带利率的风险模型的破产问题
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摘要
本论文主要解决了破产周期里的若干问题。事物的发展是波浪式的，呈周期性变化。保险公司是实践风险论的典型企业，当然不能例外，尽管实际上市场规律不能放任一个公司盈余、破产、盈余……周而复始进行，但是理论上是可以的。
精确地刻画盈余与破产的程度，即破产前最大盈余与破产后最大赤字问题。特别是破产后最大赤字问题的刻画可以使保险公司清楚认识到破产后的财务状况。进而使公司在盈余时居安思危，不易沉溺于志得意满。破产前最大盈余的刻画使保险公司进可能有效利用保险基金，使保险公司获得更大利润。
还描述了盈余持续时间的分布、破产持续时间的分布、破产前最大盈余与破产后最大赤字的联合分布、破产周期的分布等问题，这些问题清楚了，就可以绘出一张保险公司周期的轨迹。进而了解保险公司是处在风险厌恶还是在风险追求、风险中立中。投保人可根据保险公司定格在破产周期的轨迹的位置决定是否投保。
本论文首先探讨常利息率下的完全离散的经典风险模型，利用递推形式得出破产周期里的上述问题。其次探讨常利息力下的更新风险模型，利用马氏性，助转移概率把问题转化为离散形式，从而解决了破产周期里的上述问题。两模型不同之处，在常利息下的完全离散的经典风险模型里，探讨的是破产后盈余首次为非负时余额的分布、破产前一时刻的盈余分布，以及破产前一时刻与破产时余额的联合分布。但在常利息力下的更新风险模型里，探讨的是破产后盈余首次为零时余额的分布、破产前瞬间余额分布，以及破产前和破产时瞬间余额的联合分布。这体现了离散与连续的区别。离散相当于蒙太奇的基本单位——镜头，而连续是把镜头组合在一起形成蒙太奇效应，给观众讲述一个生动活泼的故事。连续问题可借助离散而得到解决。在常利息力下的更新风险模型中体现了连续转化为离散的完美性。这为我们解决连续问题提供了一个好的方法。
在常利息率下的完全离散的经典风险模型中，把第1周期里的破产问题推广在第i周期里，而在常利息力下的更新风险模型中，可以类似地把第1周期里的破产问题推广在第i周期里，只不过不同之处是它的第i(j＞1)个周期里的初始准
备金u二0o
在常利息率下的完全离散的经典风险模型中，
根据破产持续时间的定义，在常利忌攀卜阴元王网
产持续n期的n多1;而在常利息力下的更新风险模型中
破
，破产持续n期的n多0
定义名虽同，但也有区别。
我们靠逻辑去证明，但是靠直觉去发现。
有周期，但只有证明了破产周期里的问题后，
择决策方法，对自己的公司负责。
虽然直觉告诉我们，破产问题肯定
我们才能运用到实际生活中去，选
The paper solves mainly the problem in ruin circle . No straight road but circular road leads to development . An insurance serves as a typical example of practising its risk . So it is not an exception, in practice the rules of the market don't let a insurance do circularly the same thing , but in theory it can do .
I deal with specifically the degree of surplus and ruin - the distribution of the supremum surplus before ruin and the supremum deficit. In a world, an insurance can see the light about its capital, it will think hard before ruin in order not to indulge in. And it uses its capital well so that it can get it maxmium interest.
I obtain the distribution of the time in the red which describe the severity of ruin, the time in the green which describe the good of surplus, ruin circle and the joint distribution of the supremum surplus before ruin and supremum deficit .So I can make a drawing of ruin circle of the insurance. However, a policy-holder can decide whether he buys after he knows the ruin circle.
The paper firstly I discussed ruin circle problems in the completely discrete time insurance risk model with constant interest rate. The recursive expressions of the distribution of the ruin circle are obtained. Secondly the renewal risk model with constant iterest force is discussed. The surplus U (Tk) at claim occurrence times Tk is a MarKov chain with transition probability Q(x, B). The series expansions of ruin circle probability are obtained .
There are many differences between the completely discrete time insurance risk model with constant interest rate and the renewal risk model with constant interest force . For one thing, the former has not instant time but the latter has. For another thing , in the times ruin circle, the former has u≥0, howerer the latter has u=0. Finally, when they have the time in the red which describe the severity of ruin , the former has n>0 and the latter has n≥0.
It is by logic that we prove, but by intuition that we discover. Soafterweproved the discovery, we can work out decisive means.
精确地刻画盈余与破产的程度，即破产前最大盈余与破产后最大赤字问题。特别是破产后最大赤字问题的刻画可以使保险公司清楚认识到破产后的财务状况。进而使公司在盈余时居安思危，不易沉溺于志得意满。破产前最大盈余的刻画使保险公司进可能有效利用保险基金，使保险公司获得更大利润。
还描述了盈余持续时间的分布、破产持续时间的分布、破产前最大盈余与破产后最大赤字的联合分布、破产周期的分布等问题，这些问题清楚了，就可以绘出一张保险公司周期的轨迹。进而了解保险公司是处在风险厌恶还是在风险追求、风险中立中。投保人可根据保险公司定格在破产周期的轨迹的位置决定是否投保。
本论文首先探讨常利息率下的完全离散的经典风险模型，利用递推形式得出破产周期里的上述问题。其次探讨常利息力下的更新风险模型，利用马氏性，助转移概率把问题转化为离散形式，从而解决了破产周期里的上述问题。两模型不同之处，在常利息下的完全离散的经典风险模型里，探讨的是破产后盈余首次为非负时余额的分布、破产前一时刻的盈余分布，以及破产前一时刻与破产时余额的联合分布。但在常利息力下的更新风险模型里，探讨的是破产后盈余首次为零时余额的分布、破产前瞬间余额分布，以及破产前和破产时瞬间余额的联合分布。这体现了离散与连续的区别。离散相当于蒙太奇的基本单位——镜头，而连续是把镜头组合在一起形成蒙太奇效应，给观众讲述一个生动活泼的故事。连续问题可借助离散而得到解决。在常利息力下的更新风险模型中体现了连续转化为离散的完美性。这为我们解决连续问题提供了一个好的方法。
在常利息率下的完全离散的经典风险模型中，把第1周期里的破产问题推广在第i周期里，而在常利息力下的更新风险模型中，可以类似地把第1周期里的破产问题推广在第i周期里，只不过不同之处是它的第i(j＞1)个周期里的初始准
备金u二0o
在常利息率下的完全离散的经典风险模型中，
根据破产持续时间的定义，在常利忌攀卜阴元王网
产持续n期的n多1;而在常利息力下的更新风险模型中
破
，破产持续n期的n多0
定义名虽同，但也有区别。
我们靠逻辑去证明，但是靠直觉去发现。
有周期，但只有证明了破产周期里的问题后，
择决策方法，对自己的公司负责。
虽然直觉告诉我们，破产问题肯定
我们才能运用到实际生活中去，选
The paper solves mainly the problem in ruin circle . No straight road but circular road leads to development . An insurance serves as a typical example of practising its risk . So it is not an exception, in practice the rules of the market don't let a insurance do circularly the same thing , but in theory it can do .
I deal with specifically the degree of surplus and ruin - the distribution of the supremum surplus before ruin and the supremum deficit. In a world, an insurance can see the light about its capital, it will think hard before ruin in order not to indulge in. And it uses its capital well so that it can get it maxmium interest.
I obtain the distribution of the time in the red which describe the severity of ruin, the time in the green which describe the good of surplus, ruin circle and the joint distribution of the supremum surplus before ruin and supremum deficit .So I can make a drawing of ruin circle of the insurance. However, a policy-holder can decide whether he buys after he knows the ruin circle.
The paper firstly I discussed ruin circle problems in the completely discrete time insurance risk model with constant interest rate. The recursive expressions of the distribution of the ruin circle are obtained. Secondly the renewal risk model with constant iterest force is discussed. The surplus U (Tk) at claim occurrence times Tk is a MarKov chain with transition probability Q(x, B). The series expansions of ruin circle probability are obtained .
There are many differences between the completely discrete time insurance risk model with constant interest rate and the renewal risk model with constant interest force . For one thing, the former has not instant time but the latter has. For another thing , in the times ruin circle, the former has u≥0, howerer the latter has u=0. Finally, when they have the time in the red which describe the severity of ruin , the former has n>0 and the latter has n≥0.
It is by logic that we prove, but by intuition that we discover. Soafterweproved the discovery, we can work out decisive means.
引文
1 孙立娟，顾岚。离散时间保险风险模型的破产问题。应用概率统计。2002，18(8)：293-299
2 孙立娟，顾岚、刘立新。离散时间模型下最大赤字问题。经济数学。2001，18(4)：1-9
3 吴荣，杜勇宏。常利率下的更新风险模型。工程数学学报。2002,19(1)：46-54
4 成世学。破产论研究综述。数学进展。2002,31(5)：404-422．
5 Willian Feller，李志阐、郑元禄译。概率论及其应用。北京，科学出版社。2003，56-58
6 钱敏平，龚光鲁。随机过程论。(第一版)北京，北京大学出版社。2000，207-267
7 潘华、高檩、李娜等。企业财务管理学。北京，中国人民公安大学出版社。2002，42-86
8 武超标。保险精算学。北京，中国统计出版社。1999，160-171
9 甘华鸣，陈宝明，姜钦华等。管理方法(下)。北京，中国国际广播出版社。2002，425-431
10 H.Yang, Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal. 1(1998), 66-79
11 T. Rolski, H. Schmidli, V. Schmidt and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley and Sons, New York, 1999.
12 Gerber, Goovaerts, Kass. On the probability and severity of ruin. ASTIN Bulletin. 1987(17), 151-163
13 Dickson D.C.M., Dos Reis A.D.E. Ruin problems and dual events. Insurance :Mathematics and Econcmics. 1994(14), 51-60
15 Sundt B, Teugels J. L. Ruin estimates under interest force. Insurance:Mathematics and Econcmics. 1995(16), 7-22
2 孙立娟，顾岚、刘立新。离散时间模型下最大赤字问题。经济数学。2001，18(4)：1-9
3 吴荣，杜勇宏。常利率下的更新风险模型。工程数学学报。2002,19(1)：46-54
4 成世学。破产论研究综述。数学进展。2002,31(5)：404-422．
5 Willian Feller，李志阐、郑元禄译。概率论及其应用。北京，科学出版社。2003，56-58
6 钱敏平，龚光鲁。随机过程论。(第一版)北京，北京大学出版社。2000，207-267
7 潘华、高檩、李娜等。企业财务管理学。北京，中国人民公安大学出版社。2002，42-86
8 武超标。保险精算学。北京，中国统计出版社。1999，160-171
9 甘华鸣，陈宝明，姜钦华等。管理方法(下)。北京，中国国际广播出版社。2002，425-431
10 H.Yang, Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal. 1(1998), 66-79
11 T. Rolski, H. Schmidli, V. Schmidt and J. Teugels, Stochastic Processes for Insurance and Finance, Wiley and Sons, New York, 1999.
12 Gerber, Goovaerts, Kass. On the probability and severity of ruin. ASTIN Bulletin. 1987(17), 151-163
13 Dickson D.C.M., Dos Reis A.D.E. Ruin problems and dual events. Insurance :Mathematics and Econcmics. 1994(14), 51-60
15 Sundt B, Teugels J. L. Ruin estimates under interest force. Insurance:Mathematics and Econcmics. 1995(16), 7-22