Modeling and Computation of CO2 Allowance Derivatives Under Jump-Diffusion Processes

In this paper, we study carbon emission trading whose market is gaining popularity as a policy instrument for global climate change. The mathematical model is presented for pricing options on $CO_2$ emission allowance futures with jump diffusion processes, and a so-called fitted finite volume method is proposed to solve the pricing model for the spatial discretization, in which the Crank-Nicolson is employed for time stepping. In addition, the stability and the convergence of the fully discrete scheme are given, and some numerical results, which are compared with the closed form solution and the Monte Carlo simulation solution, are provided to demonstrate the rates of convergence and the robustness of the numerical method.     

Analysis of materials selected for multilayer diffractive optical elements

An analysis model to optimize the materials selected for multilayer diffractive elements (MLDOEs) is presented with approximate Cauchy dispersion formula of refractive index and the maximum polychromatic integral diffraction efficiency (PIDE). The analysis model presents that the maximum PIDE of MLDOEs consisting of two materials with large Abbe number difference and small partial dispersion difference can be generated. The scope of application and the relationship between diffraction efficiencies of MLDOEs with different material pairs and different design wavelength pairs are presented and simulated with the analysis model of MLDOEs.     

Generalized pricing formulas for stochastic volatility jump diffusion models applied to the exponential Vasicek model

Path integral techniques for the pricing of financial options are mostly based on models that can be recast in terms of a Fokker-Planck differential equation and that, consequently, neglect jumps and only describe drift and diffusion. We present a method to adapt formulas for both the path-integral propagators and the option prices themselves, so that jump processes are taken into account in conjunction with the usual drift and diffusion terms. In particular, we focus on stochastic volatility models, such as the exponential Vasicek model, and extend the pricing formulas and propagator of this model to incorporate jump diffusion with a given jump size distribution. This model is of importance to include non-Gaussian fluctuations beyond the Black-Scholes model, and moreover yields a lognormal distribution of the volatilities, in agreement with results from superstatistical analysis. The results obtained in the present formalism are checked with Monte Carlo simulations.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

A new approach for modelling the damped Helmholtz oscillator:applications to plasma physics and electronic circuits

In this paper,a new approach is devoted to find novel analytical and approximate solutions to the damped quadratic nonlinear Helmholtz equation(HE)in terms of the Weiersrtrass elliptic function.The exact solution for undamped HE(integrable case)and approximate/semi-analytical solution to the damped HE(non-integrable case)are given for any arbitrary initial conditions.As a special case,the necessary and sufficient condition for the integrability of the damped HE using an elementary approach is reported.In general,a new ansatz is suggested to find a semi-analytical solution to the non-integrable case in the form of Weierstrass elliptic function.In addition,the relation between the Weierstrass and Jacobian elliptic functions solutions to the integrable case will be derived in details.Also,we will make a comparison between the semi-analytical solution and the approximate numerical solutions via using Runge-Kutta fourth-order method,finite difference method,and homotopy perturbation method for the first-two approximations.Furthermore,the maximum distance errors between the approximate/semi-analytical solution and the approximate numerical solutions will be estimated.As real applications,the obtained solutions will be devoted to describe the characteristics behavior of the oscillations in RLC series circuits and in various plasma models such as electronegative complex plasma model.     

高温超导磁通跳跃过程中的磁致伸缩效应

文中基于超导磁通动力学理论,考虑电磁力与热激活对磁通运动的影响,基本模型包括由等效电阻率随超导体温度和磁场变化的磁通扩散方程,以及比热随超导体温度变化的热传导方程组成.在此基础上,用数值方法求解了这组非线性磁热耦合方程,主要研究了有磁通跳跃状发生状态时环境温度和外磁场速度对于高温超导磁致伸缩的影响.结果表明:磁通进入超...     

一种基于特征理论模拟凝聚炸药爆轰的单元中心型Lagrange方法

提出一种数值模拟凝聚炸药爆轰问题的单元中心型Lagrange方法.利用有限体积离散爆轰反应流动方程组,基于双曲型偏微分方程组的特征理论获得离散网格节点的速度与压力,获得的网格节点速度与压力用于更新网格节点位置以及计算网格单元边的数值通量.以这种方式获得的网格节点解是一种"真正多维"的理论解,是一维Godunov格式在二维Riemann问题的推广.有限体积离散得到的爆轰反应流动的半离散系统使用一种显-隐Runge-Kutta格式来离散求解：显式格式处理对流项,隐式格式处理化学反应刚性源项.算例表明,提出的单元中心型Lagrange方法能够较好地模拟凝聚炸药的爆轰反应流动.     

Master equations for subordinated processes

We study Markov jump processes constructed by subordination of diffusion processes. The procedure can be viewed as a randomization or a coarse graining of time. We construct the master equation for the cases of finite and infinite total jump rates, and give a collection of explicitly solvable examples.     

Influence of the finite lifetime of hydrogen jump phase on the quasi-elastic neutron scattering by hydrogen in metals†

A complex jump diffusion model assuming a finite lifetime for hydrogen both in the jump and in the residence phases is developed as a straightforward extension of the known Rowe et al. theory [1], and also of the Multi-Sublattice Jump Diffusion Model [2]. The residence-residence, jump-jump, residence-jump and vice versa transitions of hydrogen are explicitly included in the model. This approach is a method for calculating the double differential, d²σ/dΩdω, and hence the differential, dσ/dΩ, cross section for quasi-elastic neutron scattering from metal hydrides, MeH_x. In terms of the model we have calculated the diffusion coefficient, which reveals explicitly the contributions of the residence and jump phases.     

Nonlinear Schrödinger equation with a Dirac delta potential: finite difference method*

The nonlinear Schrödinger equation with a Dirac delta potential is considered in this paper. It is noted that the equation can be transformed into an equation with a drift-admitting jump. Then following the procedure proposed in Chen and Deng (2018 Phys. Rev. E 98 033302), a new second-order finite difference scheme is developed, which is justified by numerical examples.     

Initial-boundary value problem for the two-component Gerdjikov-Ivanov equation on the interval

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of three matrix-value spectral functions s (λ), S (λ) and S_L(λ), which arising from the initial values at t = 0, boundary values at x = 0 and boundary values at x = L, respectively. Moreover, The associated Dirichlet to Neumann map is analyzed via the global relation. The relevant formulae for boundary value problems on the finite interval can reduce to ones on the half-line as the length of the interval tends to infinity.     

电磁耦合实验平台系统线缆束的电磁拓扑分析

复杂线缆束系统的电磁耦合分析,随着电磁环境的日趋复杂化,显得越来越困难。而高功率电磁辐射技术的发展,对具有复杂电缆束网络和电子设备的系统,带来了严重的电磁威胁。这从电磁攻击与电磁防护两方面,对复杂线缆束的电磁耦合分析,都提出了更加迫切的需求。由于复杂线缆束网络所涉及的几何空间的边界条件十分复杂,所以很难用时域有限差分(FDTD)或频域有限差分(FDFD)方法求解复杂线缆束网络系统的电磁耦合问题。我们在基于传输线理论,采用拓扑学中将空间按照拓扑结构进行分解的思想,建立了线缆束网络电磁耦合的拓扑模型,得出了计算复杂线缆束网络系统终端耦合电压与电流的计算方法,并给出了仿真计算实例用以验证电磁拓扑法处理线缆网络电磁耦合效应的有效性。     

A matched interface and boundary method for solving multi-flow Navier–Stokes equations with applications to geodynamics

We have developed a second-order numerical method, based on the matched interface and boundary (MIB) approach, to solve the Navier–Stokes equations with discontinuous viscosity and density on non-staggered Cartesian grids. We have derived for the first time the interface conditions for the intermediate velocity field and the pressure potential function that are introduced in the projection method. Differentiation of the velocity components on stencils across the interface is aided by the coupled fictitious velocity values, whose representations are solved by using the coupled velocity interface conditions. These fictitious values and the non-staggered grid allow a convenient and accurate approximation of the pressure and potential jump conditions. A compact finite difference method was adopted to explicitly compute the pressure derivatives at regular nodes to avoid the pressure–velocity decoupling. Numerical experiments verified the desired accuracy of the numerical method. Applications to geophysical problems demonstrated that the sharp pressure jumps on the clast-Newtonian matrix are accurately captured for various shear conditions, moderate viscosity contrasts and a wide range of density contrasts. We showed that large transfer errors will be introduced to the jumps of the pressure and the potential function in case of a large absolute difference of the viscosity across the interface; these errors will cause simulations to become unstable.     

Pseudo-spectral methods and linear instabilities in reaction-diffusion fronts

We explore the application of a pseudo-spectral Fourier method to a set of reaction-diffusion equations and compare it with a second-order finite difference method. The prototype cubic autocatalytic reaction-diffusion model as discussed by Gray and Scott [Chem. Eng. Sci. 42, 307 (1987)] with a nonequilibrium constraint is adopted. In a spatial resolution study we find that the phase speeds of one-dimensional finite amplitude waves converge more rapidly for the spectral method than for the finite difference method. Furthermore, in two dimensions the symmetry preserving properties of the spectral method are shown to be superior to those of the finite difference method. In studies of plane/axisymmetric nonlinear waves a symmetry breaking linear instability is shown to occur and is one possible route for the formation of patterns from infinitesimal perturbations to finite amplitude waves in this set of reaction-diffusion equations. (c) 1996 American Institute of Physics.     

A numerical method for the solution of the bidomain equations in cardiac tissue

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.     

Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows

In this Letter, a three-dimensional (3D) lattice-Boltzmann model is presented following the non-free-parameter lattice-Boltzmann method of Qu et al. [K. Qu, C. Shu, Y.T. Chew, Phys. Rev. E 75 (2007) 036706]. A simple function, which satisfies the zeroth- through third-order moments of the Maxwellian distribution function, is introduced to replace the Maxwellian distribution function as the continuous equilibrium distribution function in 3D space. The function is then discretized to discrete-velocity directions via a 25-point Lagrangian interpolation polynomial. To simulate compressible flows with shock waves, an implicit-explicit finite-difference scheme based on the total variation diminishing flux limitation is adopted to solve the discrete Boltzmann-BGK equation in order to capture the shock waves in compressible flows with a finite number of grid points. The model is validated by its application to some typical inviscid compressible flows ranging from 1D to 3D, and the numerical results are found to be in excellent agreement with the analytical solutions and/or other numerical results.     

Asymptotic Analysis of Lattice Boltzmann Boundary Conditions

In this article, we use a general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions. Several link based boundary conditions for commonly used lattice Boltzmann BGK models are considered. With our method, the accuracy of the algorithms can be exactly predicted. Moreover, the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity. The analysis is applicable to general geometries and instationary flows     

Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

This work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number is high. For Asian options, additional similar problems arise when the "spatial" variable, the stock price, is close to zero.Here we focus on three methods: the exponentially fitted scheme, a modification of Wang's finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put on the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.     

原子分子光子系统的耗散相互作用和退相干

原子分子系统与量子化的电磁场或光子模式耦合的系统是非相对论量子力学理论研究和实验研究的主要对象和模型. 现实系统必然与外界环境耦合,且即便原子隔绝较好、光学腔壁品质因子足够高,原子系统也不等价于少数几个能级构成的简单模型：它仍然有不为零的几率跃迁到不可控的能级空间、与原子相互作用的自由空间真空场的量子效应也必须考虑. 本文将结合开放量子系统理论的基本要素与原子光子的基本模型,对原子分子系统在电磁场中发生的耗散以及量子退相干过程做简单综述,并重点介绍描述量子系统退相干过程的主流理论工具——主方程.