求解金融工程中整数和非整数模型的新解析法
发布时间：2024-06-07 04:34
在本文中,我们提出一些新的高效的解析方法来求解金融,应用物理科学和工程中一些重要的分数阶(非整数)和非分数阶(整数)模型,包括Cantor集上出现的不可微问题.本文讨论了所提方法的具体推导步骤及其收敛性分析和误差估计.所有提出的解析方法都可应用于金融与工程领域的一些实际模型,如分数阶和非分数阶扩散方程,分数阶和非分数阶热方程,分数阶Black-Scholes期权定价方程,分数阶和非分数阶波动方程,不可微热方程,波方程和扩散方程.在第二章中,我们重点介绍了论文所需的基础只是,并简要回顾了现存文献中整数阶和非整数阶导数和积分的历史.在分数阶导数中,我们简要讨论了著名的Caputo和Riemann-Liouville分数阶导数和积分.此外,我们还讨论了分数阶微积分的一些最新进展,如Caputo-Fabrizio和Atangana-Baleanu分数阶导数,其中Caputo-Fabrizio和Atangana-Baleanu是具有非奇异核的新型分数.在第七章中,我们成功地将Caputo-Fabrizio和Atangana-Baleanu分数阶导数以及Laplace型积分变换应用于金融中分数阶B...
【文章页数】：265 页
【学位级别】：博士
【文章目录】：
摘要
Abstract
附件
Chapter 1 Introduction
Chapter 2 Brief history of fractional calculus
§2.1 Integer and Non-integer Order Derivatives
§2.1.1 Basic Definitions of Non-integer Order Derivative
§2.1.2 Some Basic Definition of Fractional Integrals
Chapter 3 Integral transform and their applications
§3.1 New Integral Transforms for Solving Ordinary and Partial DifferentialEquations
§3.2 J-transform Properties and Its Applications
§3.2.1 Applications of J-transform to Partial Differential Equations
§3.2.2 Applications of J-transform to Ordinary Differential Equations
§3.2.3 Is J-transform more efficient than the Sumudu transform and the natural transform?
§3.2.4 Is J-transform more efficient than the Laplace transform?
§3.3 Shehu Transform Properties and Its Applications
§3.3.1 Properties of Shehu Integral transform
§3.3.2 Applications of Shehu transform to Ordinary Differential Equations
§3.3.3 Applications of Shehu transform to Partial Differential Equations
§3.4 Background of Fuzzy Function and Fuzzy sets
§3.5 Fuzzy Shehu Transform and Its Applications
§3.5.1 Some Basic Properties of Fuzzy Shehu Integral Transform
§3.5.2 Application of Fuzzy Shehu Transform to Second-Order Fuzzy Initial Value Problem
§3.5.3 Application of Fuzzy Shehu Transform to Fuzzy Volterra Integral Equation of the Second Kind of the Form
Chapter 4 Fractal models on Cantor sets
§4.1 Preliminaries of Local Fractional Calculus
§4.1.1 Local Fractal Derivative
§4.1.2 Local Fractal Integral
§4.1.3 Some Integral Transform on Fractal Space
§4.1.4 Local Fractal Natural Transform and Its Properties
§4.2 Applications of Local Fractal Natural Transform
§4.2.1 Application on signal defined on a Cantor sets
§4.2.2 Application of Non-differentiable Ordinary Differential Equations
§4.2.3 Application of Non-differentiable Volterra Integral Equation of the Second Kind
§4.2.4 Application of Non-differentiable Heat Equation Defined on Cantor Sets
§4.2.5 Application of Non-differentiable Wave Equation Defined on Cantor Sets
Chapter 5 Analytical methods for fractal models
§5.1 The Homotopy Analysis Method (HAM)
§5.2 Local Fractional Homotopy Analysis Method
§5.2.1 Convergence Analysis of the Local Fractional Homotopy Analysis Method
§5.2.2 Application of the Local Fractional Homotopy Analysis Method to Non-differentiable Fractional Heat Equation
§5.3 Fractal Laplace Homotopy Analysis Method
§5.3.1 Convergence Analysis of the Local Fractional Laplace Homotopy Analysis Method
§5.3.2 Application of the LFLHAM and Its Comparison with LFHAM on Non-differentiable Linear and Nonlinear Fractional Wave E-quations
§5.4 Fractal Natural Decomposition Method
§5.4.1 Convergence Analysis of the Local Fractional Natural Decomposition Method
§5.5 Applications of the LFNDM
Chapter 6 Analytical techniques for fractional models
§6.1 Homotopy Analysis Shehu Transform Method
§6.1.1 Convergence Analysis of the Homotopy Analysis Shehu Transform Method
§6.1.2 Absolute Error Analysis of the HASTM
§6.1.3 Homotopy Perturbation Laplace Transform Technique (HPLTT)
§6.1.4 Applications of the Homotopy Analysis Shehu Transform Method to Linear and Nonlinear Fractional Diffusion Equations
§6.2 Homotopy Analysis Transform Algorithm
§6.2.1 Convergence Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
§6.2.2 Error Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
§6.2.3 Applications of the Homotopy Analysis Fuzzy Shehu Transform Algorithm to Fuzzy Fractional Partial Differential Equations
§6.3 Homotopy Perturbation Transform Method
§6.3.1 Application of the Homotopy Perturbation Method Shehu Transform Method to Fractional Models
Chapter 7 Analytical methods for Black-Scholes equation
§7.1 Homotopy Perturbation Method (HPM)
§7.2 Analytical solutions for Option Pricing Equation
§7.3 Application of NHPM on option pricing equation
§7.4 New fractional option pricing equations
§7.4.1 Modelling of Fractional Black-Scholes European option pricing equations with Atangana-Baleanu fractional derivative
§7.5 The Existence and Uniqueness Analysis
§7.6 New Q-Homotopy Analysis Transform Method
§7.6.1 Q-Homotopy Analysis Transform Method via Caputo,Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives for Option Pricing Equation in Finance
§7.7 Application of the q-homotopy analysis method to new fractional option pricing equations
7.8 Numerical Results and Discussion
Chapter 8 Conslusions
§8.1 Conslusions
Major Achievement in this Dissertation
Appendix
References
Publications
Acknowledgement
学位论文评阅及答辩情况表
本文编号：3990787
【文章页数】：265 页
【学位级别】：博士
【文章目录】：
摘要
Abstract
附件
Chapter 1 Introduction
Chapter 2 Brief history of fractional calculus
§2.1 Integer and Non-integer Order Derivatives
§2.1.1 Basic Definitions of Non-integer Order Derivative
§2.1.2 Some Basic Definition of Fractional Integrals
Chapter 3 Integral transform and their applications
§3.1 New Integral Transforms for Solving Ordinary and Partial DifferentialEquations
§3.2 J-transform Properties and Its Applications
§3.2.1 Applications of J-transform to Partial Differential Equations
§3.2.2 Applications of J-transform to Ordinary Differential Equations
§3.2.3 Is J-transform more efficient than the Sumudu transform and the natural transform?
§3.2.4 Is J-transform more efficient than the Laplace transform?
§3.3 Shehu Transform Properties and Its Applications
§3.3.1 Properties of Shehu Integral transform
§3.3.2 Applications of Shehu transform to Ordinary Differential Equations
§3.3.3 Applications of Shehu transform to Partial Differential Equations
§3.4 Background of Fuzzy Function and Fuzzy sets
§3.5 Fuzzy Shehu Transform and Its Applications
§3.5.1 Some Basic Properties of Fuzzy Shehu Integral Transform
§3.5.2 Application of Fuzzy Shehu Transform to Second-Order Fuzzy Initial Value Problem
§3.5.3 Application of Fuzzy Shehu Transform to Fuzzy Volterra Integral Equation of the Second Kind of the Form
Chapter 4 Fractal models on Cantor sets
§4.1 Preliminaries of Local Fractional Calculus
§4.1.1 Local Fractal Derivative
§4.1.2 Local Fractal Integral
§4.1.3 Some Integral Transform on Fractal Space
§4.1.4 Local Fractal Natural Transform and Its Properties
§4.2 Applications of Local Fractal Natural Transform
§4.2.1 Application on signal defined on a Cantor sets
§4.2.2 Application of Non-differentiable Ordinary Differential Equations
§4.2.3 Application of Non-differentiable Volterra Integral Equation of the Second Kind
§4.2.4 Application of Non-differentiable Heat Equation Defined on Cantor Sets
§4.2.5 Application of Non-differentiable Wave Equation Defined on Cantor Sets
Chapter 5 Analytical methods for fractal models
§5.1 The Homotopy Analysis Method (HAM)
§5.2 Local Fractional Homotopy Analysis Method
§5.2.1 Convergence Analysis of the Local Fractional Homotopy Analysis Method
§5.2.2 Application of the Local Fractional Homotopy Analysis Method to Non-differentiable Fractional Heat Equation
§5.3 Fractal Laplace Homotopy Analysis Method
§5.3.1 Convergence Analysis of the Local Fractional Laplace Homotopy Analysis Method
§5.3.2 Application of the LFLHAM and Its Comparison with LFHAM on Non-differentiable Linear and Nonlinear Fractional Wave E-quations
§5.4 Fractal Natural Decomposition Method
§5.4.1 Convergence Analysis of the Local Fractional Natural Decomposition Method
§5.5 Applications of the LFNDM
Chapter 6 Analytical techniques for fractional models
§6.1 Homotopy Analysis Shehu Transform Method
§6.1.1 Convergence Analysis of the Homotopy Analysis Shehu Transform Method
§6.1.2 Absolute Error Analysis of the HASTM
§6.1.3 Homotopy Perturbation Laplace Transform Technique (HPLTT)
§6.1.4 Applications of the Homotopy Analysis Shehu Transform Method to Linear and Nonlinear Fractional Diffusion Equations
§6.2 Homotopy Analysis Transform Algorithm
§6.2.1 Convergence Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
§6.2.2 Error Analysis of the Homotopy Analysis Fuzzy Shehu Transform Algorithm
§6.2.3 Applications of the Homotopy Analysis Fuzzy Shehu Transform Algorithm to Fuzzy Fractional Partial Differential Equations
§6.3 Homotopy Perturbation Transform Method
§6.3.1 Application of the Homotopy Perturbation Method Shehu Transform Method to Fractional Models
Chapter 7 Analytical methods for Black-Scholes equation
§7.1 Homotopy Perturbation Method (HPM)
§7.2 Analytical solutions for Option Pricing Equation
§7.3 Application of NHPM on option pricing equation
§7.4 New fractional option pricing equations
§7.4.1 Modelling of Fractional Black-Scholes European option pricing equations with Atangana-Baleanu fractional derivative
§7.5 The Existence and Uniqueness Analysis
§7.6 New Q-Homotopy Analysis Transform Method
§7.6.1 Q-Homotopy Analysis Transform Method via Caputo,Caputo-Fabrizio, and Atangana-Baleanu fractional derivatives for Option Pricing Equation in Finance
§7.7 Application of the q-homotopy analysis method to new fractional option pricing equations
7.8 Numerical Results and Discussion
Chapter 8 Conslusions
§8.1 Conslusions
Major Achievement in this Dissertation
Appendix
References
Publications
Acknowledgement
学位论文评阅及答辩情况表
本文编号：3990787
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