摘要
在保险数学中,破产理论是保险风险理论研究的重要问题之一,它可以为保险公司决策者提供一个非常有用的早期风险预警手段.因此对其进行研究具有非常重要的理论和现实意义.
本文从跳扩散过程出发,一方面利用随机过程以及随机微分方程的相关知识和理论,考虑了首次通过时间问题.首次通过时间,即第一次通过（向下或者向上）某个边界的时间.对于单边跳和双边跳的扩散过程来说,首次通过时间关系到其破产问题, Gerber-Shiu函数,以及破产前期望折现分红,期权定价等问题.
另一方面,分红策略也是风险理论中重要的研究问题.“分红,是指将公司的（部分）盈余作为红利分发给公司所有者或股份参与者.”其现实意义使得分红策略的研究备受关注.对于这些受益人而言,他们不仅关心公司目前的经济状态,更关心的是采取何种分红策略才能使自己的收益以一定的折现率折现后尽可能的大,即所谓的最优分红问题.根据不同的客户要求,或者说在不同的分红要求下,最优的分红策略自然是不同的.现在常用的策略有两种,一种是障碍分红策略,另外一种是阈值分红策略,他们己经被证明在相应的限制下是最优的.这两种分红策略在第二章、第三章和第五章都有讨论.
第一章主要介绍了本论文的研究背景,包括基本的风险模型,分红策略,以及L′evy过程的基本知识.
第二章研究了超指数跳跃（扩散）过程对水平边界的首次通过时间问题,获得了首次通过时间、首次通过时间与undershoot (overshoot)、过程与最大值（最小值）等量的分布或联合分布的Laplace变换的明确解.这一过程覆盖了复合Poisson风险模型、扩散干扰的复合Poisson风险模型及其它们的对偶模型.并且给出了在障碍分红策略及阈值分红策略下的分红公式的精确表达式.（本章研究结果已经发表在Journal of Computational andApplied Mathematics.）
第三章考察了当保险公司的非控制的盈余过程是一个谱负的L′evy过程时的最优分红问题.假设分红按照常数比例分给客户,当L′evy测度有一个完全单调的密度时,证明了阈值策略是最优分红策略.（本章研究结果已经发表在Acta Mathematicae Applicatae Sinica,English Series.）
第四章研究了混合指数跳扩散过程下的常数界的首次通过时间问题.得到了首次通过时间、首次通过时间与undershoot (overshoot)分布或联合分布的Laplace变换的明确解.并且得到了双边跳盈余过程的Gerber-Shiu函数的明确表达式,给出了路径依赖的期权的解析解,回望和障碍期权的Laplace变换,带跳的结构性信贷风险模型的零息贷款的闭合表达式.（已投稿.）
第五章研究了广义复合泊松模型(其计数过程是一个广义泊松过程)的最优分红问题,并以经典风险模型和Po′lya-Aeppli风险模型为例讨论其性质.本章的目标是找到一种分红策略,实现最大限度地给股东分红,直至公司破产.最后证明了在一定条件下的最优分红策略是阈值策略.（本章研究结果已经发表在the Applied Mathematics.）
In insurance mathematics, ruin theory is one of the mainly contents in insurance risktheory, and it can supply a very useful early-warning measure for the risk of the insur-ance company. So it has important theoretical and practical significance for the insurancecompany.
In this thesis, we pay attention to the jump difusion process: for one hand, with theknowledge of stochastic process and stochastic diferential equation, we research on the firstpassage time which is the first time pass (downward or upward) a flat boundary. Thefirst passage time relate to the classical ruin problem and the expected discounted penaltyfunction or the Gerber-Shiu function, the expected total discounted dividends up to ruin aswell as the pricing options.
On the other hand, dividend strategies become an important branch of risk theory.Dividends are payments made to stockholders from a firm’s surplus (or part). Due to itspractical importance, people pay more attentions to it. It is desirable to find a fixed rulewhich produces the largest possible expected sum of discounted dividend, and that is theoptimal dividend problem. There are two common strategies, the barrier dividend strategyand the threshold dividend strategy, and they were proved to be the “optimal” under theircorresponding constraints in their risk models. Therefore, we consider risk models with thepresence of these dividend strategies in Chapter2, Chapter3and Chapter5.
In Chapter1, we mainly introduce the basis background, include some basic risk models,dividend strategies, and the basic knowledge of L′evy processes.
In Chapter2, we investigate the first passage time to flat boundaries for hyper-exponentialjump (difusion) processes. Explicit solutions of the Laplace transforms of the distribution ofthe first passage time, the joint distribution of the first passage time and undershoot (over-shoot), the joint distribution of the process and running supreme (infima) are obtained. Theprocesses recover many models appearing in the literature such as the compound Poissonrisk models, the difusion perturbed compound Poisson risk models, and their dual models.As applications, we present explicit expressions of the dividend formulae for barrier strategyand threshold strategy.(Published on Journal of Computational and Applied Mathematics.)
In Chapter3, we consider the optimal dividend problem for an insurance company whose uncontrolled surplus precess evolves as a spectrally negative L′evy process. We assume thatdividends are paid to the shareholders according to admissible strategies whose dividend rateis bounded by a constant. We shown that a threshold strategy forms an optimal strategyunder the condition that the L′evy meansure has a completely monotone density.(Publishedon Acta Mathematicae Applicatae Sinica, English Series.)
In Chapter4, we consider the first passage time to constant boundaries for mixed-exponential jump difusion processes. Explicit solutions of the Laplace transforms of thedistribution of the first passage time, the joint distribution of the first passage time andundershoot (overshoot) are obtained. As applications, we present explicit expression ofthe Gerber-Shiu functions for surplus processes with two-sided jumps,present the analyticalsolutions for popular path-dependent options such as lookback and barrier options in termsof Laplace transforms and give a closed-form expression on the price of the zero-coupon bondunder a structural credit risk model with jumps.(Submitted.)
In Chapter5, we study the optimal dividend problem for a company whose surplusprocess, in the absence of dividend payments, evolves as a generalized compound Poissonmodel in which the counting process is a generalized Poisson process. This model includingthe classical risk model and the P′olya-Aeppli risk model as special cases. The objectiveis to find a dividend policy so as to maximize the expected discounted value of dividendswhich are paid to the shareholders until the company is ruined. We show that under someconditions the optimal dividend strategy is formed by a barrier strategy.(Published on theApplied Mathematics.)
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