随机方程及其在信用风险中的应用
发布时间：2022-10-21 13:11
本文分五个部分来研究反射随机微分方程,随机偏微分方程以及它们在信用风险理论中的应用。 反射随机微分方程可视为一个Skorohod问题。在李普希兹条件下,其强解的存在唯一性被Lions和Sznitman所证明。随后李普希兹条件被分别拓展到了单面李普希兹和Yamada-Watanabe条件。Le Gall,Bass和Chen,Zhang,Marin和Real分别证明了在这些条件下,其强解的存在唯一性。本文第一章1.1节将延续这方面的研究。与现有文献的不同之处在于,我们考虑在Fang和Zhang的非李普希兹条件下,其强解的存在唯一性。特别地,在一维情形下,我们结合Fang和Zhang非李普希兹条件的特点以及局部时的性质证明了一个强比较定理。由于反射随机微分方程的解能被限制在某些特定的凸区域上,这个特点使这类方程在排队论、金融建模中有重要应用。其中Harrison用双边反射布朗运动建立了存储模型,随后Ata,Harrison和Shepp用双边反射O-U过程描述了布朗网络优化问题。Goldstein和Keirstead用零点单边反射随机微分方程建模瞬时利率过程,并导出了当利率过程建模为...
【文章页数】：288 页
【学位级别】：博士
【文章目录】：
Abstract
摘要
1 Reflected Stochastic Differential Equations and Applications
1.1 Strong comparison for RSDEs with non-Lipschitzian coefficients
1.1.1 Motivation
1.1.2 Pathwise uniqueness
1.1.3 Strong comparison theorem
1.2 First passage time of the reflected O-U process with two-sided barriers
1.2.1 Motivation
1.2.2 Laplace transform of the first passage time
1.2.3 An extended case
1.2.4 Applications in financial modelings
1.3 Large deviations for perturbed reflected diffusion processes
1.3.1 Motivation and method
1.3.2 LDP for perturbed diffusion processes
1.3.3 LDP for perturbed reflected diffusion processes
1.4 Hedging for a defaultable claim with recovery and dividend under local risk minimization
1.4.1 Motivation
1.4.2 The model and local risk minimization
1.4.3 Hedging for H under local risk minimization
2 Optimal Portfolio with Defaultable Risk and HJB Equations
2.1 Optimal portfolio with defaultable risk-log utility
2.1.1 Motivation and method
2.1.2 The optimization with a defaultable bond
2.1.3 Verification theorem
2.1.4 A numerical analysis example
2.2 Optimal portfolio with defaultable risk-power utility
2.2.1 The optimal portfolio with non-log HARA utility
2.2.2 The HJB equation
2.2.3 Solutions to the HJB equation
2.2.4 The verification theorem
2.2.5 Sensitivity analysis
2.3 Optimal portfolio and consumption with defaultable risk-a viscosity solution approach
2.3.1 Motivation
2.3.2 Price dynamics of the defaultable bond
2.3.3 The value function and HJB equation
2.3.4 The viscosity solution
3 Parabolic Type Stochastic Partial Differential Equations
3.1 On solutions of Cahn-Hilliard SPDE with Levy space-time white noise
3.1.1 Motivation
3.1.2 The definition of Levy space-time white noise
3.1.3 A new version of Burkholder-Davis-Gundy inequality and the definition of the solution
3.1.4 Existence and uniqueness of local solutions
3.2 On solutions of Cahn-Hilliard SPDE with fractional noise-a weak convergence approach
3.2.1 Motivation and main result
3.2.2 Fractional noise and embedding theorem
3.2.3 Weak convergence of local solutions
3.3 Support theorem for stochastic Cahn-Hilliard equation
3.3.1 Motivation and main result
3.3.2 Difference approximation to white noise
3.3.3 Localization framework
3.3.4 Auxiliary lemmas
3.3.5 The proof of(C1)
3.3.6 The proof of(C2)
3.4 The higher-order Ito and Skorokhod Anderson models
3.4.1 Lyapunov exponent estimates of fourth-order Ito Anderson model
3.4.2 Skorokhod fourth-order Anderson models with fractional noises
3.5 Stochastic nonlocal Kuramoto-Sivashinsky equation with jumps
3.5.1 Motivation
3.5.2 Preliminaries and hypothesis
3.5.3 Existence and uniqueness of the weak solution
3.5.4 Invariant measure
4 Hyperbolic Type Stochastic Partial Differential Equations
4.1 Explosive solutions of stochastic wave equations with damping
4.1.1 Motivation
4.1.2 Preliminaries
4.1.3 Explosive solutions to Eq.(4.1.5)
4.1.4 Explosive solutions to Eq.(4.1.6)
4.2 Stochastic wave equations driven by compensated Poisson random measures
4.2.1 Motivation
4.2.2 Preliminaries and hypothesis
4.2.3 Existence and uniqueness
4.2.4 Markov property
4.2.5 Invariant measure
4.3 Stochastic wave equation with non-Gaussian Levy perturbation
4.3.1 Equation formulation and motivation
4.3.2 Preliminaries and hypothesis
4.3.3 Existence and uniqueness
4.3.4 Invariant measure
5 Discontinuous Galerkin Method for the Elliptic SPDE
5.1 Introduction
5.2 Regular Approximation to White Noise
5.3 LDG Approximation of Regularized Problem
5.4 Numerical Test
Further Work
Appendix
References
Acknowledgements
个人简介与学术成果
本文编号：3695712
【文章页数】：288 页
【学位级别】：博士
【文章目录】：
Abstract
摘要
1 Reflected Stochastic Differential Equations and Applications
1.1 Strong comparison for RSDEs with non-Lipschitzian coefficients
1.1.1 Motivation
1.1.2 Pathwise uniqueness
1.1.3 Strong comparison theorem
1.2 First passage time of the reflected O-U process with two-sided barriers
1.2.1 Motivation
1.2.2 Laplace transform of the first passage time
1.2.3 An extended case
1.2.4 Applications in financial modelings
1.3 Large deviations for perturbed reflected diffusion processes
1.3.1 Motivation and method
1.3.2 LDP for perturbed diffusion processes
1.3.3 LDP for perturbed reflected diffusion processes
1.4 Hedging for a defaultable claim with recovery and dividend under local risk minimization
1.4.1 Motivation
1.4.2 The model and local risk minimization
1.4.3 Hedging for H under local risk minimization
2 Optimal Portfolio with Defaultable Risk and HJB Equations
2.1 Optimal portfolio with defaultable risk-log utility
2.1.1 Motivation and method
2.1.2 The optimization with a defaultable bond
2.1.3 Verification theorem
2.1.4 A numerical analysis example
2.2 Optimal portfolio with defaultable risk-power utility
2.2.1 The optimal portfolio with non-log HARA utility
2.2.2 The HJB equation
2.2.3 Solutions to the HJB equation
2.2.4 The verification theorem
2.2.5 Sensitivity analysis
2.3 Optimal portfolio and consumption with defaultable risk-a viscosity solution approach
2.3.1 Motivation
2.3.2 Price dynamics of the defaultable bond
2.3.3 The value function and HJB equation
2.3.4 The viscosity solution
3 Parabolic Type Stochastic Partial Differential Equations
3.1 On solutions of Cahn-Hilliard SPDE with Levy space-time white noise
3.1.1 Motivation
3.1.2 The definition of Levy space-time white noise
3.1.3 A new version of Burkholder-Davis-Gundy inequality and the definition of the solution
3.1.4 Existence and uniqueness of local solutions
3.2 On solutions of Cahn-Hilliard SPDE with fractional noise-a weak convergence approach
3.2.1 Motivation and main result
3.2.2 Fractional noise and embedding theorem
3.2.3 Weak convergence of local solutions
3.3 Support theorem for stochastic Cahn-Hilliard equation
3.3.1 Motivation and main result
3.3.2 Difference approximation to white noise
3.3.3 Localization framework
3.3.4 Auxiliary lemmas
3.3.5 The proof of(C1)
3.3.6 The proof of(C2)
3.4 The higher-order Ito and Skorokhod Anderson models
3.4.1 Lyapunov exponent estimates of fourth-order Ito Anderson model
3.4.2 Skorokhod fourth-order Anderson models with fractional noises
3.5 Stochastic nonlocal Kuramoto-Sivashinsky equation with jumps
3.5.1 Motivation
3.5.2 Preliminaries and hypothesis
3.5.3 Existence and uniqueness of the weak solution
3.5.4 Invariant measure
4 Hyperbolic Type Stochastic Partial Differential Equations
4.1 Explosive solutions of stochastic wave equations with damping
4.1.1 Motivation
4.1.2 Preliminaries
4.1.3 Explosive solutions to Eq.(4.1.5)
4.1.4 Explosive solutions to Eq.(4.1.6)
4.2 Stochastic wave equations driven by compensated Poisson random measures
4.2.1 Motivation
4.2.2 Preliminaries and hypothesis
4.2.3 Existence and uniqueness
4.2.4 Markov property
4.2.5 Invariant measure
4.3 Stochastic wave equation with non-Gaussian Levy perturbation
4.3.1 Equation formulation and motivation
4.3.2 Preliminaries and hypothesis
4.3.3 Existence and uniqueness
4.3.4 Invariant measure
5 Discontinuous Galerkin Method for the Elliptic SPDE
5.1 Introduction
5.2 Regular Approximation to White Noise
5.3 LDG Approximation of Regularized Problem
5.4 Numerical Test
Further Work
Appendix
References
Acknowledgements
个人简介与学术成果
本文编号：3695712
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