Forecasting duration in high-frequency financial data using a self-exciting flexible residual point process

This paper presents a method for forecasting limit order book durations using a self-exciting flexible residual point process. High-frequency events in modern exchanges exhibit heavy-tailed interarrival times, posing a significant challenge for accurate prediction. The proposed approach incorporates the empirical distributional features of interarrival times while preserving the self-exciting and decay structure. This work also examines the stochastic stability of the process, which can be interpreted as a general state-space Markov chain. Under suitable conditions, the process is irreducible, aperiodic, positive Harris recurrent, and has a stationary distribution. An empirical study demonstrates that the model achieves strong predictive performance compared with several alternative approaches when forecasting durations in ultra-high-frequency trading data.

💡 Research Summary

The paper proposes a novel statistical framework for forecasting the durations between mid‑price changes in a limit order book (LOB) using a self‑exciting point process that incorporates flexible residuals. High‑frequency trading data exhibit inter‑arrival times with heavy tails, containing both extremely short and very long intervals, which makes conventional exponential‑based models inadequate. The authors introduce a general observation‑driven construction (Definition 1) where an i.i.d. innovation εₙ with an arbitrary continuous distribution drives the inter‑arrival time τₙ through a strictly increasing transformation Φ(t, x). The latent state Xₙ evolves deterministically via an update function Ψ(t, x), yielding a Markovian state‑space representation of the point process.

A specific instantiation, called the “self‑exciting exponentially decaying flexible residual point process” (Definition 6), defines the latent intensity Λₙ as
Λₙ = μ + (Λₙ₋₁ − μ + α) e^{−β τₙ},
with μ, α, β > 0. The cumulative intensity function Φ(t, Λₙ₋₁) = μ t + (Λₙ₋₁ − μ + α)(1 − e^{−β t})/β links the innovation εₙ to the observed duration τₙ via τₙ = Φ^{-1}(εₙ, Λₙ₋₁). Because εₙ can follow any heavy‑tailed distribution, the model preserves the empirical shape of the residuals while retaining the classic Hawkes‑type self‑excitation and exponential decay.

The authors rigorously analyze stochastic stability by interpreting {Λₙ} as a general‑state Markov chain. Under the condition α < β E