Ruin and dividend measures in the renewal dual risk model

TL;DR

This paper extends the dual risk model with renewal processes, especially Erlang distributions, to analyze dividends and ruin probabilities, providing explicit formulas and numerical examples for these quantities.

Contribution

It introduces a method for analyzing the dual risk model with renewal processes, including explicit formulas for dividends and ruin probabilities, applicable beyond the classical Poisson case.

Findings

Explicit formulas for dividend probabilities and distributions.

Analysis of ruin and dividend counts in renewal models.

Numerical examples illustrating the theoretical results.

Abstract

In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under study have been the calculation of expected discounted future dividends and optimal strategies, where the dividend calculation have been done on aggregate. We can find works, at first using the classical compound Poisson model, then some examples of other renewal Erlang models. Knowing that ruin is ultimately achieved, we find important that dividends should be evaluated on an individual basis, where the early dividend contribution for the aggregate are of utmost importance. From our calculations we can really see how much important is the contribution of the first dividend. Afonso et al. (2013) had worked similar problems for the classical compound Poisson dual model. Besides that we find explicit formulae for both the probability of getting a dividend and the distribution of the amount of a single dividend. We still work the probability distribution of the number of gains to reach a given upper target (like a constant dividend barrier) as well as for the number of gains down to ruin. We complete the study working some illustrative numerical examples that show final numbers for the several problems under study.