Some limit theorems for locally stationary Hawkes processes

TL;DR

This paper establishes limit theorems for multivariate Hawkes processes with time-varying rates, combining martingale methods with new ideas, and applies these results to financial statistics, including price distortion analysis.

Contribution

It introduces a law of large numbers and a functional central limit theorem for non-stationary Hawkes processes, advancing theoretical understanding and practical applications.

Findings

Proved law of large numbers for multivariate Hawkes processes.

Established functional central limit theorem with time-dependent rates.

Derived closed-form expressions for price distortions under liquidity constraints.

Abstract

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by recombining classical martingale methods introduced in Bacry et al. [3] with novel ideas proposed by Kwan et al. [19]. The asymptotic theory we obtain yields useful applications in financial statistics. As an illustration, we derive closed-form expressions for price distortions under liquidity constraints.