Space-Time Risk Processes with Applications in Non-Life Insurance

保险业务的本质是保险人承诺支付所有承保的索赔，以换取保单保费。此外，保险公司还会投入一些资金，以确保即使在灾难性事件或金融危机等特殊情况下也能兑现承诺。我研究了保险公司内部投资组合的准备金，在考虑的期间开始时，有一个正分配的初始资金。风险准备金是指按固定利率记入的保费和到货时到期的债权的借方金额。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模型。保险公司和监管机构担心保险公司可能没有足够的资金来支付其债务，这是风险准备金低于零或某低水平时的情况。更确切地说，当这种情况发生时，重要的是要知道在有限的时间内发生这种情况的几率是多少，以及最坏的情况是什么。这些是保险从业人员感兴趣的基本风险相关数量的例子。**在实践中，随着时间的推移，保费率是恒定的，索赔到达率是恒定的（例如，遵循泊松过程），这种情况不太现实。必须考虑到一些实际的考虑，特别是考虑到某些不均匀性。例如，在模拟与灾难有关的保险风险，如地震和飓风的数量和损失时，通常开发的保险模型既考虑时间和地点的波动，也可能考虑背景风险因素。**在本提案中，我打算通过评估保险公司风险准备金随时空变化的动态，研究潜在风险在时间和地点上的波动对保险公司准备金的影响，以及偿付能力依赖问题。在这里，空间建模将包括汽车保险中基于位置的风险因素和农业保险中地理上不同的损失。我还打算利用完善的时空统计数据来建立保险风险模型，这可以增强建模和参数估计的灵活性。**预计研究结果将为分析真实保险索赔数据提供有用的建模方法，这可能对保险公司特别重要。例如，对受空间影响的飓风、地震和海啸进行随机建模，可能有助于非寿险公司更准确地评估气候和空间相关风险，从而公平地为其产品定价。本研究的理论或数值结果将对精算科学中的风险理论和可信度理论以及应用概率论领域做出新的贡献。这代表了时空统计的一种新应用，并将导致精算科学应用的创新。

The nature of the insurance business is the promise by the insurer to pay all covered claims in exchange for a policy premium. In addition, the insurer commits some capital to assure that the promise will be kept even under special circumstances such as catastrophic events or financial crises. I study the reserve of a portfolio within the insurance company with a positive allocated initial fund at the beginning of the considered period. The risk reserve is credited premiums at a constant rate and debited amounts of claims due at the time of arrival. When the claim arrivals are modeled by a Poisson process, the risk reserve process is the so-called classical risk model. Insureds and regulators are concerned about the possibility that the insurer does not have enough funds to pay its liabilities, which is the case when the risk reserve falls below zero, or a certain low level. More precisely, when this happens it is important 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 examples of basic risk-related quantities of insurance practitioners' interest. **In practice, the cases where the premium rate is constant over time and the claim arrivals according to a constant intensity rate (e.g., following a Poisson process) are not so realistic. Some practical considerations, in particular to allow for certain inhomogeneity, have to be taken into account. For example, in modeling catastrophe-related insurance risks such as earthquake and hurricane counts and losses, it is common to develop insurance models that take into account both fluctuations in time and location and perhaps also background risk factors. **In this proposal, I intend to study the effect of the underlying risk fluctuations in time and location to the insurer's reserve, and the solvency-dependent problems by evaluating the dynamics of the risk reserve of the insurance company subject to spatial and temporal variations. Here the modeling in space will incorporate such as the location-based risk factors in automobile insurance and geographically varying losses in agricultural insurance. I also intend to make the use of the well-established spatial-temporal statistics for the proposed insurance risk models that could enhance the flexibility of the modeling and parameter estimations. **It is anticipated that the research outcomes would provide a useful modeling approach for analyzing real insurance claims data that may be of particular importance to insurance companies. For example, stochastically modeling of spatially affected hurricanes, earthquakes and tsunamis, may help non-life insurance companies in assessing climatological and spatial related risks more accurately and hence fairly pricing their products. Theoretical or numerical results obtained from this research would make novel contributions to risk theory and credibility theory in actuarial science as well as the applied probability field. This represents a novel application of spatial-temporal statistics and should lead to innovation for use in actuarial science.