On risk models under Markovian environments and their applications

保险业务的本质是保险人承诺支付被保险人所承保的所有索赔，以换取保费。此外，保险公司还承诺一定的资本（盈余），以确保即使在特殊情况下也能信守承诺。我们研究了保险公司内的一个投资组合，该投资组合在所考虑的时期开始时分配了正的初始风险准备金（基金）。风险准备金按固定利率记入贷方保费，并借方到达时到期的索赔金额。当索赔到达采用泊松过程建模时，风险准备金过程就是所谓的经典风险过程。我们担心，当风险准备金低于零时，保险公司可能没有足够的资金来支付其负债（索赔）。更准确地说，当这种情况发生时，我们想知道在有限的时间内发生这种情况的可能性有多大，以及最坏的情况会是什么。这些是我感兴趣的基本风险相关数量。

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.