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

保险业务的性质是保险公司承诺支付所有承保索赔，以换取保单保费。此外，保险公司承诺投入一些资本，以确保即使在灾难性事件或金融危机等特殊情况下也会信守承诺。我研究了保险公司投资组合的准备金，在考虑的期间开始时，初始资金是正分配的。风险准备金按不变费率记入保费贷方，并记入到达时到期索赔金额的借方。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没有足够的资金来偿还债务的可能性，当风险准备金降至零以下或某个较低水平时就是这种情况。更准确地说，当这种情况发生时，重要的是知道这种情况在有限的时间内发生的可能性有多大，以及最糟糕的情况会是什么。这些都是保险从业者兴趣中与风险相关的基本数量的例子。**在实践中，保险费率随着时间的推移是恒定的，索赔按照恒定的强度费率到达的情况(例如，遵循泊松过程)并不那么现实。必须考虑到一些实际的考虑因素，特别是考虑到某些不同质性。例如，在对地震和飓风计数和损失等与巨灾有关的保险风险进行建模时，通常会开发既考虑时间和地点的波动，又可能还考虑背景风险因素的保险模型。**在这项建议中，我打算通过评估保险公司风险准备金随时间和时间变化的动态，来研究潜在风险在时间和时间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的地理变化损失。我还打算为所建议的保险风险模型利用成熟的时空统计，以增强建模和参数估计的灵活性。**预期研究成果将为分析可能对保险公司特别重要的真实保险索赔数据提供有用的建模方法。例如，对受空间影响的飓风、地震和海啸进行随机建模，可能有助于非人寿保险公司更准确地评估与气候和空间有关的风险，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领域的风险理论和可信度理论以及应用概率领域做出新的贡献。这代表了时空统计的一种新的应用，并应导致在精算科学中使用的创新。

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.