This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.

本文考虑了 Cramér-Lundberg 模型，其附加特征是客户数量会随时间波动。客户根据泊松过程到达，他们在系统中花费的时间形成一系列独立且同分布的非负随机变量。在系统中，每个客户都会生成索赔并支付保费。为了描述模型的罕见事件行为，我们建立了样本路径大偏差原则。这描述了储备级别进程和客户端人口大小进程的联合罕见事件行为。大偏差原理可用于确定随时间变化的破产概率的衰减率以及最有可能破产的路径。