【作者】 赵翔华；

【导师】 尹传存；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2008， 博士

【摘要】 Levy过程是一具有独立平稳增量的随机过程，具有如马尔可夫性，无穷可分性等许多良好的性质，在金融数学中一直扮演着重要的角色．另一方面，风险理论中的许多风险模型，如经典风险模型（复合泊松过程），带干扰的经典风险模型，布朗运动等，均是一些特殊的Levy过程．这时，不禁要问若风险模型中描述保险风险的基本盈余过程为一般的Levy过程，其破产问题会是什么样子的?近几年来，许多学者对基本盈余过程为谱负的Levy过程的风险模型进行了研究．在本文第二至六章中，利用逼近的的方法对具有正、负跳跃的Levy风险模型及具有投资收益的Levy风险模型进行了初步的研究并且讨论了两类由某特殊Levy过程决定的并且具有某分红策略的Ornstein-Uhlenbeck型风险过程的分红问题．渐近估计方法是研究破产问题的一种常用方法．在第七、八章中，首先给出了随机游动阶梯高度及极大值的局部估计和尾估计以及一瑕疵更新方程解的渐近估计，然后将结果应用到风险理论中，得到了一些新结论．迄今，大多数风险模型都假定索赔时间间隔与索赔量是独立的．若两者关，会对破产概率及其他相关的量产生怎样的影响．在本文的最后一章，我们研究了一类索赔时间间隔与索赔量相关且带干扰的风险模型．根据内容本文分为以下九章：第一章：我们介绍了Levy过程及一些轻尾，重尾分布族的定义以及Levy过程的一些基本定理．对于论文中用到的一些符号也给出了规定．第二章：我们推广了Garrido and Morales[47]的Levy风险过程，引入了既有正跳跃又有负跳跃的Levy风险过程，其具有下面的形式U（t）=u+ct-S（t），t≥0，过程{S（t），t≥0}为一具有正、负跳跃的Levy过程，它包含了非连续收取的保费和索赔总量．在本章中，利用逼近的方法，得到了此风险过程的罚金折现函数Φ满足的更新方程及其Φ的一个级数表达式．在第四节中，我们还讨论了上风险过程在初始盈余值趋于无穷大时，其破产概率的一些渐近形式．第三章：设U_t为一个具有风险投资回报的风险过程在t时刻的盈余值，具有以下形式U_t=e^Mt(u+∫₀^te^-MsdR_s),t≥0，U₀=u．其中，保险公司的基本盈余过程为一Levy风险过程：R_t=ct-J_t+σB_t．索赔总量过程{J_t，t≥0)为一无漂移和干扰项的Levy过程．过程{e^Mt=e^δt+rWt，t≥0}是一几何布朗运动，δ>0，r是两个固定的参数．{B_t，t≥0}和{W_t，t≥0}为两个相互独立的一维标准布朗运动．在本章中，利用逼近的方法得到了破产概率满足的积分-微分方程．第四章：假设某保险公司采用这样的财政策略：当公司的资产大于某一水平△（>0）时，超过△的部分进行投资，取得利息率为r（≥0）的收益；当公司的资产为负值但不低于某一特定的值（-c／δ）时，通过贷款来继续维持公司的运作，贷款利率为δ（>0）．设U（t）为采用上述策略后某保险公司在t时刻的盈余，满足下面的随机微分方程：索赔总量过程Z=：{Z（t），t≥0}是一个无漂移部分的从属过程（Subordinator）．通过研究绝对破产时刻，绝对破产前的瞬间盈余及绝对破产时的赤字三者的联合分布来研究上模型的绝对破产问题．首先，得到了当索赔总量过程为复合泊松过程时，联合分布满足的积分-微分方程，并给出了方程的一般解．利用上述结果我们得到了当索赔总量过程为从属过程时，联合分布的一个表达式．第五章：设具有某一分红策略的保险公司在t时刻的盈余值过程R={R_t，t≥0)为一Ornstein-Uhlenbeck型风险过程，满足下面的随机微分方程dR_t=（μ+ρR_t-l（t））dt+σdW_t，其中，{W_t，t≥0}为一标准的布朗运动，μ，ρ，σ>0为三个固定的常数，l（t）表示在t时刻的分红率函数．本章，讨论了当l（t）为有界函数时，分红策略的最优分红问题．第六章：设带有固定分红策略的α-平稳Ornstein-Uhlenbeck（1<α≤2）型风险过程X：={X_t，t≥0}满足下面的随机微分方程：dX_t=-λX_tdt+dZ_t-dD_t，t≥0，其中，λ>0为一固定的参数，{Z_t，t≥0)为谱负α-平稳过程．D：={D_t，t≥0)为到时刻t为止的分红总量．利用广义Wright’s超几何函数给出此模型分红总量折现值各阶矩的表达式．第七章：在本章中，讨论了实数域上具有负均值的随机游动，得到了在指数估计不成立的条件下，此随机游动的阶梯高度和极大值的局部估计以及尾估计．随后，我们将这些结果应用到风险理论中的Sparre Andersen模型中．第八章；本章，我们对一类瑕疵更新方程解进行了研究，得到了其解的非指数渐近表达式，并将结果应用到风险理论中．第九章：在本章中，讨论了一个索赔相依且带干扰的风险模型，利用Laplace变换研究了破产时刻，破产前的瞬间盈余及破产时的赤字的联合分布，得到了此联合分布Laplace变换的一个表达式．当上风险模型具有部分分红策略时，研究了分红折现期望及矩母函数，得到了他们的一个表达式．更多还原

【Abstract】 Levy processes are stochastic processes with independent and stationary increments, and play a fundamental role in Mathematical Finance. On the other hand, many important risk processes are also special cases of Levy processes, such as Brownian motion, the compound Poisson process, the compound Poisson process perturbed by Brownian motion. In Chapters 2-4 of this thesis, we study ruin problems for a Levy processes with positive and negative jumps and for some Levy processes with investment. In Chapters 5-6, we introduce two Ornstein-Uhlenbeck type risk processes driven by some Levy process. The optimal dividend problem and the moments of the cumulative dividend are studied, respectively.The asymptotic problems of ruin problems are important topics in risk theory. In.Chapters7-8, when the conditions for the exponential estimate are not satisfied, a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk are derived and non-exponential asymptotic forms for solutions of defectiverenewal equations are obtained. All the results are applied to risk models. In last chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size.Organization and outline of this thesisChapter 1: We introduce the definition of and some important theorems related to Levy processes, and the definitions of some light-tailed and heavy-tailed distributions are introduced also. Some notations which will be used throughout the thesis are set down in this chapter.Chapter 2: Garrido and Morales [47] introduced a Levy risk process only with negative jumps, in which they studied the Gerber-Shiu function. The aim of this chapter is to extend their work to a general Levy risk process with positive and negative jumps, that is, the Levy risk process has the formU（t）=u + ct-S（t）,t≥0,{S（t）} is a jump Levy process with positive and negative jumps. We prove that the GerberShiu functionΦsatisfies a renewal equation and a infinite series expression ofΦis obtained. Some asymptotic behaviors of the ruin probability as u→∞are discussed.Chapter 3: In this chapter, we assume the surplus of the insurer at time t under some investment assumption is denoted by Ut:U_t=e^Mt(u+∫₀^te^-MsdR_s),t≥0,U₀=u.where R_t=ct-J_t+σB_t is a Levy risk process. The aggregate claims process {Jt,t≥0}is a jump Levy process without drift and diffusion coefficient, started at 0. The process{e^Mt=e^δt+rWt,t≥0}is a geometric Brownian motion,δ> 0 and r are constants, and{W_t,t≥0} is an one-dimensional Brownian motion independent of {Rt, t≥0}. In thischapter, it is shown that the ruin probabilities （by a jump or by oscillation） of the resulting surplus process satisfy certain integro-differential equations.Chapter 4: Let U（t） be the surplus of insurer at time t with the liquid reserve level,. the credit interest force r≥0 and debit interest forceδ> 0. Then the surplus process {U（t）, t≥0} satisfies the following stochastic differential equation:where u≥0 is the initial value and c > 0 is a constant premium rate, defined as c = （1 +θ）EZ（1）, whereθ≥0 is the security loading factor. {Z（t）, t≥0} is a subordinator with zero drift.We study the absolute ruin questions by defining the joint distributed function of the absolute ruin time, the surplus immediately before absolute ruin and the deficit at absolute ruin. Using an approximation scheme, a general expression for the joint distributed function of the risk process driven by a subordinator is obtained.Chapter 5: In this chapter, a controlled Ornstein-Uhlenbeck type model is studied whose reserve R_t is assumed to be governed by the stochastic differential equationdR_t=（μ+ρR_t-l（t））dt+σdW_t,where {W_t,t≥0} is a standard Brownian motion,μ,ρ,σ>0 are constants and l（t） is the rate of dividend payment at time t. It is shown how the optimal return function and the optimal dividend-payment strategy can be calculated when the rate function l（t） is restricted so that the function l（t） varies in [0, M] for some M <∞.Chapter 6: In this chapter, we present a spectrally negative a-stable Ornstein-Uhlenbeck （1 <α≤2） type risk process X := {X_t,t≥0} with dividend barrier, which is the solution to the linear stochastic differential equationdX_t=-λX_tdt+dZ_t-dD_t,with X₀ = x, and the parameterλ> 0, Z := {Z_t,t≥0} be a spectrally negativeα-stableprocess, withα∈(1,2]. D := {D_t, t≥0} be the cumulative dividends process. The momentsof the present value of dividend payments until ruin are provided in terms of the Wright’s generalized hypergeometric function 2ψ1.Chapter 7: For a random walk on the real line with negative mean, we obtain a local asymptotic estimate and a tail asymptotic estimate for the distributions of ladder height and supremum for the random walk when the conditions for the exponential estimate are not satisfied. The results are applied to the Sparre Andersen model, some new results on the probability of ruin are presented.Chapter 8: In this chapter, we derive non-exponential asymptotic forms for solutions of defective renewal equations. Applications of this result is given to the Gerber-Shiu discounted penalty function in the classical risk model.Chapter 9: In this chapter, we consider a jump-diffusion model with a dependent setting, where the claim inter-occurrence times depend on the previous claim size. We study the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin, and an exact analytical expression for the Laplace transform of the the joint distribution is derived. We also study the dividend problem for the above model with a dividend strategy. A system of integro-differential equations with certain boundary conditions satisfied by the expected discounted dividend payments prior to ruin is derived and an infinite series expression of it is obtained.更多还原