A System-Theoretic Approach to Hawkes Process Identification with Guaranteed Positivity and Stability
This addresses a critical bottleneck in modeling self-exciting event streams like financial transactions or social media activity, though it appears incremental as it builds on existing identification methods.
The authors tackled the problem of identifying Hawkes process parameters while guaranteeing positivity and stability, overcoming issues with ill-conditioned matrices in standard methods. They introduced a system-theoretic framework using a Laguerre basis and semidefinite programming, achieving a well-conditioned Gram matrix independent of model order.
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.