Research On Total Claims Distribution And Surplus Process In Risk Theory

There are three risk models in classical risk theory, namely short-time individualrisk model, short-time collective risk model and long-time collective risk model.The paperintroduced the definition and background of three risk models, emphatically on the last tworisk models.In short-time risk model,how to calculate the distribution of the total claimsamount is an important and centred topic.The type or property of the total distributionhas direct guidance and reference on how to rating,consulting premium and developingnew kinds of insurance.The classical calculation of the total claims amount distribution in short-time riskmodels have the method of convolution, moment generating function,character function,whichare exact methods.The approximate methods include normal distribution approxima-tion,Gamma distribution appproximation.But since Panjer introduced the recursive al-gorithm of the total claims amount distribution in 1980,many magazines began to do thiswork. The recursive algorithm has collective property to the too much policy or greatclaims amount.Because the recursive evaluation which we have refereed is suit for fewclaim number distribution,people begin to generalize the type of the claim number distri-bution.The thesis extends the Panjer(a,b,k)-class basically on previous work,then find anew distribution which named EX-Panjer(a,b,k) and the differential equation of the gen-erating moment.Besides of the type of distribution, we also have interested in multivariatecases.The thesis derive the recursive formula of the total claims distribution with one-dimension claim number and m-dimension claim severity which developed the previouspaper.and get convenient formula in practice.The recursive algorithm is useful in prac-tice.We also take a numberical example about calculation of net premium of a stop-lossinsurance contract.In long-time collective risk model,we mostly discuss the surplus process,which hasgreat value for insurance company to distribute fund and develop reinsurance plan.Thepaper discussed the surplus process in the classical model,using martingales and randomWalks we got the approximate formula about probability of that the insurer's surplusreaches an upper barrier before the ruin and its precise estimate. And we also got theexact formula of expectation of hitting times, unconditionally on the event that ruin doesnot occur before the surplus exceeds.At the same time, upper barrier of the conditionalexpectation was given if we considered ruin in practice. At last,all the works in this thesis are summarized and some prospects are alsoproposed.