A risk model for insurance companies on time scales

This article deals with the problems of constructing and analyzing a collective risk model for an insurance company when the time evolution is defined on a general time scale. The relevance of the study is determined by the need to describe premium accumulation and claim payments occurring at discrete or irregular time instants within a unified analytical framework. The characteristic features of the classical risk model and its extension to time scales are analyzed, and the need to investigate the behavior of the non-ruin probability under such a generalization is identified and justified. On the basis of the study, the authors construct an analogue of the classical model on time scales and derive a dynamic equation for the distribution of the number of claims. An integral equation on a time scale for the non-ruin probability is formulated. Conditions ensuring the correctness of the constructed model are established. It is proved that the non-ruin probability defined on a family of time scales converges pointwise to the corresponding probability in the classical continuous-time risk model as the graininess function tends to zero. It is shown that the proposed approach provides a rigorous justification of the transition from discrete to continuous risk models.