马氏调节过程在保险与金融中的应用

发布时间：2021-06-17 02:14

　　相对于经典的金融保险模型而言,马氏调节的金融保险模型似乎更能适应现实中的金融保险数据。在风险理论中,马氏调节的风险模型有这样一个优点:保险公司可以随外界环境（天气,经济,政府政策等）的改变而调节自身的保险政策。举个例子来说吧,在汽车保险中,天气环境的好坏是影响事故发生的重要因素。在不同的天气环境下,汽车保险中索赔的分布以及索赔在一定时间内发生的强度将会有很大的不同。因此,不同的天气环境下,保险公司的保险政策也将会有很大的不同,比如说,保费的收取将会随天气环境的变化而变化。在金融理论中,著名的Black-Scholes-Merton金融市场是基于几何布朗运动来描述标的资产（股票）的价格变化的。但是越来越多的实证研究表明,几何布朗运动并不能描述一些标的资产价格数据中的重要实证结果,比如,标的资产价格分布的重尾性质,标的资产价格的方差应该是随时间变化而变化的等等。Hardy[75]对马氏调节的金融市场模型与其他模型对现实金融数据的适合程度进行了比较并得出结论,马氏调节的金融市场模型对现实金融数据的适合程度明显优于其他一些比较常用金融市场模型。基于以上原因,马氏调节模型在金融保险理论中正变得越... 

【文章来源】：南开大学天津市 211工程院校 985工程院校 教育部直属院校

【文章页数】：221 页

【学位级别】：博士

【文章目录】：
摘要
Abstract
1 Introduction
    1.1 Background
    1.2 Organization and Main Contents of This Dissertation
2 Preliminaries
    2.1 Representation of Double Martingales
    2.2 Markov Chain
    2.3 Markov-Modulated Lévy Processes
    2.4 It(o|
)'s Formula for Generalized Markov-Modulated SDEs
3 Ruin Problems in a Markov-Modulated Compound Poisson Model
    3.1 Introduction
    3.2 The Markov-Modulated Compound Poisson Model
    3.3 Gerber-Shiu Discounted Penalty Functions
        3.3.1 Integro-differential Equations of Gerber-Shiu Discounted Penalty Function
        3.3.2 The Value of φ(O)
    3.4 Explicit Expressions of Gerber-Shiu Discounted Penalty Functions
        3.4.1 Formulas for φ_1(O) and φ_2(O)
        3.4.2 Explicit Expressions of α_1(u) and φ_2(u) for K_n-family Claim Size Distributions
    3.5 Numerical Illustrations
4 Complete Markovian Regime-Switching Market via Double Martingales
    4.1 Introduction
    4.2 Markovian Regime-Switching Market
    4.3 Some Results for Markov-Modulated Brownian Motion
        4.3.1 Characterization of Markov-Modulated Brownian Motion
        4.3.2 A Measure Change for Markov-modulated Brownian Motion
    4.4 Complete the Markovian Regime-Switching Market and Hedging
    4.5 Equivalent Martingale Measures
5 Portfolio Selection in the Enlarged Markovian Regime-Switching Market
    5.1 Introduction
    5.2 Enlarged Markovian Regime-Switching Market
        5.2.1 Markovian Regime-Switching Market
        5.2.2 Enlarging the Markovian Regime-Switching Market
    5.3 Arbitrage-Free and Completeness of the Enlarged Market
    5.4 Optimization Problems
        5.4.1 Logarithmic Utility
        5.4.2 Power Utility
    5.5 Comparisons of Optimization Problems in Enlarged and Original Market
6 Portfolio Optimization in Extended Markovian Regime-Switching Market
    6.1 Introduction
    6.2 Problem Formulation
    6.3 No Shorting Constraint
        6.3.1 Logarithmic Utility
        6.3.2 Power Utility
    6.4 No Constraint
        6.4.1 Logarithmic Utility
        6.4.2 Power Utility
7 Reinsurance and Investment in Extended Regime-Switching Market
    7.1 Introduction
    7.2 Extended Markovian Regime-Switching Market and Insurance Model
    7.3 Maximize the Utility and HJB Equation
    7.4 Solution of HJB Equation for Exponential Utility Function
    7.5 The Verification Theorem
    7.6 Some Special Cases of Our Model
    7.7 Numerical Example
8 Mean-Variance Problem in the Markov-Switching Jump-Diffusion Market
    8.1 Introduction
    8.2 Markov-Switching Jump-Diffusion Market and Feasibility of M-V Problem
    8.3 Solution to Unconstrained Markov-Modulated Stochastic LQ Problem
    8.4 Efficient Portfolio and Efficient Frontier
9 Optimal Reinsurance and Investment in a Hidden Markov Model
    9.1 Introduction
    9.2 Insurance Risk Model and Hidden Markovian Regime Switching Market
    9.3 Separation Principle
    9.4 HJB-equation Approach
Bibliography
Index
Resume and Publications
Acknowledgements



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