17.5.3 LARGE CLAIM NUMBER LOSS PROCESSES: GENERIC CLOSED-FORM LDA MODELS WITH INSURANCE MITIGATION
The interest in geometric stable models lies in the fact that they act as exact asymptotic representations to geometric compound sums of i.i.d. random variables. Under this class of models, one may obtain closed-form results for any LDA model and any insurance policy asymptotically in the limit of large mean number of annual losses. Therefore, the application of this closed-form model is particularly of interest in risk process model settings in which one has high numbers of claims arriving each year. In OpRisk, this would typically arise in settings such as credit card fraud. An alternative place where this may be of interest is in approximation of an overall business unit or even financial institutions LDA model, which really comprises many loss processes and many different insurance policies; this can also be a setting in which we may adopt a geometric stable approximation for the overall grouped data loss process.
In these cases, we may obtain a closed-form expression (asymptotically) for the annual loss distribution of the insured process. To proceed, we first present a basic characterization of the geometric stable limiting model and then present the representations that are available for closed-form expressions of the resulting insured annual loss density and distribution, as well as tail asymptotics.