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

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 uniquely reproduces the volatility signature plot's characteristic upward slope by capturing the "local super-criticality" triggered during disequilibrium states. Crucially, we identify Marketable Limit Orders (MLO) as the primary catalyst that forces the LOB into these unstable states. Comparative analysis reveals that models lacking physical constraints (e.g., standard SD-Hawkes) suffer from explosive branching ratios 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 uniquely maintains statistical integrity where unconstrained models succumb to structural bias.