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，以换取保险费。此外，保险公司还承诺一定的资本，以确保即使在灾难性事件或金融危机等特殊情况下也能履行承诺。我研究了保险公司内部的投资组合的准备金，在所考虑的时期开始时具有正的初始资金分配。风险准备金按固定费率贷记保险费，借记到达时到期的索赔金额。当索赔到达服从泊松过程时，风险准备金过程就是所谓的经典风险模型。投保人和监管机构担心保险公司可能没有足够的资金来支付其负债，这是当风险准备金福尔斯零或某一低水平时的情况。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量的例子。在实践中，保险费率随时间恒定且索赔到达根据恒定强度率（例如，Poisson过程）并不现实。必须考虑到一些实际因素，特别是考虑到某些不均匀性。例如，在建模与灾难相关的保险风险（例如地震和飓风计数和损失）时，通常会开发考虑时间和位置波动以及背景风险因素的保险模型。在这个建议中，我打算研究的影响，潜在的风险波动的时间和地点的保险公司的准备金，和偿付能力依赖的问题，通过评估的动态的风险准备金的保险公司的空间和时间的变化。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的建议保险风险模型，可以提高建模和参数估计的灵活性。预计研究成果将提供一个有用的建模方法，分析真实的保险索赔数据，可能是特别重要的保险公司。例如，对受空间影响的飓风、地震和海啸进行随机建模，可以帮助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。本文的研究成果将对精算学中的风险理论和可信性理论以及应用概率领域做出新的贡献。这是时空统计的一种新应用，应导致精算学的创新。

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.