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.

翻译：受近期神经启发建模热潮的驱动，点过程在系统与控制领域获得了显著关注。尽管霍克斯过程是刻画具有记忆性的随机事件序列的标准模型，但其未知核函数的辨识常受非线性问题阻碍。采用指定基核函数的方法应运而生，以实现线性参数化，然而这类方法通常依赖于迭代似然方法，且缺乏在模型误设下的严格分析。本文为具有指定核函数的霍克斯过程论证了一种闭式最小二乘辨识框架。我们通过经验格拉姆矩阵的几乎必然正定性保证了估计器的存在性，并证明了在模型正确设定下估计器收敛于真实参数，在误设下收敛于定义良好的伪真实参数。此外，我们推导了两种情形下显式的中心极限定理，提供了一个完整且可解释的渐近理论。我们通过对比数值模拟验证了这些理论发现。