Tail behavior of supremum of a random walk when Cramér’s condition fails

We investigate tail behavior of the supremum of a random walk in the case that Cramer＇s condition fails, namely, the intermediate case and the heavy-tailed ease. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.     

On the distribution of cumulative Parisian ruin

We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyse for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.     

A note on the convexity of ruin probabilities

Conditions for the convexity of compound geometric tails and compound geometric convolution tails are established. The results are then applied to analyze the convexity of the ruin probability and the Laplace transform of the time to ruin in the classical compound Poisson risk model with and without diffusion. An application to an optimization problem is given.     

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A Markov observation model with dividend is defined and the interpretation of the practical significance is given. We try to use an irreducible and homogeneous discrete-time Markov chain to modulate the inter-observation times and embed a dividend strategy. In the Markov observation model with dividend, a system of liner equations for the expected discounted value of dividends until ruin time is derived. Moreover, an explicit expression is obtained and proved. Finally, some interesting properties are illustrated by numerical analysis and by comparing with the complete compound binomial model with dividend.     

残余应力对压力容器破坏模式的影响分析

破前漏(简称LBB)是压力容器、核电站设备结构设计与评价中的一个重要准则.表面裂纹准静态扩展的几何形貌变化规律的预测是破前漏(LBB)评判十分重要的课题之一.本文对特定焊接残余应力场加载作用下,含三维表面裂纹的压力容器模型,用有限元软件(ABAQUS)进行了表面裂纹准静态扩展模拟计算,得到在此残余应力场作用下应力强度因子沿裂纹前缘的分布规律.结合外载引起的应力强度因子,就可以判别裂纹的扩展形貌,从而判断结构是否满足LBB要求.     

On a partial integrodifferential equation of Seal’s type

In this paper we generalize a partial integrodifferential equation satisfied by the finite time ruin probability in the classical Poisson risk model. The generalization also includes the bivariate distribution function of the time of and the deficit at ruin. We solve the partial integrodifferential equation by Laplace transforms with the help of Lagrange’s implicit function theorem. The assumption of mixed Erlang claim sizes is then shown to result in tractable computational formulas for the finite time ruin probability as well as the bivariate distribution function of the time of and the deficit at ruin. A more general partial integrodifferential equation is then briefly considered.     

带连续变利率风险模型最终破产概率上界

考虑了带二元连续变利息力的Sparre Andersen风险模型.研究了积累值盈余过程的表达式与性质;在利率递增环境下,利用推广后的调节系数方程组与递归技术推导了最终破产概率的上界,结论表明得到的破产概率上界是更为一般的Lundberg指数上界.     

Erlang(2)扩散风险模型的Gerber-Shiu期望折现惩罚函数

将由布朗运动刻画的随机干扰项加入到Erlang(2)风险模型中,在模型中引入了由Gerber和Shiu定义的期望折现惩罚函数,并给出了这类模型的Gerber-Shiu函数所满足的积分微分方程.     

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In this paper, for a kind of risk models with heavy-tailed and delayed claims, we derive the asymptotics of the infinite-time ruin probability and the uniform asymptotics of the finite-time ruin probability. The numerical simulation results are also presented. The results of theoretical analysis and numerical simulation show that the influence of the delay for the claim payment is nearly negligible to the ruin probability when the initial capital and running-time are all large.     

On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times

The structural properties of the moments of the time to ruin are studied in dependent Sparre Andersen models. The moments of the time to ruin may be viewed as generalized versions of the Gerber–Shiu function. It is shown that structural properties of the Gerber–Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in the model of Willmot and Woo (2012), which has Coxian interclaim times and arbitrary time-dependent claim sizes. Structural quantities needed to determine the moments of the time to ruin are specified under this model. Numerical examples illustrating the methodology are presented.