Asymptotic analysis of Poisson shot noise processes, and applications

TL;DR

This paper provides a comprehensive asymptotic analysis of Poisson shot noise processes, including deviations, fluctuations, and stable approximations, with applications to various stochastic models in different fields.

Contribution

It extends and refines existing theoretical results on Poisson shot noise processes and applies these findings to complex models like Hawkes processes and risk processes.

Findings

Derived sharp deviation results for Poisson shot noise processes

Established stable probability approximations for fluctuations

Applied theoretical results to practical models in multiple disciplines

Abstract

Poisson shot noise processes are natural generalizations of compound Poisson processes that have been widely applied in insurance, neuroscience, seismology, computer science and epidemiology. In this paper we study sharp deviations, fluctuations and the stable probability approximation of Poisson shot noise processes. Our achievements extend, improve and complement existing results in the literature. We apply the theoretical results to Poisson cluster point processes, including generalized linear Hawkes processes, and risk processes with delayed claims. Many examples are discussed in detail.