A Novel Method for Linear Systems of Fractional Ordinary Differential Equations with Applications to Time-Fractional PDEs

This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations(FODEs)which have been widely used in modeling various phenomena in engineering and science.An approximate solution of the system is sought in the formof the finite series over the Müntz polynomials.By using the collocation procedure in the time interval,one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure.This technique also serves as the basis for solving the time-fractional partial differential equations(PDEs).The modified radial basis functions are used for spatial approximation of the solution.The collocation in the solution domain transforms the equation into a system of fractional ordinary differential equations similar to the one mentioned above.Several examples have verified the performance of the proposed novel technique with high accuracy and efficiency.

On discrete time hedging errors in a fractional Black-Scholes model

In this paper we investigate asymptotic behavior of error of a discrete time hedging strategy in a fractional Black-Scholes model in the sense of Wick-ItS-Skorohod integration. The rate of convergence of the hedging error due to discrete-time trading when the true strategy is known for the trader, is investigated. The result provides new statistical tools to study and detect the effect of the long-memory and the Hurst parameter for the error of discrete time hedging.

变分分数阶PDE的自适应图像保边缘去噪方法

为了优化图像去噪过程中的边缘保真问题,本文以ROF模型及其改进模型为基础,以分数阶变分偏微分方程方法为工具,通过构造自适应的边缘检测函数,提出了一种能更好地保护图像的边缘特征,有利于纹理细节保持和“阶梯效应”抑制的基于自适应分数阶变分PDE的图像去噪模型,并以标准图像测试了新模型的去噪效果。实验结果表明:与同类方法相比,新方法可以更加有效地提高图像的信噪比;从去噪前后的图像差可以看出,新方法在去噪的同时更好地保护了图像的边缘特征及纹理细节信息。文中提出的方法可以用于遥感成像、医学图像处理及地震图像处理等领域。

Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations

Flocking refers to collective behavior of a large number of interacting entities,where the interactions between discrete individuals produce collective motion on the large scale.We employ an agent-based model to describe the microscopic dynamics of each individual in a flock,and use a fractional partial differential equation(fPDE)to model the evolution of macroscopic quantities of interest.The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model.Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics,we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations.We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one-and two-dimensional nonlocal flocking dynamics.In particular,a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual,while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities.The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally.They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization.We show in one-and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method.The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models,and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.

Pricing Formulae of Asian Options under the Fractional Brownian Motion

In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equation(PDE).The call-put parity for the geometric average Asian options is given.The results are generalization of option pricing under standard Brownian motion.

Pricing Perpetual American Put Option in theMixed Fractional Brownian Motion

Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing formula of perpetual American put option bythis partial differential equation theory.

A unique solution to a semilinear Black-Scholes partial differential equation for valuing multi-assets of American options

In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation （PDE） was invesigated in a fixed domain for valuing two assets of American （call-max/put-min） options. From the viscosity solution of a PDE, a unique viscosity solution was obtained for the semilinear Black-Scholes PDE.

Boundary Control of Coupled Non-Constant Parameter Systems of Time Fractional PDEs with Different-Type Boundary Conditions

This paper addresses a boundary state feedback control problem for a coupled system of time fractional partial differential equations(PDEs)with non-constant(space-dependent)coefficients and different-type boundary conditions(BCs).The BCs could be heterogeneous-type or mixed-type.Specifically,this coupled system has different BCs at the uncontrolled side for heterogeneous-type and the same BCs at the uncontrolled side for mixed-type.The main contribution is to extend PDE backstepping to the boundary control problem of time fractional PDEs with space-dependent parameters and different-type BCs.With the backstepping transformation and the fractional Lyapunov method,the Mittag-Leffler stability of the closed-loop system is obtained.A numerical scheme is proposed to simulate the fractional case when kernel equations have not an explicit solution.

A Convex Approximation for a PDE Constrained Fractional Optimization Problem with an Application to Photonic Crystal Design

Based on a subspace method and a linear approximation method,a convex algorithm is designed to solve a kind of non-convex PDE constrained fractional optimization problem in this paper.This PDE constrained problem is an infinitedimensional Hermitian eigenvalue optimization problem with non-convex and low regularity.Usually,such a continuous optimization problem can be transformed into a large-scale discrete optimization problem by using the finite element methods.We use a subspace technique to reduce the scale of discrete problem,which is really effective to deal with the large-scale problem.To overcome the difficulties caused by the low regularity and non-convexity,we creatively introduce several new artificial variables to transform the non-convex problem into a convex linear semidefinite programming.By introducing linear approximation vectors,this linear semidefinite programming can be approximated by a very simple linear relaxation problem.Moreover,we theoretically prove this approximation.Our proposed algorithm is used to optimize the photonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals as an application.The results of numerical examples show the effectiveness of our proposed algorithm,while they also provide several optimized photonic crystal structures with a desired wide-band-gap.In addition,our proposed algorithm provides a technical way for solving a kind of PDE constrained fractional optimization problems with a generalized eigenvalue constraint.

A FAST AND HIGH ACCURACY NUMERICAL SIMULATION FOR A FRACTIONAL BLACK-SCHOLES MODEL ON TWO ASSETS

In this paper,a two dimensional(2D)fractional Black-Scholes(FBS)model on two assets following independent geometric Lévy processes is solved numerically.A high order convergent implicit difference scheme is constructed and detailed numerical analysis is established.The fractional derivative is a quasidifferential operator,whose nonlocal nature yields a dense lower Hessenberg block coefficient matrix.In order to speed up calculation and save storage space,a fast bi-conjugate gradient stabilized(FBi-CGSTAB)method is proposed to solve the resultant linear system.Finally,one example with a known exact solution is provided to assess the effectiveness and efficiency of the presented fast numerical technique.The pricing of a European Call-on-Min option is showed in the other example,in which the influence of fractional derivative order and volatility on the 2D FBS model is revealed by comparing with the classical 2D B-S model.

基于有限差分方法的雪球式期权产品定价研究

本文针对雪球式期权的内在结构,将其分解为多个更易进行定价的不同期权,采用PDE和Black-Scholes模型对各期权结构进行求解,通过价格累加以实现雪球式期权的定价。为验证所提思路的可行性,选择具有不同波动率(20.81%~44.83%)的股票,构建不同合约期限和敲出水平的雪球期权合约,将所计算得到的票息与市场报价进行对比分析。结果表明,本文所述方法基本能够反映各标的期权价格,所计算得到的票息与市场报价较为接近,因此,可认为所述方法是可行有效的。

分数次布朗运动的欧式障碍期权定价

假定标的股票服从分数次布朗运动,应用偏微分方程的方法求出下降敲出欧式看涨障碍期权价格显示解,以及看涨-看跌的平价关系式.最后,通过有限差分法比较了显示解的准确性,分析了Hurst参数对期权价格和风险特征参数的影响.

利用变分迭代法求Riesz分数阶偏微分方程近似解

变分迭代法是一种有效的求解分数阶偏微分方程的迭代方式。将其应用到求解Riesz分数阶偏微分方程中,给出Riesz分数阶偏微分方程相应的修正泛函方程,对修正泛函方程进行求解;确定拉格朗日乘子,给出初值,通过迭代即可求出方程的解。与其他方法相比,变分迭代法不需要进行变换和数值逼近,计算更加简洁。

Vasicek利率下混合分数布朗运动的欧式期权定价

假设无风险利率遵循Vasicek模型,运用混合分数布朗运动的It公式,将欧式期权的定价转化成一个偏微分方程的求解问题.最后,通过求解偏微分方程获得了欧式期权的定价公式.

分数次布朗运动下脆弱欧式期权定价的新解法

在股票价格、公司价值均服从分数次布朗运动且相关的条件下,利用△对冲方法导出脆弱欧式期权的定价模型,并利用偏微分方程方法得到其显式定价公式.

一类积分不等式及其变分计算

利用Hardy-Littlewood-Sobolev不等式和Wolff型积分不等式得到了Wolff型位势的Lp估计,并利用变分方法得到了较加权的HLS型更一般的不等式最佳函数满足的Euler-Lagrange方程.

一种基于分数阶偏微分方程的图像放大算法

针对传统图像放大算法的不足之处,将物理意义鲜明的分数阶偏微分理论引入到图像放大算法中,提出一种新的基于分数阶偏微分方程的图像放大算法,使得放大图像的轮廓更加清晰,同时能够有效保留放大图像的细节边缘特征。仿真实验结果表明,该方法对比传统图像放大算法在放大图像的同时也增强图像的清晰度和对比度。

A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment

Cancer belongs to the class of discascs which is symbolized by out of control cells growth.These cells affect DNAs and damage them.There exist many treatments avail-able in medical science as radiation therapy,targeted therapy,surgery,palliative care and chemotherapy.Cherotherapy is one of the most popular treatments which depends on the type,location and grade of cancer.In this paper,we are working on modeling and prediction of the effect of chemotherapy on cancer cells using a fractional differen-tial equation by using the differential operator in Caputos sense.The presented model depicts the interaction between tumor,norrnal and immune cells in a tumor by using a system of four coupled fractional partial differential equations(PDEs).For this system,initial conditions of tumor cells and dimensions are taken in such a way that tumor is spread out enough in size and can be detected easily with the clinical machines.An operational matrix method with Genocchi polynomials is applied to study this system of fractional PDFs(FPDEs).An operational matrix for fract.ional differentiation is derived.Applying the collocation method and using this matrix,the nonlinear system is reduced to a system of algebraic equations,which can be solved using Newton iteration method.The salient features of this paper are the pictorial presentations of the numerical solution of the concerned equation for different particular cases to show the effect of fractional exponent on diffusive nature of immune cells,tumor cells,normal cells and chemother-apeutic drug and depict the interaction among immune cells,normal cells and tumor cells in a tumor site.

Exact Solitary Wave Solutions of Nonlinear Evolution Equations with a Positive Fractional Power Term

The bounded and smooth solitary wave solutions of 10 nonlinear evolution equations with a positive fractional power term of dependent variable are successfully obtained by homogeneous balance principle and with the aid of sub-ODEs that admits a solution of sech-power or tanh-power type.In the special cases that the fractional power equals to 1 and 2,the solitary wave solutions of more than 10 important model equations arisen from mathematical physics are easily rediscovered.

一类时间变换的强马氏过程

该文考虑了一类时间变换的强马氏过程,时间变换是截断从属过程的逆过程,这是对文章(Chen Zhenqing.Time fractional equations and probabilistic representation.Chaos Solitons and Fractals,2017,102:168-174)中结论的推广.该文建立了一种从一般Bernstein函数到广义时间分数阶偏微分方程的对应关系.