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幾類風(fēng)險(xiǎn)模型的破產(chǎn)理論及分紅問題的研究

發(fā)布時(shí)間:2020-12-06 14:52
  本文分別從絕對(duì)破產(chǎn),Gerber-Shiu期望折扣罰金函數(shù)(簡(jiǎn)稱Gerber-Shiu函數(shù))和最優(yōu)分紅三個(gè)方面來研究了保險(xiǎn)中的若干問題。我們研究的風(fēng)險(xiǎn)模型大致分為兩類,一類是具有利率的風(fēng)險(xiǎn)模型,另一類是L(?)vy風(fēng)險(xiǎn)模型。一、具有利率的風(fēng)險(xiǎn)模型:對(duì)于絕對(duì)破產(chǎn)問題的研究我們一般是借助于對(duì)Gerber-Shiu函數(shù)的研究來展開的。而對(duì)于Gerber-Shiu函數(shù)的研究,則是通過隨機(jī)過程及隨機(jī)微分方程的知識(shí)得到它滿足的積分-微分方程及邊值問題,然后得到了它在指數(shù)索賠下的明確表達(dá)式以及Erlang(2)索賠下滿足的微分方程,并通過數(shù)值分析得到貸款利息及存款利息對(duì)它的影響。對(duì)于最優(yōu)分紅問題的研究是通過研究折現(xiàn)分紅總量均值的矩母函數(shù)/高階矩,最優(yōu)分紅策略以及最優(yōu)分紅界幾個(gè)方面展開的。通過概率的手段推導(dǎo)出折現(xiàn)分紅總量均值的矩母函數(shù),高階矩滿足的積分-微分方程及邊值條件,或者通過粘性解理論來刻化最優(yōu)值函數(shù)。進(jìn)一步我們通過數(shù)據(jù)分析得到存款利息及貸款利息對(duì)折現(xiàn)分紅總量均值函數(shù)及最優(yōu)分紅界的影響。二,L(?)vy風(fēng)險(xiǎn)模型:我們通過研究L(?)vy風(fēng)險(xiǎn)盈余過程的L(?)vy測(cè)度對(duì)應(yīng)的密度函數(shù)π的log-凸性... 

【文章來源】:曲阜師范大學(xué)山東省

【文章頁數(shù)】:134 頁

【學(xué)位級(jí)別】:博士

【文章目錄】:
中文摘要
ABSTRACT
Chapter 1 Preliminaries
    §1.1 Some basic risk models
    §1.2 About optimal dividend problems
    §1.3 Confluent hypergeometric equation
Chapter 2 Dividend payments in the classical risk model under absolute ruin
    §2.1 Introduction
u,b">    §2.2 Moment generating function of Du,b
  •     §2.3 Moments of Du,b
  •     §2.4 Explicit expressions for exponential claims
        §2.5 Optimal dividend barrier for exponential claims
        §2.6 Numerical analysis for Erlang(2) claim sizes
        §2.7 The Gerber-Shiu expected discounted penalty function
    Chapter 3 Optimal dividends in the classical risk model with credit and debit interests under absolute ruin
        §3.1 Introduction
    u,b">    §3.2 Moment generating function of Du,b
  •     §3.3 Moments of Du,b
  •     §3.4 Explicit expressions of Mu, y; b and Vn(u, b)
        §3.5 Optimal choice of dividend barrier for exponential claims
        §3.6 The Laplace transform of absolute ruin time
    Chapter 4 The perturbed compound Poisson risk process with in vestment and debit interest
        §4.1 Introduction
        §4.2 The stochastic Dirichlet problem
        §4.3 Integro-differential equations
        §4.4 Integral equations
    +">    §4.5 A renewal equation and asymptotic results for Φ+
  •     §4.6 Explicit results for exponential claims Φ+
  • Chapter 5 On the perturbed compound Poisson risk model under absolute ruin with debit interest and a constant dividend barrier
        §5.1 Introduction
    1(u, b)">    §5.2 Integro-differential equations for V1(u, b)
    u,b">    §5.3 Moment generating function and higher moments of Du,b
  •     §5.4 The Gerber-Shiu expected discounted penalty function
    Chapter 6 Optimal dividend strategy in the perturbed compound Poisson risk model with investment interest
        §6.1 Introduction
        §6.2 Hamilton-Jacobi-Bellman equation
        §6.3 Construction of the optimal strategy
        §6.4 Examples
    Chapter 7 Optimality of the barrier strategy for spectrally negative Levy risk processes
        §7.1 Introduction
        §7.2 Preliminaries on log-convex functions and related functions
        §7.3 Convex solutions for integro-differential equations
        §7.4 The optimality of the barrier strategy
    References
    Acknowledgements



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