一索赔到达间隔为离散相形分布的Sparre Andersen模型的破产问题

Ruin probabilities of a continuous-time Sparre Andersen model with inter-claim times distributed as discrete phase-type

本文研究的是索赔到达时间间隔服从离散相形分布的连续时间Sparre Andersen模型的破产问题,其中索赔额分布也是离散的.首先利用向前马尔可夫技巧,把此风险过程化成逐段决定马尔可夫过程(PDMP)过程,然后借助于带有离散部分的广义生成算子得到了一个指数鞅.随之,利用鞅方法和测度变换的思想,求出了破产概率的一般表达式,破产概率的Lundberg界和Cramér-Lundberg逼近这些与经典风险模型和连续时间复合二项模型中相平行的结果.对索赔额分布为几何分布情形,得到了破产概率的明确表达式.

This paper considers a continuous-time Sparre Andersen risk model where the distribution of inter-claim times is discrete phase-type and the distribution of claim sizes is discrete. Firstly, the risk process is made into a piecewise deterministic Markov process （PDMP） by adding supplementary components from forward Markovization technique. Then we get an exponential martingale by virtue of the extended generator with a discrete part which is proposed. The theory about change of probability measure is developed for this model. The general expressions, Lundberg bounds and CramerLundberg approximations to the ruin probabilities paralleling to the ones in the compound Poisson model are obtained. And the explicit expression of the ruin probability is derived when the claim sizes distributed as geometric distribution.