On a penalty function in the Erlang renewal dual risk model under independent randomised observations

We consider the dual risk model with financial applications, where random gains (or claims, in the context of the primal insurance risk model) occur according to a renewal process. Owing to the mathematical robustness of the model, we do not need to impose the usual income condition, as shown by Alcoforado et al. (Ruin and dividend measures in the renewal dual risk model. Methodology and Computing in Applied Probability, 24(2), 537–569, (2022)), despite the ruin probability being equal to one if the condition is not met. We introduce a Gerber–Shiu type of penalty function applied to the dual model and assume that randomised observations are in place, as in Albrecher et al. (Randomized observation periods for the compound Poisson risk model: The discounted penalty function. Scandinavian Actuarial Journal, 2013(6), 424–452, (2013)) for the primal or classical compound Poisson insurance risk model. We go further than these authors by studying a renewal risk process using a different and improved approach, specifically mentioning the Erlang(n,λ) renewal model with an independent observational Poisson process. Under this model, as with Albrecher et al. (2013), ruin can only occur if it is observed; that is, the risk process may fall below zero, but ruin does not occur if the surplus recovers before it is observed. For our new developments, we derive integral and integro-differential equations, from which we analyse solutions in various cases, and, in particular, solve some examples analytically in full. We also present figures and graphs based on selected numerical examples. Where applicable, we draw connections to the models and solutions developed by the aforementioned authors. In relation to that primal model, we provide further developments and demonstrate how our approach can be used to derive corresponding equations and highlight similarities in their solutions.