From Pseudo-Poisson-Type Processes to Trawl Processes: Structural Insights and the Markov Property of Exponential Trawls

We study the limiting behavior of the sum of independent copies of a pseudo-Poisson-type process, also known as the Poisson stochastic index (PSI) process. Under a series scheme satisfying the infinitesimality condition, we prove that the sum of independent PSI processes with random intensity converges, in the sense of finite-dimensional distributions, to a trawl process. To support the theoretical convergence results, we provide a series of Monte Carlo simulations illustrating the behavior of the sum of independent PSI processes. As part of this analysis, we derive an explicit formula for the elements of the slice partition associated with monotonic trawls having an exponential structure of the trawl function. In the special case of PSI processes with nonrandom intensity, we show convergence to a trawl process with an exponential trawl. The connection between PSI processes and trawl processes highlights that the theoretical results obtained can inform the modeling of real-world phenomena, such as irregular event arrivals in communication networks or rare, extreme fluctuations in financial markets. Finally, we investigate structural properties of the trawl process with an exponential trawl, demonstrating that it is generally non-Markovian. We provide examples illustrating how the Markov property of such processes is closely related to the distributional features of the associated Lévy basis bridge.