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The probability of ruin in a kind of Markov-modulated risk model. (English) Zbl 1199.91075
Summary: In this paper, the probability of ruin is investigated in a Markov-modulated risk model, in which the claim inter-arrives and claim sizes are influenced by an external Markovian process, and the premium rate depends on the external Markovian process and the level of company’s reserves. We consider not only that stochastic environment influences the insurance company, but also that the insurance company adjusts premium according to the level of reserves to attract new customers. So the risk model is closer to reality and more accessible to be applied. By using the backward differential argument and the Markov property of the external process, we derive the integral equation satisfied by the probability of ruin. Further, we solve the equation by Laplace transforms. Finally, an example is given to illustrate the feasibility and effectiveness of the obtained theoretical results.
MSC:
91B30Risk theory, insurance
60K37Processes in random environments