Hawkes Identification with a Prescribed Causal Basis: Closed-Form Estimators and Asymptotics

TL;DR

This paper introduces a closed-form least squares method for identifying Hawkes process kernels, providing rigorous theoretical guarantees and asymptotic analysis, which improves over iterative likelihood approaches especially under model misspecification.

Contribution

It develops a novel closed-form estimator for Hawkes processes with prescribed kernels, including asymptotic properties and robustness under misspecification.

Findings

Estimator exists under positive definite Gram matrix

Converges to true or pseudo-true parameters

Provides explicit Central Limit Theorems

Abstract

Driven by the recent surge in neural-inspired modeling, point processes have gained significant traction in systems and control. While the Hawkes process is the standard model for characterizing random event sequences with memory, identifying its unknown kernels is often hindered by nonlinearity. Approaches using prescribed basis kernels have emerged to enable linear parameterization, yet they typically rely on iterative likelihood methods and lack rigorous analysis under model misspecification. This paper justifies a closed-form Least Squares identification framework for Hawkes processes with prescribed kernels. We guarantee estimator existence via the almost-sure positive definiteness of the empirical Gram matrix and prove convergence to the true parameters under correct specification, or to well-defined pseudo-true parameters under misspecification. Furthermore, we derive explicit Central Limit Theorems for both regimes, providing a complete and interpretable asymptotic theory. We demonstrate these theoretical findings through comparative numerical simulations.