On risk models under Markovian environments and their applications

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The nature of the insurance business is the promise by the insurer to pay all claims of the insured that are covered in exchange for a policy premium. In addition, the insurer commits some capital (surplus) to assure that the promise will be kept even under special circumstances. We study a portfolio within the insurance company with a positive allocated initial risk reserve (fund) at the beginning of the considered period. The risk reserve is credited premiums with a constant rate and debited amounts of claims due at the time of the arrival. When the claim arrivals are modeled by a Poisson process, the risk reserve process is the so-called classical risk process. We are concerned about the possibility that the insurer does not have enough funds to pay its liabilities (claims), which is the case when the risk reserve falls below zero. More precisely, when this happens we want to know what the odds are for this to happen within a finite time period and what the worse-case scenario would be. These are the basic risk-related quantities of my interest.