Innovations in medicine provide us longer and healthier life, leading lower mortality. Sooner rather than later, much greater longevity would be possible for us due to artificial intelligence advances in health care. Similarly, Advanced Driver Assistance Systems (ADAS) in highly automated vehicles may reduce or even eventually eliminate accidents by perceiving dangerous situations, which would minimize the number of accidents and lead to fewer loss claims for insurance companies. To model the survivor function capturing greater longevity as well as the number of claims reflecting less accidents in the long run, in this paper, we study a Cox process whose intensity process is piecewise-constant and decreasing. We derive its ultimate distributional properties, such as the Laplace transform of intensity integral process, the probability generating function of point process, their associated moments and cumulants, and the probability of no more claims for a given time point. In general, this simple model may be applicable in many other areas for modeling the evolution of gradually disappearing events, such as corporate defaults, dividend payments, trade arrivals, employment of a certain job type (e.g., typists) in the labor market, and release of particles. In particular, we discuss some potential applications to insurance.

医学创新为我们提供了更长寿、更健康的生命，从而降低了死亡率。由于人工智能在医疗保健领域的进步，我们迟早会有更长的寿命。同样，高度自动化车辆中的高级驾驶员辅助系统 (ADAS) 可以通过感知危险情况来减少甚至最终消除事故，这将最大限度地减少事故数量并减少保险公司的损失索赔。为了对幸存者函数进行建模，以捕获更长的寿命以及从长远来看反映更少事故的索赔数量，在本文中，我们研究了一个 Cox 过程，其强度过程是分段恒定且递减的。我们推导出其最终的分布特性，例如强度积分过程的拉普拉斯变换，点过程的概率生成函数，它们的相关时刻和累积量，以及给定时间点不再索赔的概率。一般来说，这个简单的模型可能适用于许多其他领域，用于模拟逐渐消失的事件的演变，例如公司违约、股息支付、贸易到达、劳动力市场中某种工作类型（例如打字员）的就业，以及粒子释放。特别是，我们讨论了一些潜在的保险应用。劳动力市场中某种工作类型（例如打字员）的就业，以及颗粒物的释放。特别是，我们讨论了一些潜在的保险应用。劳动力市场中某种工作类型（例如打字员）的就业，以及颗粒物的释放。特别是，我们讨论了一些潜在的保险应用。