Extended State-dependent Hawkes Process for Limit Order Books: Mathematical Foundation and the Reproduction of Volatility Signature Plots

Abstract

This paper proposes an Extended State-Dependent Hawkes Process (ExsdHawkes) to model the intricate dynamics of Limit Order Books (LOBs). Our theoretical contribution lies in relaxing traditional constraints by allowing for state disappearances---a phenomenon frequently observed in high-frequency trading. We mathematically prove, using Karush--Kuhn--Tucker (KKT) conditions, that the maximum likelihood estimation remains separable, justifying an efficient two-step procedure. In the empirical section, we apply our model to three months of high-frequency tick data of Mitsubishi UFJ Financial Group (8306). We demonstrate that ExsdHawkes successfully replicates the characteristic upward slope of the volatility signature plot by capturing the ``local super-criticality'' triggered during disequilibrium states. Crucially, we clarify that the transition out of equilibrium is deterministically triggered by Aggressive Market Orders (AMS/AMB), while Marketable Limit Orders (MLO) function as a critical liquidity-depletion catalyst within the expanded spread. Comparative analysis reveals that models lacking physical constraints (e.g., standard SD-Hawkes) suffer from explosive spectral radii and fail to maintain simulation stability. Our findings suggest that physical consistency is not merely a mathematical nicety, but a prerequisite for accurately modeling macro-level volatility. By enforcing the physical geometry to `pause' the residual accumulation during inadmissible periods, ExsdHawkes maintains statistical integrity where unconstrained models succumb to structural bias and simulation instability.

20 pages, 8 figures. This work was supported by JSPS KAKENHI Grant Number JP20K14366 and CREST, JST