The Hawkes process models self-exciting event streams, requiring a strictly non-negative and stable stochastic intensity. Standard identification methods enforce these properties using non-negative causal bases, yielding conservative parameter constraints and severely ill-conditioned least-squares Gram matrices at higher model orders. To overcome this, we introduce a system-theoretic identification framework utilizing the sign-indefinite orthonormal Laguerre basis, which guarantees a well-conditioned asymptotic Gram matrix independent of model order. We formulate a constrained least-squares problem enforcing the necessary and sufficient conditions for positivity and stability. By constructing the empirical Gram matrix via a Lyapunov equation and representing the constraints through a sum-of-squares trace equivalence, the proposed estimator is efficiently computed via semidefinite programming.

翻译：霍克斯过程用于建模自激励事件流，要求其随机强度函数严格非负且稳定。传统辨识方法通过非负因果基函数强制满足这些性质，导致参数约束过于保守，且在高阶模型下最小二乘格拉姆矩阵呈现严重病态。为克服此局限，本文提出一种基于系统理论的辨识框架，采用符号不定的标准正交拉盖尔基函数，该基函数能保证格拉姆矩阵良态且渐近独立于模型阶次。我们构建了约束最小二乘问题，其中施加了正定性与稳定性的充要条件。通过李雅普诺夫方程构建经验格拉姆矩阵，并借助平方和迹等价形式表征约束条件，所提出的估计器可通过半定规划高效求解。