随机游动的更新理论和尾渐近理论及其应用
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摘要
众所周知,随机游动理论,包括随机游动的更新理论,随机游动及相关对象的尾渐近性理论等等,历来是概率论研究的重要组成部分.它不但自身具有重要的理论研究价值,而且在排队论、风险理论、大偏差理论和分支过程等许多领域也都有重要的应用.如在风险理论中,保险公司的破产概率就可以通过随机游动的上确界大于初始资本的概率来刻画,因而随机游动理论对金融保险中的实际工作也具有一定的指导意义.
本文将从如下三个方面研究随机游动理论.
更新计数过程的基本更新定理,更一般地,随机游动生成的计数过程的基本更新定理,是更新理论的重要组成部分.对于增量是独立同分布的情形,经典的更新理论已日趋成熟.但是在实践中,相关的增量往往不是相互独立的.那么,对于带某种相依增量的随机游动生成的计数过程,是否也可以建立相应的基本更新定理?在很长一段时间内,这一问题没有实质性的进展.本文的第二章将系统地讨论这些问题.我们将给出带宽相依增量的随机游动所生成的计数过程的基本更新定理.为了证明基本更新定理,我们还将建立一个带宽相依增量的随机游动的弱大数律.在进一步的条件下,我们还给出了带宽相依增量的随机游动的强大数律.此外,我们还将使用所得的基本更新定理的结果,给出一个随机游动的随机和的精致大偏差的渐近估计,其中随机和中的随机指标是一个带宽相依增量的非标准更新计数过程.在这一章的最后部分,我们还将讨论带正飘移的随机游动生成的首超时过程的强大数律,所得结果较大程度地推广和改进了一些已有的结果.
另一方面,随机游动的尾渐近理论与其增量的分布理论有着密切的关系.分布理论,特别是重尾分布理论中的一个重要事实是,当随机游动的增量相互独立且服从相同的次指数分布时,这些增量是最大-和等价的.即若干随机变量和的尾分布等价于它们中最大一个随机变量的尾分布.如在风险理论中,这一性质可解释为保险公司破产往往是由一个大额索赔导致的.因此,这一性质受到了广泛的关注.在这一问题的研究中,人们通常要求随机游动的增量的分布属于某个特定的分布族.然而这个条件显得比较苛刻.Li和Tang (2010)[1]注意到了这个问题,他们将相应的条件加到了随机变量的最大值的分布上,从而扩大了应用的范围.但他们的研究仍如前人那样,要求最大值的分布属于次指数分布族.由于存在大量的分布属于长尾分布族和。-次指数分布族的交集,但不属于次指数分布族,所以在第三章中,我们的工作是将上述问题的研究范围进一步扩大到长尾分布族和O-次指数分布族的交集.值得指出的是,当随机游动增量的最大值的分布属于次指数分布族时,它们的和产生的振荡是最小的；或者说,当增量的最大值的分布属于长尾分布族和O-次指数分布族的交集,但不属于次指数分布族时,就可能产生大的振荡或风险.因此很有必要研究增量的最大值的分布属于长尾分布族和O-次指数分布族的交集的随机游动的相关理论.
最后,在第四章中,我们将给出随机游动理论在一类同时带有金融风险和保险风险的非标准离散时风险模型中的应用.我们将给出该类风险模型中有限时破产概率的渐近估计,其中保险风险或其与金融风险的乘积的分布可以不属于卷积等价分布族.为此,我们将先研究一个比卷积等价分布族更大的分布族,即指数分布族的性质.我们将给出指数分布族中分布的卷积封闭性和卷积的尾渐近等价的几个充要条件和一些便于验证的充分条件.由于以往的研究只给出了充分性的结果,所以我们的工作推广和改进了以往的工作.另外,以往人们得到的一些渐近等价关系,都需要相关分布属于卷积等价分布族的条件.而我们的这部分工作,从理论上并通过实例,说明上述条件不是必要的,从而较大幅度地扩大了所得结果的应用面,丰富了相应的分布理论和随机游动的尾渐近理论.
It is well known that random walk theory, including the renewaltheory, tail asymptotic theory of random walks and related objects, hasalways been the important part of probability theory. It has not only im-portant theoretical value, but also important applications in many fieldssuch as queuing system, risk theory, branching process, etc. For exam-ple, in risk theory, we may describe the ruin probability of an insurancecompany through the probability of the supremum of a random walkexceeds the initial capital. So, random walk theory also has practicalsignificance in the finance work.
This paper will investigate random walk theory from the followingthree aspects.
The elementary renewal theorems of renewal counting processes,more generally, of counting processes generated by random walks, areone of the important parts of renewal theory. For the case where theincrements are independent and identically distributed, the classical re-newal theory has been mature. However, in practice, related randomvariables are usually not independent to each other. So, for the countingprocesses generated by random walks with some dependent increments,is it possible to establish corresponding results? For a long time, therehas not been any substantial progress in this field. In Chapter2, wewill discuss these problems systematically. We will deliver elementaryrenewal theorems of counting processes generated by random walks withwidely dependent increments. In proving them, we will first establisha weak law of large numbers for random walks with widely dependentincrements. Under some further conditions, a strong law of large num-bers for the same object will also be established. Besides, the obtainedresults will be applied to presenting an asymptotic estimate for precise large deviation of a random sum, where the random index is a nonstan-dard renewal counting process with widely dependent increments. Atthe end of this chapter, we will discuss a strong law of large numbersfor the first passage time process of random walks with a tend. Theobtained results will extend and improve the existing results to a greatextent.
On the other hand, tail asymptotic theory of random walks hasclose relation with distribution theory. An important fact of distribu-tion theory, in particular, heavy-tailed distribution theory is that, whenthe increments of a random walk are independent and have commonsubexponential distribution, these increments are max-sum equivalent,which means that the tail probability of the sum of some random vari-ables is equivalent to that of the maximum of these random variables. Inrisk theory, this property may be interpreted as the ruin of an insurancecompany is usually caused by a big claim. Hence this property attractsmuch attention. In the study, the distributions of the increments areusually assumed to be in some specific distribution class. However, thiscondition seems too strict. Li and Tang (2010)[1]noticed this problemand they imposed the corresponding condition on the distribution of themaximum of the increments, thus expanding the scope of application.However, as usual, they assumed that the distribution of the maximumof the increments were subexponential. Since there exist many distribu-tions are long-tailed and O-subexponential, but not subexponential, sowe will in Chapter3extend the above-mentioned investigation to the in-tersection of the long-tailed distribution class and the O-subexponentialdistribution class. It is worth mentioning that when the distributionof the maximum of the increments is subexponential, the oscillationcaused by the sum of the increments is the smallest. In other words,when the distribution of the maximum of the increments is long-tailedand O-subexponential, but not subexponential, there may appear bigoscillation or risk. Thus, it is necessary to investigate related theory of random walks with the distribution of the maximum of the incrementsis long-tailed O-subexponential.
At last, in Chapter4, we will apply the random walk theory to aclass of non-standard discrete-time risk models, which have both financeand insurance risks. We will present asymptotic estimates for finite-timeruin probabilities in this kind of risk models, where the distributions ofthe insurance risk or the product of the two risks may not belong tothe convolution equivalence distribution class. In doing so, we will firststudy the properties of a larger distribution class than the convolutionequivalence distribution class, namely the exponential distribution class.We will derive some equivalent conditions and some sufcient conditionseasy to check for closure property under convolution and tail equivalenceof distributions from the exponential distribution class. Since the pastwork presents only sufcient results, so our work generalizes some exist-ing sufcient conditions. Besides, in the study of tail equivalence, it iscommon to assume that the related distributions belong to the convo-lution equivalence distribution class. Our results show that the abovecondition is not necessary, thus expanding its application and enrich-ing the corresponding distribution theory and tail asymptotic theory ofrandom walks substantially.
本文将从如下三个方面研究随机游动理论.
更新计数过程的基本更新定理,更一般地,随机游动生成的计数过程的基本更新定理,是更新理论的重要组成部分.对于增量是独立同分布的情形,经典的更新理论已日趋成熟.但是在实践中,相关的增量往往不是相互独立的.那么,对于带某种相依增量的随机游动生成的计数过程,是否也可以建立相应的基本更新定理?在很长一段时间内,这一问题没有实质性的进展.本文的第二章将系统地讨论这些问题.我们将给出带宽相依增量的随机游动所生成的计数过程的基本更新定理.为了证明基本更新定理,我们还将建立一个带宽相依增量的随机游动的弱大数律.在进一步的条件下,我们还给出了带宽相依增量的随机游动的强大数律.此外,我们还将使用所得的基本更新定理的结果,给出一个随机游动的随机和的精致大偏差的渐近估计,其中随机和中的随机指标是一个带宽相依增量的非标准更新计数过程.在这一章的最后部分,我们还将讨论带正飘移的随机游动生成的首超时过程的强大数律,所得结果较大程度地推广和改进了一些已有的结果.
另一方面,随机游动的尾渐近理论与其增量的分布理论有着密切的关系.分布理论,特别是重尾分布理论中的一个重要事实是,当随机游动的增量相互独立且服从相同的次指数分布时,这些增量是最大-和等价的.即若干随机变量和的尾分布等价于它们中最大一个随机变量的尾分布.如在风险理论中,这一性质可解释为保险公司破产往往是由一个大额索赔导致的.因此,这一性质受到了广泛的关注.在这一问题的研究中,人们通常要求随机游动的增量的分布属于某个特定的分布族.然而这个条件显得比较苛刻.Li和Tang (2010)[1]注意到了这个问题,他们将相应的条件加到了随机变量的最大值的分布上,从而扩大了应用的范围.但他们的研究仍如前人那样,要求最大值的分布属于次指数分布族.由于存在大量的分布属于长尾分布族和。-次指数分布族的交集,但不属于次指数分布族,所以在第三章中,我们的工作是将上述问题的研究范围进一步扩大到长尾分布族和O-次指数分布族的交集.值得指出的是,当随机游动增量的最大值的分布属于次指数分布族时,它们的和产生的振荡是最小的；或者说,当增量的最大值的分布属于长尾分布族和O-次指数分布族的交集,但不属于次指数分布族时,就可能产生大的振荡或风险.因此很有必要研究增量的最大值的分布属于长尾分布族和O-次指数分布族的交集的随机游动的相关理论.
最后,在第四章中,我们将给出随机游动理论在一类同时带有金融风险和保险风险的非标准离散时风险模型中的应用.我们将给出该类风险模型中有限时破产概率的渐近估计,其中保险风险或其与金融风险的乘积的分布可以不属于卷积等价分布族.为此,我们将先研究一个比卷积等价分布族更大的分布族,即指数分布族的性质.我们将给出指数分布族中分布的卷积封闭性和卷积的尾渐近等价的几个充要条件和一些便于验证的充分条件.由于以往的研究只给出了充分性的结果,所以我们的工作推广和改进了以往的工作.另外,以往人们得到的一些渐近等价关系,都需要相关分布属于卷积等价分布族的条件.而我们的这部分工作,从理论上并通过实例,说明上述条件不是必要的,从而较大幅度地扩大了所得结果的应用面,丰富了相应的分布理论和随机游动的尾渐近理论.
It is well known that random walk theory, including the renewaltheory, tail asymptotic theory of random walks and related objects, hasalways been the important part of probability theory. It has not only im-portant theoretical value, but also important applications in many fieldssuch as queuing system, risk theory, branching process, etc. For exam-ple, in risk theory, we may describe the ruin probability of an insurancecompany through the probability of the supremum of a random walkexceeds the initial capital. So, random walk theory also has practicalsignificance in the finance work.
This paper will investigate random walk theory from the followingthree aspects.
The elementary renewal theorems of renewal counting processes,more generally, of counting processes generated by random walks, areone of the important parts of renewal theory. For the case where theincrements are independent and identically distributed, the classical re-newal theory has been mature. However, in practice, related randomvariables are usually not independent to each other. So, for the countingprocesses generated by random walks with some dependent increments,is it possible to establish corresponding results? For a long time, therehas not been any substantial progress in this field. In Chapter2, wewill discuss these problems systematically. We will deliver elementaryrenewal theorems of counting processes generated by random walks withwidely dependent increments. In proving them, we will first establisha weak law of large numbers for random walks with widely dependentincrements. Under some further conditions, a strong law of large num-bers for the same object will also be established. Besides, the obtainedresults will be applied to presenting an asymptotic estimate for precise large deviation of a random sum, where the random index is a nonstan-dard renewal counting process with widely dependent increments. Atthe end of this chapter, we will discuss a strong law of large numbersfor the first passage time process of random walks with a tend. Theobtained results will extend and improve the existing results to a greatextent.
On the other hand, tail asymptotic theory of random walks hasclose relation with distribution theory. An important fact of distribu-tion theory, in particular, heavy-tailed distribution theory is that, whenthe increments of a random walk are independent and have commonsubexponential distribution, these increments are max-sum equivalent,which means that the tail probability of the sum of some random vari-ables is equivalent to that of the maximum of these random variables. Inrisk theory, this property may be interpreted as the ruin of an insurancecompany is usually caused by a big claim. Hence this property attractsmuch attention. In the study, the distributions of the increments areusually assumed to be in some specific distribution class. However, thiscondition seems too strict. Li and Tang (2010)[1]noticed this problemand they imposed the corresponding condition on the distribution of themaximum of the increments, thus expanding the scope of application.However, as usual, they assumed that the distribution of the maximumof the increments were subexponential. Since there exist many distribu-tions are long-tailed and O-subexponential, but not subexponential, sowe will in Chapter3extend the above-mentioned investigation to the in-tersection of the long-tailed distribution class and the O-subexponentialdistribution class. It is worth mentioning that when the distributionof the maximum of the increments is subexponential, the oscillationcaused by the sum of the increments is the smallest. In other words,when the distribution of the maximum of the increments is long-tailedand O-subexponential, but not subexponential, there may appear bigoscillation or risk. Thus, it is necessary to investigate related theory of random walks with the distribution of the maximum of the incrementsis long-tailed O-subexponential.
At last, in Chapter4, we will apply the random walk theory to aclass of non-standard discrete-time risk models, which have both financeand insurance risks. We will present asymptotic estimates for finite-timeruin probabilities in this kind of risk models, where the distributions ofthe insurance risk or the product of the two risks may not belong tothe convolution equivalence distribution class. In doing so, we will firststudy the properties of a larger distribution class than the convolutionequivalence distribution class, namely the exponential distribution class.We will derive some equivalent conditions and some sufcient conditionseasy to check for closure property under convolution and tail equivalenceof distributions from the exponential distribution class. Since the pastwork presents only sufcient results, so our work generalizes some exist-ing sufcient conditions. Besides, in the study of tail equivalence, it iscommon to assume that the related distributions belong to the convo-lution equivalence distribution class. Our results show that the abovecondition is not necessary, thus expanding its application and enrich-ing the corresponding distribution theory and tail asymptotic theory ofrandom walks substantially.
引文
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