Analytic Form And Inequalities For Ruin Probabilities

The distribution of ruin time, the surplus before ruin, the deficit at ruin as well as their joint distribution, ruin probabilities and other ruin related quantities are the main research problems in ruin theory. Ruin probability is the main concern in this paper. It is often hard to obtain an analytic form of ruin probability. Upper bounds or approximation of it are commonly derived, but for some special claim size distribution, closed expression of ruin probability can be given.It is well-known that the ruin probability of the classical risk model, ordinary renewal risk model with phase-type claims is of the tail of another phase-type dis-tribution. Here in chapter4, a new proof of this result is presented through the application of the closure properties of phase-type distribution. We extend the re-sult to the risk model with batch arrival.In chapter5, ruin probability of risk model with a Markov chain interest and excess of loss reinsurance is studied. Recursive and integral equations for ruin prob-abilities are given through probabilistic argument. Methods of induction and mar-tingale are used to obtain two upper bounds for the ruin probabilities. Both bounds are sharper than the classical Lundberg one.