Research On Ruin Probability In Markov-Modulated Risk Models

After the continuous efforts of many actuarial scholars made, ruin theory, one ofthe three major technical of modern risk management and control, has been developingprofoundly in recent years. Today, the traditional risk models can not portray thepractical modes of financial operations, and various risk modeling and riskmeasurement techniques have been coming into being. Joining the consideration of avariety of economic factors, based on various generalized risk models, has been one ofthe main research focus in modern ruin theory. In this paper, we mainly study theimpacts that constant interest force and heavy-tailed claims have on ruin probability, ina kind of semi-Markov risk models.In Chapter2, this paper mainly considers a kind of Homogeneous semi-Markovrisk models, which has been introducted in Albrecher and Boxma（2005）, with constantinterest force. By means of basic probabilistic analysis, we first derive a matrixintegro-differential equation satisfied by the survival probabilities, and then deriveanother corresponding matrix integral equation, whose format is much conciser.In Chapter3, this paper considers a simplified case of Chapter2, that is a two-statesemi-Markov risk model, with heavy-tailed claims simultaneously. When the systemreaches the state1, the claim amounts will follow the exponential distribution withparameterβ, and the following inter-claim time will follow the exponential distributionwith parameter λ₁; while the claim sizes will follow the heavy-tailed distribution F₂, aslong as the system arrives at the state2, and the following inter-claim time will followthe exponential distribution with parameter λ₂.By means of Laplace-Transforms, wefirst convert the system of integro-differential equations, satisfied by the ruin probabilit-ities, into a second variable coefficient linear ordinary differential equation（LODE）, andthen derive the corresponding solutions in series form. Finally, we make use of theKaramata Tauberian Theorem and the Heaviside Operational Principle to obtain theasymptotic behaviors of the ultimate ruin probabilities. It is shown that the asymptoticbehaviors of ruin probabilities depend only on the state2with heavy-tailed claims, noton the state1with exponential claim sizes. Moreover, these results can also be extendedto more states case. At that point, the Laplace Transforms of the ruin probabilities,corresponding to any states, will satisfy a variable coefficient LODE with higher orders,at the same time, the solving processes will become more cumbersome. In Chapter4, we consider a stationary SMRM, which is a particular case of Chapt-er2, with additional assumptions on the distribution of the initial state and the transitionprobabilities of the imbedded Markov chain. Taking into account of the conditionalindependence between the claim sizes and the inter-claim times, it is shown that this sta-tionary SMRM belongs to a renewal risk model, in essential. Thus, it directly derivesthe consistent asymptotic behavior of the finite time ruin probability of this model, bydrawing some relative results of Hao and Tang（2008）. In addition, we can also obtainthe consistent asymptotic expression of the ultimate ruin probability, by adding somemild constraints on the distributions of the claim sizes and the inter-claim times.