Convergence of nonhomogeneous Hawkes processes and Feller random measures

Abstract

We consider a sequence of Hawkes processes whose excitation measures may depend on the generation, and study its scaling limits in the near-unstable limiting regime. The limiting random measures, characterized via a nonlinear convolutional equation, form a family parameterized by a pair consisting of a locally finite measure and a geometrically infinitely divisible probability distribution on the positive real line. These measures can be interpreted as generalizations of the Feller diffusion and fractional Feller (CIR) processes, but also allow for a "driving noise" associated with general L\' evy-type operators of order at most $1$, including fractional derivatives of any order $\alpha>0$ (formally corresponding to possibly negative Hurst parameters).