Scaling limits for interactive Hawkes shot noise processes

Abstract

We introduce an interactive Hawkes shot noise process, in which the shot noise process has a Hawkes arrival process whose intensity depends on the state of the shot noise process via the self-exciting function. Namely, the shot noise process and the Hawkes process are interactive. We prove a functional law of large numbers (FLLN) and a functional central limit theorem (FCLT) for the joint dynamics of shot noise process and the Hawkes process, and characterize the effect of the interaction between them. The FLLN limit is determined by a nonlinear function determined through an integral equation. The diffusion limit is a two-dimensional interactive stochastic differential equation driven by two independent time-changed Brownian motions. The limit of the CLT-scaled shot noise process itself can be also expressed equivalently in distribution as an Ornstein–Uhlenbeck process with time-dependent parameters, unlike being a Brownian motion in the standard case without interaction. The limit of the CLT-scaled Hawkes counting process can be expressed as a sum of two independent terms, one as a time-changed Brownian motion (just as the standard case), and the other as a (Volterra type) Gaussian process represented by an Itô integral with another time-changed Brownian motion, capturing the effect of the interaction in the self-exciting function with the state of the shot noise process. To prove the joint convergence of the co-dependent Hawkes and shot noise processes, the standard techniques for Hawkes processes using the immigration-birth representations and the associated renewal equations are no longer applicable. We develop novel techniques by constructing representations for the LLN and CLT scaled processes that resemble the limits together with the associated residual terms, and then use a localization technique together with some martingale properties to prove the residual terms converge to zero and hence the joint convergence of the scaled processes. We also consider an extension of our model, an interactive marked Hawkes shot noise process, where the intensity of the Hawkes arrivals also depends on an exogenous noise, and present the corresponding FLLN and FCLT limits.