Research On The Ruin Probability On Two Risk Models Respectively With Heavy-tailed And Light-tailed Distribution

In the actuarial industry,the ruin probability,as an important index to measure the operating status of a company,is widely concerned by experts and scholars.In this paper,we study the ruin probability of two kinds of risk models.The first kind of risk model assumes that both premium and claim random variables follow a light-tailed distribution and the counting process is Poisson-Geometric process,includes two one-dimensional risk models with Brown perturbation and without Brown perturbation(after degeneration).The second model is a two dimensional renewal risk model,assumes that both the primary claim and the delayed claim follow the heavy-tailed distribution(subexponential distribution),and the primary claim has FGM dependent structure.Both models are further supplements and innovations to risk models on the basis of existing researches.On the one hand,the upper bound estimate of the ruin probability of the first class risk model with Brown perturbation and the corresponding adjustment coefficient(also known as Lundberg index)are obtained by using the related theory of martingale and the stopping time technique.On the other hand,the property of ruin probability of the first model without Brown perturbation is analyzed from the point of view of the equation,and a differential and integral equation which satisfies the survival probability is obtained,and the exact solution of ruin probability is obtained under the condition that the random variables such as claim follow the exponential distribution.Because of the special property of the heavy-tailed distribution,the adjustment coefficient of the risk model often does not exist,and the traditional method of studying the ruin probability using martingale and stopping time theory is invalid.For the above reasons,many scholars turn to study the speed of ruin probability approaching 0 when the initial capital tends to infinity,and analyze the nature of ruin probability from another perspective.Therefore,for the second type of risk model,the paper considers the claim under the heavy-tailed distribution(subexponential distribution)to study the asymptotic expression of the ruin probability when the initial capital tends to infinity.By using the properties of subexponential distribution,relevant preliminary lemmas and some analytical techniques,the asymptotic formula of ruin probability is successfully derived.