若干风险模型中期望折现罚金函数和最优分红的研究

发布时间：2018-10-13 12:07

【摘要】：风险理论是精算学的重要组成部分.它研究的内容主要有两点：一是公司面临的风险,二是公司的收益.公司的风险可以用一些精算量来刻画,如破产概率、破产前盈余和破产时赤字等,而期望折现罚金函数将这几个破产量统一起来,成为风险理论中重要的精算量.除了风险外,公司还关心其收益,衡量公司收益最具代表性的量是破产前分红的总量,如何使公司的收益最大化已成为风险理论研究的热点问题.于是,本文致力于研究更新风险模型中的期望折现罚金函数和对偶风险模型中的最优分红问题,具体内容如下. 首先,研究两类更新风险模型中的期望折现罚金函数.一类是具有泊松随机收益的更新模型,利用拉格朗日插值公式求出了期望折现罚金函数满足的更新方程和有理分布索赔下的具体表达式.另一类是有投资和债务利率的更新模型,得到了期望折现罚金函数满足的积分-微分方程,并用超几何函数表示出了绝对破产概率. 其次,在复合泊松对偶风险模型中,研究了带有注资的最优分红问题,其中考虑了固定交易税和比例交易税的影响.这里主要基于两种情况：一种是若破产发生,公司将受到一定的惩罚,即考虑破产惩罚；另一种是由于受到外界因素的影响,盈余过程可能被随机的终止,即考虑随机时间界.在这两种情况下,我们都得到了破产前红利与注资成本之差的折现期望的最大值和相应的最优分红策略. 最后,在谱正Levy风险模型中,研究了三个问题.根据分红速率是否有限制,我们考虑两种最优分红问题,在这两个问题中研究了带有注资的最优分红,其中也考虑了交易税的影响.我们用Levy过程的尺度函数表示出了破产前红利与注资成本之差的折现期望的最大值,并得到了相应的最优分红策略.第三个问题研究了随机离散时间的分红,亦用Levy过程的尺度函数表示出了破产前红利折现期望的最大值,并得到了相应的最优分红策略.
[Abstract]:Risk theory is an important part of actuarial science. It has two main contents: the risk that the company faces, and the profit of the company. The risk of a company can be described by some actuarial quantities, such as the probability of bankruptcy, the surplus before bankruptcy and the deficit at the time of bankruptcy, and the expected discounted penalty function unifies these broken yields and becomes an important actuarial quantity in risk theory. Besides risks, companies also care about their earnings. The most representative amount of corporate earnings is the total amount of dividends before bankruptcy. How to maximize the profits of companies has become a hot issue in risk theory. Therefore, this paper is devoted to the study of the expected discounted penalty function in the updated risk model and the optimal dividend in the dual risk model, the details of which are as follows. First, we study the expected discounted penalty function in two kinds of updated risk models. One is the renewal model with Poisson's stochastic income. By using Lagrangian interpolation formula, the renewal equation of the expected discounted penalty function and the concrete expression under the rational distribution claim are obtained. The other is the renewal model with interest rate of investment and debt. The integro-differential equation of expected discounted penalty function is obtained, and the absolute ruin probability is expressed by hypergeometric function. Secondly, in the compound Poisson dual risk model, the optimal dividend problem with capital injection is studied, in which the effects of fixed transaction tax and proportional transaction tax are considered. There are mainly two kinds of cases: one is that if bankruptcy occurs, the company will be punished, that is, considering bankruptcy punishment; the other is due to the influence of external factors, the surplus process may be randomly terminated, that is, consider random time bound. In both cases, we obtain the maximum expected discounted expectation and the corresponding optimal dividend strategy for the difference between the pre-bankruptcy dividend and the capital injection cost. Finally, three problems are studied in the spectral positive Levy risk model. According to whether the rate of dividend is limited or not, we consider two kinds of optimal dividend problem. In these two problems, we study the optimal dividend with capital injection, and consider the effect of transaction tax. We use the scale function of the Levy process to express the maximum value of the discounted expectation of the difference between the dividend before bankruptcy and the cost of capital injection, and obtain the corresponding optimal dividend strategy. In the third problem, the dividend of random discrete time is studied, and the maximum expectation of dividend discounted before bankruptcy is expressed by using the scale function of Levy process, and the corresponding optimal dividend strategy is obtained.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：F840;O211.6

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