Recurrence of the fitted spread process

Determine whether the fitted spread process in the spread-gated Hawkes-flocking limit order book model is recurrent.

Background

The paper proves that the constrained Hawkes-flocking limit order book process is non-explosive, meaning that only finitely many bid and ask price movements occur on every finite time interval. The authors explicitly distinguish this finite-horizon property from recurrence of the spread or existence of a stationary distribution for the joint spread–intensity system.

For the fitted models, the total compensator over the estimation window indicates more spread-widening than spread-narrowing events. The authors therefore leave unresolved whether the fitted spread process is recurrent, identifying this as an empirical question rather than establishing it theoretically or empirically.

References

The proposition does not establish recurrence of the spread or a stationary distribution of the joint spread--intensity system. Such properties have been proved for simple state-dependent spread models: \citet{ruan2023selfexciting} show ergodicity for one-tick jumps and a single exponential kernel, using an intensity for downward jumps that grows with the spread. By contrast, \citet{sfendourakis2020lob} describe the stability of Hawkes processes with a state-dependent factor as open. In our fitted models the total compensator over the estimation window implies more spread-widening than spread-narrowing events (Section~3.3), so whether the fitted spread process is recurrent is an empirical question that we do not settle.

— A Spread-Gated Hawkes-Flocking Model for Best Bid and Ask Dynamics, with an Application to Limit Order Placement  (2609.36631 - Lee et al., 29 Sep 2026) in Remark “Non-explosion is not recurrence,” Section 2.3 (immediately after Proposition 2.3)