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的所有索赔，以换取保单保费。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兑现承诺。我们研究了保险公司内部的一个投资组合，在所考虑的期间开始时，该投资组合的初始风险准备金（基金）为正。风险准备金是以固定利率记入保费，并借记到货时到期的索赔金额。当索赔到达用泊松过程建模时，风险准备金过程就是所谓的经典风险过程。我们担心保险公司可能没有足够的资金来支付其债务（索赔），这是风险准备金低于零的情况。更准确地说，当这种情况发生时，我们想知道它在有限时间内发生的几率是多少以及最坏的情况是什么。这些是我的兴趣中与风险相关的基本数量。

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.