本文分五個部分來研究反射隨機微分方程,隨機偏微分方程以及它們在信用風(fēng)險理論中的應(yīng)用。 反射隨機微分方程可視為一個Skorohod問題。在李普希茲條件下,其強解的存在唯一性被Lions和Sznitman所證明。隨后李普希茲條件被分別拓展到了單面李普希茲和Yamada-Watanabe條件。Le Gall,Bass和Chen,Zhang,Marin和Real分別證明了在這些條件下,其強解的存在唯一性。本文第一章1.1節(jié)將延續(xù)這方面的研究。與現(xiàn)有文獻(xiàn)的不同之處在于,我們考慮在Fang和Zhang的非李普希茲條件下,其強解的存在唯一性。特別地,在一維情形下,我們結(jié)合Fang和Zhang非李普希茲條件的特點以及局部時的性質(zhì)證明了一個強比較定理。由于反射隨機微分方程的解能被限制在某些特定的凸區(qū)域上,這個特點使這類方程在排隊論、金融建模中有重要應(yīng)用。其中Harrison用雙邊反射布朗運動建立了存儲模型,隨后Ata,Harrison和Shepp用雙邊反射O-U過程描述了布朗網(wǎng)絡(luò)優(yōu)化問題。Goldstein和Keirstead用零點單邊反射隨機微分方程建模瞬時利率過程,并導(dǎo)出了當(dāng)利率過程建模為... 

【文章頁數(shù)】：288 頁

【學(xué)位級別】：博士

【文章目錄】：
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
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