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

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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.