Convergence of nonhomogeneous Hawkes processes and Feller random measures

TL;DR

This paper studies the scaling limits of nonhomogeneous Hawkes processes with generation-dependent excitation measures, revealing a family of limiting random measures that generalize classical diffusions and incorporate Levy-type noise.

Contribution

It introduces a novel framework for analyzing the convergence of Hawkes processes with generation-dependent excitation measures to a family of generalized Feller-type diffusions.

Findings

Limiting measures are characterized by a nonlinear convolutional equation.

The family includes generalizations of Feller diffusion and fractional Feller processes.

The framework accommodates Levy-type noise with derivatives of order up to 1.

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).