Critical thresholds in a limit order book with mutually exciting order flow

In the Kelly–Yudovina limit order book, independent Poisson bid and ask arrivals generate critical prices κ_b<κ_a: bids below κ_b and asks above κ_a accumulate, while orders in the active interval are eventually matched. We replace the arrival streams by a stationary bivariate linear Hawkes process. Writing Φ=(φ_{ij}) for the integrable kernel matrix and K=∫_0^∞Φ(s)ds, we assume ρ(K)<1; price marks remain independent conditional on order side. The stationary arrival-rate vector is Λ=(I−K)⁻¹μ.

Our design separates mean activity from temporal clustering. We compare the Hawkes book with an auxiliary Poisson book having the same rates Λ, derive a fluid-limit characterisation of K↦(κ_b(K),κ_a(K)), obtain its first-order expansion at K=0, and simulate symmetric and asymmetric excitation against the matched-rate benchmark. This contrast isolates changes attributable to dependence rather than average order flow. The novelty is therefore not Hawkes-driven order flow itself, but an excitation-to-threshold map for the Kelly–Yudovina matching mechanism and a decomposition of mean-rate and clustering effects.

The thresholds have a direct liquidity-risk interpretation: they delimit price regions in which passive orders face persistent non-execution and order accumulation. An upward shift of κ_b or a downward shift of κ_a enlarges the corresponding non-execution region. Simulations with symmetric and asymmetric subcritical kernels quantify one-sided threshold displacement and transient spread excursions under clustered flow. Large movements of these diagnostics identify high-frequency regimes with severe non-execution exposure and potential liquidity dry-ups. The framework converts Hawkes kernels estimated from event data, or specified for stress testing, into interpretable execution- and liquidity-risk indicators.