Stochastic risk processes with economic factors

在本研究计划中，我计划遵循最近的一些工作，将保险业务的盈余过程建模为两个半鞅的复合。其中，一个过程被视为没有经济因素的世界中的风险过程，另一个过程描述风险和/或无风险投资的回报。提出的随机模型涵盖了大多数有用的保险风险模型。此外，它还为统计和概率的许多结果和方法建立了一个广阔的平台。首先，我打算调查不同类型的税收对保险业务的定量影响。我的研究将主要通过有和没有纳税盈余过程的破产概率来进行。我将研究周期税、亏损结转税以及溢价税的缴纳对盈余过程破产概率的影响。我还将研究盈余过程的随机折现净损失的右尾行为，其中随机折现净损失是一段时间内通过随机折现因子的总损失的现值。本研究对我们把握金融风险对保险盈余过程的影响具有重要意义。此外，损失分布的右尾概率在风险度量中也有其应用，如风险价值和风险尾部价值，以及违约风险估计。因为除了一些非常特殊的情况，不可能找到我的问题的明确解决方案，我将使用边界和渐近作为主要工具来近似期望的概率。为此，在保险和/或金融风险的分布上施加一些条件（如次指数性、指数性和可靠性属性）将非常有帮助。

In this research program I plan to follow some recent works to model the surplus process of an insurance business as a compound of two semimartingales. Among them, one process is regarded as a risk process in a world without economic factors and the other describes returns on risky and/or risk-free investments. This proposed stochastic model covers most of the useful insurance risk models. Also, it builds a broad platform for many results and methods from statistics and probability.First I plan to investigate the quantitative impacts of different types of taxation on insurance business. My study will be conducted mainly via the ruin probabilities of the surplus process with and without tax payment. I will study periodic taxation, loss-carry-forward taxation, as well as premium tax payment on their influences on the ruin probability of the surplus process.I will also investigate the right tail behavior of the stochastic discounted net loss of the surplus process, where the stochastic discounted net loss is the present value of the aggregate loss in a period of time by a stochastic discount factor. This study is important for us to grasp the influence of financial risks on insurance surplus process. Also, the right tail probability of loss distribution has its applications in risk measures, such as Value-at-Risk and Tail-Value-at-Risk, as well as default risk estimation.Since except for some very specific cases it is impossible to find out explicit solutions to my questions, I will use bounds and asymptotics as main tools to approximate the desired probabilities. To do so, it will be very helpful to impose some conditions (such as subexponentiality, exponentiality, and reliability properties) on the distributions of the insurance and/or financial risks.