🤖 AI Summary
This work addresses the challenge of constructing a tractable self-exciting point process that retains the long-memory characteristics of power-law decay while admitting a finite-dimensional Markovian representation. To this end, the authors propose a novel self-exciting point process in which a nonlinear Markov chain replaces the dependence on the entire event history, thereby enabling a finite-dimensional Markovian characterization of the intensity dynamics while preserving the local jump-and-slope update structure. This approach uniquely integrates power-law long memory with a finite state space, achieving both analytical tractability and computational feasibility. Theoretical analysis demonstrates that the latent state process reproduces long-memory behavior near the stability boundary and rigorously establishes the existence of a unique invariant distribution, along with key ergodic properties including irreducibility, aperiodicity, T-chain structure, and positive Harris recurrence.
📝 Abstract
We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.