage time (FPT) of diffusions with holding and jumping boundary (DHJ) in one dimensional case. We first show that the Laplace transform of FPT of DHJ can be represented explicitly by the behavior of the killed process for one holding and jumping point. The results are then extended to the Laplace transform of FPT of DHJ with two end points. Finally, we demonstrate how the results are applied to a Wiener-type neuronal model in the presence of exponential refractoriness.     

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump‐diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonparametric evaluation of the first passage time of degradation processes

This paper discusses a nonparametric method to approximate the first passage time (FPT) distribution of the degradation processes incorporating random effects if the process type is unknown. The FPT of a degradation process is unnecessarily observed since its density function can be approximated by inverting the empirical Laplace transform using the empirical saddlepoint method. The empirical Laplace transform is composed of the measured increments of the degradation processes. To evaluate the performance of the proposed method, the approximated FPT is compared with the theoretical FPT assuming a true underlying process. The nonparametric method discussed in this paper is shown to possess the comparatively small relative errors in the simulation study and performs well to capture the heterogeneity in the practical data analysis. To justify the fitting results, the goodness‐of‐fit tests including Kolmogorov‐Smirnov test and Cramér‐von Mises test are conducted, and subsequently, a bootstrap confidence interval is constructed in terms of the 90th percentile of the FPT distribution.     

Network representation of dynamical systems: Connectivity patterns,information and predictability

The present work elaborates on predictability and information aspects of dynamical systems, in connection with the connectivity features of their network representation. The basic idea underlying this work is to map the set of coarse-grained states of a dynamical system onto a set of network nodes and transitions between them onto a set of network links. Based on the vertex centrality of these nodes, we define (a) a local indicator of predictability, (b) a measure of the information that is available about the state of the system after one transition occurring within an arbitrary long time window and (c) an upper bound for the time horizon of predictability. We address the cases of the tent and the cusp maps, as representative examples of Markov and non-Markov processes. An analytical exact result for the horizon of predictability is obtained for the tent map, as well as for its higher iterates, and its connection with the corresponding network diameters is discussed. Similarly, analytical expressions are derived for the bounds of the predictability horizon in the case of the cusp map.     

A case study in applying neural networks to predicting insolvency for property and casualty insurers

This paper presents a neural network artificial intelligence model developed in cooperation with the Texas Department of Insurance as part of an early warning system for predicting insurer insolvency. A feed-forward back-propagation methodology is utilised to compute an estimate of insurer propensity towards insolvency. The results are then applied to a collection of all Texas domestic property and casualty insurance companies which became insolvent between 1987 and 1990 and the goal of predicting insolvency three years ahead of time. The results show high predictability and generalisability of results for the purpose of insolvency prediction, suggesting that neural networks may be a useful technique for this and other purposes.     

A comparative assessment of classification methods for resonance frequency prediction of Langevin piezoelectric transducers

A Langevin piezoelectric transducer is used as a physical element for transmitting and receiving sound waves. The operating frequency of a transducer determines the distance that the sound wave can travel, so it is important to measure it. Due to the fact the structure of a transducer is quite complicated, it is quite difficult to estimate the precise physical parameters for the simulation model. Therefore, it takes a long time to measure the resonance frequency in the laboratory and fix the parameters by trial and error methods. This study applies a learning method to estimate a transducer frequency instead by trial and error experiments. The learning methods applied and compared including artificial neural network, support vector machine, C4.5, neuro-fuzzy, and ega-fuzzification. Compared with the theoretical one-dimensional model (simple lump element model), the results indicate that a learning method is an efficient way to estimate the piezoelectric transducer resonance frequency. The mega-fuzzification method is the best compared with other methods in this study.     

Local stability and attractive basin of Cohen–Grossberg neural networks

The local stability analysis of a neural network is essential in evaluating the performance of this network when it acts as associative memories. This paper addresses the local stability of the Cohen–Grossberg neural networks (CGNNs). A sufficient condition for the local exponential stability of an equilibrium point is presented, and the size of the attractive basin of a locally exponentially stable equilibrium is estimated. The proposed condition and estimate are easily checkable and applicable, because they are phrased only in terms of the network parameters, the nonlinearities of the neurons, and the relevant equilibrium point. To our knowledge, this is the first time that such an estimate for CGNNs has been presented. The utility of our results is illustrated via a numerical example.     

A cobweb model with gradient adjustment mechanism: nonlinear dynamics and multistability

A cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empirical estimates of the marginal profits. Analytical conditions for local stability of the market equilibrium are provided, showing that the stability loss of the market equilibrium may give rise to chaotic dynamic as well. When memory is introduced in the production adjustment mechanism, a locally stabilizing effect is revealed as well as a globally qualitatively destabilizing role for memory. This is related to the occurrence of period doubling and Neimark–Sacker bifurcations, the latter being of supercritical nature as analytically proved. Endogenous fluctuations and multistability, with consequent loss of predictability in the long run dynamics, are observed.     

低维混沌时序非线性动力系统的预测方法及其应用研究

主要研究由低维混沌时序所确定的非线性动力系统的预测方法及其应用。在国外学者研究工作的基础上,应用一种非线性混沌模型在相空间内对时序进行重构工作,先通过改进的最小二乘方法来估计模型的参数,满足一定精度后,再采用最优化方法来估计模型的参数,并用所求得的混沌时序模型在其相空间内对时序的未来值进行预测。给出了非常有代表性的实例对文中模型和算法进行验证。结果发现采用该算法能较准确地求得模型的参数,在相空间中对混沌时序进行预测,将传统方法中的外推变成了相空间中的内插,及选取最佳的模型阶数等工作都能增加预测的准确程度,且混沌时序不可能进行长期的预测。     

The cluster deletion problem for cographs

The min-edge clique partition problem asks to find a partition of the vertices of a graph into a set of cliques with the fewest edges between cliques. This is a known NP-complete problem and has been studied extensively in the scope of fixed-parameter tractability (FPT) where it is commonly known as the Cluster Deletion problem. Many of the recently-developed FPT algorithms rely on being able to solve Cluster Deletion in polynomial time on restricted graph structures.     

Analysis of the dynamics of a delayed HIV pathogenesis model

In this paper, considering full Logistic proliferation of CD4⁺ T cells, we study an HIV pathogenesis model with antiretroviral therapy and HIV replication time. We first analyze the existence and stability of the equilibrium, and then investigate the effect of the time delay on the stability of the infected steady state. Sufficient conditions are given to ensure that the infected steady state is asymptotically stable for all delay. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold, and investigate the existence of Hopf bifurcation by using a delay τ as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the main results.     

A new system approach in estimate theory deterministic approach and uncertainty measures

This paper deals with the analysis of models to estimate the value of buildings and goods in general. A new approach is proposed on the basis of system theory both in the deterministic and stochastic framework. The models refer both to the static and to the dynamic behaviour. The paper is developed in three parts. Linear and nonlinear models are proposed in the first part for static predictions and is followed by identification of parameters. The second part proposes the logic and methodologic paths towards a measure of uncertainty. The third part deals with the definition of estimate criteria time dependent evaluations. The paper has to be regarded as the first step towards a new estimate theory.     

Improved techniques for parametric and nonparametric evaluations of the first‐passage time for degradation processes

For degradation data in reliability analysis, estimation of the first‐passage time (FPT) distribution to a threshold provides valuable information on reliability characteristics. Recently, Balakrishnan and Qin (2019; Applied Stochastic Models in Business and Industry, 35:571–590) studied a nonparametric method to approximate the FPT distribution of such degradation processes if the underlying process type is unknown. In this article, we propose some improved techniques based on saddlepoint approximation, which enhance those existing methods. Numerical examples and Monte Carlo simulation studies are used to illustrate the advantages of the proposed techniques. Limitations of the improved techniques are discussed and some possible solutions to such are proposed. Some concluding remarks and practical recommendations are provided based on the results.     

Convergence of a finite volume scheme for coagulation-fragmentation equations

This paper is devoted to the analysis of a numerical scheme for the coagulation and fragmentation equation. A time explicit finite volume scheme is developed, based on a conservative formulation of the equation. It is shown to converge under a stability condition on the time step, while a first order rate of convergence is established and an explicit error estimate is given. Finally, several numerical simulations are performed to investigate the gelation phenomenon and the long time behavior of the solution.

     

Some properties of a simple moving average when applied to forecasting a time series

Simple (equally weighted) moving averages are frequently used to estimate the current level of a time series, with this value being projected as a forecast for future observations. A key measure of the effectiveness of the method is the sampling error of the estimator, which this paper defines in terms of characteristics of the data. This enables the optimal length of the average for any steady state model to be established and the lead time forecast error derived. A comparison of the performance of a simple moving average (SMA) with an exponentially weighted moving average (EWMA) is made. It is shown that, for a steady state model, the variance of the forecast error is typically less than 3% higher than the appropriate EWMA. This relatively small difference may explain the inconclusive results from the empirical studies about the relative predictive performance of the two methods.     

On the Cauchy problem for a semilinear fractional elliptic equation

We study, for the first time in the literature on the subject, the Cauchy problem for a semilinear fractional elliptic equation. Under an a priori assumption on the solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

Models and Complexity of Multibin Packing Problems

This paper extends the previous work on multibin packing problems and gives a deeper insight into these models and their complexity, so as to provide a strong framework for future application-oriented studies. In a multibin problem, an object requires several bins to be packed. New models are represented, including the maximum cardinality multibin packing. Their complexity is studied and several pseudo-polynomial time algorithms are described, together with a fully polynomial time approximation scheme (FPTAS) for a fixed number of bins.     

Some predictability in one-dimensional chaotic dynamical systems arising in population models

Mathematical framework is given to “resolved chaos” studied numerically by Vandermeer in population biology, which means some kind of predictability in the chaotic dynamical systems. A general theory about one-dimensional unimodal maps is constructed. A quantity called “sojourning time,” which is the duration of staying in an interval by iteration of a map, is considered. Predictability is formulated as the size of error by fluctuation from the deterministic system. Topological entropy is used as the degree of chaos and a relation between topological entropy and sojourning time is obtained. Also, some conditions for the coexistence of chaotic behavior and predictability of sojourning time are given generally. In conclusion, many of the unimodal maps with high degree of chaos are predictable on the sojourning time.     

Inverse problem for structural acoustic interaction

We consider an inverse problem of determining a source term for a structural acoustic partial differential equation (PDE) model that is comprised of a two- or a three-dimensional interior acoustic wave equation coupled to an elastic plate equation. The coupling takes place across a boundary interface. For this PDE system, we obtain uniqueness and stability estimates for the source term from a single measurement of boundary values of the “structure” (acceleration of the elastic plate). The proof of uniqueness is based on a Carleman estimate (first version) of the wave problem within the chamber. The proof of stability relies on three main points: (i) a more refined Carleman estimate (second version) and its resulting implication, a continuous observability-type estimate; (ii) a compactness/uniqueness argument; (iii) an operator theoretic approach for obtaining the needed regularity in terms of the initial conditions.