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. (Insur Math Econ, 53(3), 906–918, 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.

在这份手稿中，我们考虑了金融应用的双重风险模型，其中随机收益发生在更新过程中。我们特别使用 Erlang( n) 间隔时间的共同分布的情况，从那里很容易理解我们的方法或程序可以推广到矩阵指数家庭情况下的其他情况。我们处理涉及未来红利和破产的几个不同的问题。我们还表明，即使不满足通常的收入条件，我们的结果也是有效的。在二元模型下的大多数已知工作中，研究的主要目标是计算预期的未来折现红利和最优策略，其中红利计算是在总体上完成的。我们可以找到作品，首先使用经典的复合泊松模型，然后是其他更新的 Erlang 模型的一些示例。知道最终会毁灭，我们发现应该根据个人情况评估红利，这一点很重要，其中，对总体的早期股息贡献至关重要。从我们的计算中，我们可以真正看到第一次股息的贡献有多么重要。阿方索等人。（保险数学经济，53 (3), 906–918, 2013) 为经典复合泊松对偶模型解决了类似的问题。除此之外，我们还为获得股息的概率和单次股息金额的分布找到了明确的公式。我们仍然计算收益数量的概率分布，以达到给定的上限目标（如恒定的红利障碍）以及收益数量下降到破产的概率分布。我们通过一些说明性的数值例子完成了这项研究，这些例子显示了所研究的几个问题的最终数字。