Analysing the Mean First Exit Time for Compound Poisson Processes

Abstract

This study focuses on the mean first exit time for a compound Poisson process characterized by positive jumps and an upper horizontal boundary. The compound Poisson process is an important model for events occurring at random intervals, where the size of each jump follows a specific distribution. The study derives an explicit formula for calculating the mean first exit time, which refers to the expected time it takes for the process to hit the upper boundary for the first time. This measure is crucial for applications where the time until a specific threshold is reached is of interest. Furthermore, an application of the derived formula is provided in the context of traffic accidents, where the first exit time can model the time until a traffic system reaches a critical congestion level or failure point. This approach offers valuable insights for understanding the dynamics of complex stochastic systems, with potential applications in risk management, transportation, and other fields that involve compound Poisson processes. The results demonstrate the utility of the derived formula in practical scenarios, highlighting the significance of first exit times in modelling real-world processes.