本文分別從絕對破產,Gerber-Shiu期望折扣罰金函數(shù)（簡稱Gerber-Shiu函數(shù)）和最優(yōu)分紅三個方面來研究了保險中的若干問題。我們研究的風險模型大致分為兩類,一類是具有利率的風險模型,另一類是L（?）vy風險模型。一、具有利率的風險模型:對于絕對破產問題的研究我們一般是借助于對Gerber-Shiu函數(shù)的研究來展開的。而對于Gerber-Shiu函數(shù)的研究,則是通過隨機過程及隨機微分方程的知識得到它滿足的積分-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它的影響。對于最優(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)分紅界幾個方面展開的。通過概率的手段推導出折現(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)。進一步我們通過數(shù)據分析得到存款利息及貸款利息對折現(xiàn)分紅總量均值函數(shù)及最優(yōu)分紅界的影響。二,L（?）vy風險模型:我們通過研究L（?）vy風險盈余過程的L（?）vy測度對應的密度函數(shù)π的log-凸性... 

【文章來源】：曲阜師范大學山東省

【文章頁數(shù)】：134 頁

【學位級別】：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 D_u,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 D_u,b

    §3.3 Moments of D_u,b

    §3.4 Explicit expressions of M_{u, y; b} and V_n（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 D_u,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



本文編號：2901559

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