Insurance Risk Theory Based On Entrance Processes And Its Applications

Risk theory,one of the important branches of applied probability,plays a key role in the area of insurance.In this thesis,we propose another new insurance risk model based on the policy entrance processes,which is called LIG model for short.Basing on probability limit theory,insurance risk theory,heavy-tailed distribution and reliability,a systematic study is carried out for this new model,including its asymptotical properties, large deviations,upper bound for ruin probabilities,and estimation of ruin probabilities, etc.As an application,the relationship between LIG model and shock models is also discussed,then the weak convergence and the lifetime property of the extended shock model are investigated.The significance of our conclusions in this thesis is not only that it enriched the content of insurance risk theory,but also that achieved a valuable study on the theory and application of random sum.This thesis contains 7 chapters.Chapter 1 reviews the history of classical risk theory and gives some important results.Various extensions of the classical risk models are also discussed.By analyzing the actual situation of insurance company,it is easy to conceive that the claim number process is virtually driven by the policy entrance process,since whenever the insurer issues a policy,it will have to burden the potential claims entitled by the policy.So we construct a new risk model based on the entrance process(hereafter,LIG model),which not only records the claim process but also details the entrance process of each policy.Obviously,such model can describe the dynamics of the profit/loss process more accurately.The conclusions of this thesis also indicates that the LIG model is an extension and development of the classical risk models.In Chapter 2,we set up a LIG model with interest force and investigate its limiting properties.Assuming that the entrance process is a non-homogeneous Poisson,the risk process then has the infinitely divisible property,from which we obtain some results on the weak convergence for the model with interest rate.First,using Feller's canonical measure theory,we obtain sufficient conditions for the normalized risk process to converge weakly to stable laws.Secondly,further assuming that the claim distribution has regular varying tails,we find the conditions under which the normalized risk process converges weakly to stable laws.Finally,we apply the previously obtained results to a risk process where claim distributions with different maturities have different regular varying tails,and show that the resulting limiting distribution is ultimately determined by the one with the smallest index.This result is of both theoretical and practical significance.The precise large deviation for LIG model under the assumption that the claim distribution is heavy tailed is established in Chapter 3.As we known,precise large deviation is one of the main topics in insurance mathematics,it has been studied well for the random sum in the classical risk models.We carry a similar study for LIG model in this chapter by assuming that claim distributions belong to the C family,a broad family of heavy-tailed distributions.We first obtain precise large deviation results for a single insurance product, and then extend the results to the case of several different insurance products.Ruin probability is one of the classical themes of insurance risk,thus Chapter 4 is devoted to the ruin probabilities of LIG model in the case of that the claim distributions are light tailed.After a briefly review on the classical Lundberg-Cramer risk theory,we first employ martingale method to obtain an upper bound for ruin probability for a special class of risk models,and then discuss the upper bound of ruin probability for LIG model over the entire time horizon.Similar results over finite time horizon are also dealt with.Parallel,the ruin probabilities for LIG model in the case of that the claims have heavy-taild distributions are studied in Chapter 5.We derive the asymptotical expressions for ruin probabilities over finite time horizon when the claim distributions belong the subexponential family.Then,the corresponding results when the entrance process is replaced by a renewal process and/or when interest rate is taken into account are also achieved.The topic of Chapter 6 is the applications of LIG model to reliability.By a comparison on our LIG model and classical shock models,it shows that there is a close relation between the two different types of models,and we can naturally apply the results obtained for LIG model to shock models.Then weak convergence results for the shock process,some good estimates for the failure probabilities and the lifetime of shock model are obtained.Chapter 7 provides an outlook for further researches.