A risk model for insurance companies on time scales
DOI:
https://doi.org/10.31489/2026m2/149-165Abstract
This article deals with the problems of constructing and analyzing a collective risk model for an insur-
ance company when the time evolution is defined on a general time scale. The relevance of the study is
determined by the need to describe pvremium 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
probability of non-ruin under such 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 probability of non-ruin
is formulated. Conditions ensuring the correctness of the constructed model are established. It is proved
that the probability of non-ruin 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.
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