Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

Teletraffic analysis of cellular communication systems with general mobility based on hyper-Erlang characterization

This paper proposes a new analytical model for the performance evaluation of cellular communication systems. Our model allows more general cases of the distributions of the call time and the cell residence time. Based on the characterization of the call time by a hyper-Erlang distribution, the Laplace transform of the channel occupancy time distribution is derived as a function of the Laplace transform of the cell residence time distribution. In particular, the essential quality of service measure such as the forced termination probability is exactly obtained by using a delayed renewal process. From numerical results, it is shown that the non-exponential model has an obvious difference in performance as compared to exponential models.     

具有单重休假和修复不如新的两部件温贮备系统

研究了修理工单重休假且由两个不同型部件和一个修理工组成的可修型温贮备系统. 系统考虑了在工作故障和贮备 故障都不能 “修复如新”, 部件 1 是修复非新而部件 2 修复如新的条件下, 假设部件的工作寿命、贮备寿命、故障后的修理时间和贮备故障后的修理时间均服 从不同的指数分布, 修理工休假服从一般连续型分布. 运用几何过程理论、补充变量法、 拉普拉斯变换及拉普拉斯--司梯阶变换, 得到了系统的可用度、可靠度和系统首次故障前平均时间等可靠性指标. 最后, 通过数值模拟验证了结果的有效性.     

Impulsive displacement of a quasi-linear viscoelastic material through accurate numerical inversion of the Laplace transform

An analytical model for the one-dimensional, impulsive displacement of a quasi-linear viscoelastic material has been developed. The quasi-linear model of Fung [1] has been used successfully for a wide range of soft biological tissues. Due to the integral definition of linear viscoelastic materials, solutions are conveniently performed in the Laplace transform plane. Complex kernels like the quasilinear model are challenging to invert back to the real plane. Here, the method of Gaver [2] and Stehfest [3] is used to numerically carry Laplace space solutions to the real plane. Parametric results for a basic impulsive disturbance problem are presented. Results indicate that stress wave propagation is weakly dependent on the fast time, slow time ratio and more strongly dependent on the logarithmic damping parameter. Limitations of the numerical inversion method in the face of discontinuities are discussed as well using asymptotic methods. As an alternative to the numerical/polynomial-based Gaver-Stehfest method, a semianalytical regularization function useful near large gradient regions method is developed. A composite method that utilizes both the fully numerical and semianalytical convolution-based method is also described. The composite model provides improved results in terms of reducing computational undershoot and overshoot (wiggles) which limit both the fully numerical and the semianalytical models alone.     

Modeling process asymmetries with Laplace moving average

Many records in environmental science exhibit asymmetries: for example in shallow water and with variable bathymetry, the sea wave time series shows front–back asymmetries and different shapes for crests and troughs. In such situation, numerical models are available but their computational cost and complexity are high. A stochastic process aimed at modeling such asymmetries has recently been proposed, the Laplace moving average process, which consists in applying a linear filter on a non-Gaussian noise built using the generalized Laplace distribution. The objective is to propose a new non-parametric estimator for the kernel involved in the definition of this process. Results based on a comprehensive numerical study will be shown in order to evaluate the performances of the proposed method.     

近组分模型中主变量的选取问题研究

近组分模型有两种主变量选取方法,分别是选取自然变量作为主变量的方法和选取总属性变量作为主变量的方法。比较了这两种方法在数学模型和数值求解流程方面的差异,分析了在计算精度和计算效率方面产生的异同。二氧化碳驱概念模型的模拟结果表明,油气两相都存在时,虽然两种方法在一维驱替和二维驱替中的计算精度是一样的,但是选取自然变量作为主变量的方法不仅允许的最大时间步长更大,而且在相同时间步下运行速度更快。在相同时间步下,选取自然变量作为主变量的方法在二维驱替中提速效果有所降低。     

Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.     

两部件串联可修模型分析

本文研究典型的两部件串联可修模型.在一个部件的失效分布为Erlang分布,另一个 失效分布和两个维修分布均为连续型的假定下,利用补充变量方法求得了系统的全部可靠性 指标.这种模型能用来分析某些自动化系统的可靠性.     

Analysis of non-Fickian diffusion problems in a composite medium

The one-dimensional non-Fickian diffusion problems in a two-layered composite medium for finite and semi-infinite geometry are analyzed by using a hybrid application of the Laplace transform technique and control-volume method in conjunction with the hyperbolic shape functions, where the effect of the potential field is taken into account. The Laplace transform method used to remove the time-dependent terms in the governing differential equation and boundary conditions, and then the transformed equations are discretized by the control volume scheme. To evidence the accuracy of the present numerical method, a comparison between the present numerical results and analytical solution is made for the constant potential gradient. Results show that the present numerical results are accurate for various values of the potential gradient, relaxation time ratio, and diffusion coefficient ratio. It can be found that these values play an important role in the present problem. An interesting finding is that when the mass wave encounters an interface of the dissimilar materials, a portion of the wave is reflected and the rest is transmitted. The speed of propagation can change owing to the penetration of the mass wave into the region of the different material. The wave nature is significant only for short times and quickly dissipates with time.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.     

On dual-phase-lag magneto-thermo-viscoelasticity theory with memory-dependent derivative

A new mathematical model of generalized magneto-thermo-viscoelasticity theories with memory-dependent derivatives (MDD) of dual-phase-lag heat conduction law is developed. The equations for one-dimensional problems including heat sources are cast into matrix form using the state space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. It is also applied to a perfect conducting semi-space problem with a traction-free surface and plane distribution of heat sources located inside the medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, stress and heat flux distributions as well as the induced magnetic and electric fields are given and illustrated graphically. A comparison is made with the results obtained in the coupled theory. The impacts of the MDD heat transfer parameter and Alfven velocity on a viscoelastic material, for example, poly (methyl methacrylate) (Perspex) are discussed.

     

A state vector approach to the one dimensional consolidation of multi-layer soils

A method for determining the behaviour of a one-dimensional multi-layer soil under consolidation is described. A state vector whose components are the excess pore water pressure and discharge velocity is defined. It is shown that problems associated with the enforcement of continuity conditions at the inter-layer boundaries are thereby substantially removed. The time dependent behaviour of the settlement and excess pore water pressure is determined by numerically inverting the Laplace transform (with respect to the time) of the state vector. Examples of the numerical performance of the method are given and numerical comparisons are made with a finite element solution to the problem.     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

An FFT-based algorithm for 2D power series expansions

An effective numerical algorithm based on inverting a specialized Laplace transform is derived for computing the two-dimensional power-series expansion coefficients of a two-variable function. Due to the special structure of the constructed 2D Laplace transform, the accuracy of the inverted function values can be assured effectively by the generalized Riemann zeta function evaluation and the multiple sets of 2D FFT computation. Therefore, the algorithm is particularly amenable to modern computers having multiprocessors and/or vector processors.     

Dynamic response of plate systems by combining finite differences, finite elements and laplace transform

A numerical method for the determination of the dynamic response of large rectangular plates or plate systems to lateral loads is proposed. The method is a combination of the finite difference method, the finite element method and the Laplace transform with respect to time. The plate system is considered as an assemblage of a small number of big rectangular superelements whose stiffness matrices are derived with the aid of the finite difference method in the Laplace transform domain. These superelements are then used to formulate and solve the problem by the finite element method in the transformed domain. The dynamic response is finally obtained by a numerical inversion of the transformed solution. External viscous or internal viscoelastic damping as well as the elastic foundation interaction effect can easily be taken into account. The method is illustrated and its merits demonstrated by means of numerical examples.     

Characterisation of nonlinear and linear time-varying systems by Laplace transformation

In this article we elaborate on various characterisations of nonlinear systems. Specifically, we focus on the frequency-domain characterisation of the class of single-input single-output (SISO) linear time-varying (LTV) systems. We use the model introduced by Wiener using Volterra functional and later elaborated by others. The application of the multidimensional Laplace transform (MDLT) method as a fundamental tool in analysing variable systems and understanding system dynamic behaviours is emphasised. In particular, applications of double or two-dimensional Laplace transform (2DLT) for the analysis and synthesis of LTV circuits and systems are detailed. Examples worked out that illustrate the method and demonstrate its validity in the frequency-domain.     

Limited Information Estimation in Binary Factor Analysis: A Review and Extension

Based on the Bayes modal estimate of factor scores in binary latent variable models, this paper proposes two new limited information estimators for the factor analysis model with a logistic link function for binary data based on Bernoulli distributions up to the second and the third order with maximum likelihood estimation and Laplace approximations to required integrals. These estimators and two existing limited information weighted least squares estimators are studied empirically. The limited information estimators compare favorably to full information estimators based on marginal maximum likelihood, MCMC, and multinomial distribution with a Laplace approximation methodology. Among the various estimators, Maydeu-Olivares and Joe's (2005) weighted least squares limited information estimators implemented with Laplace approximations for probabilities are shown in a simulation to have the best root mean square errors.     

Performance analysis of wireless mobile networks with queueing priority and guard channels

The blocking probability of originating calls and forced termination probability of handoff calls are important criteria in performance evaluation of integrated wireless mobile communication networks. In this paper, we model call traffic in a single cell in an integrated voice/data wireless mobile network by a finite buffer queueing system with queueing priority and guard channels. We categorize calls into three types of service classes: originating voice calls, handoff voice calls and data calls. The arrival streams of calls are mutually independent Poisson processes, and channel holding times are exponentially distributed with different means. We describe the behavior of the system by a three‐dimensional continuous‐time Markov process and present the explicit expression for steady‐state distribution of queue lengths using a recursive approach. Furthermore, we calculate the blocking, forced termination probabilities and derive the Laplace–Stieltjes transform of the stationary distribution of actual waiting times and their arbitrary kth moment. Finally, we give some numerical results and discuss the optimization problem for the number of guard channels.     

闪电快电场信号时频特性的小波分析

应用小波变换方法将具有短时非平稳特点的闪电快电场时域波形信号转换到二维的时频平面,从时间和频率上同时描述闪电快电场信号的电场强度或相对能量分布,揭示其所包含的不同频率分量及其随时间变化的特性.分析结果表明,基于小波变换的闪电快电场信号时频特性分析可以挖掘闪电电场更深层次的变化规律和特征;也为闪电放电类型的识别研究提供了新的思路.     

State estimation for systems with sojourn-time-dependent Markovmodel switching

A switching process in which the switching probabilities depend on a random sojourn time is a class of semi-Markov processes and is encountered in target tracking, systems subject to failures, And also in the socioeconomic environment. In such a system, knowledge of the sojourn time is needed for the computation of the conditional transition probabilities. It is shown how one can infer the transition probabilities through the evaluation of the conditional distribution of the sojourn time. Subsequently, a recursive state estimation for such systems is obtained using the conditional sojourn time distribution for dynamic systems with imperfect observations and changing structures (models)