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

Paper: arXiv 2604.23961

Authors: Akitoshi Kimura

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 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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper combines sophisticated mathematical framework derivations involving KKT conditions and point process intensity decomposition with rigorous empirical validation using real-world high-frequency tick data. It successfully bridges abstract stochastic modeling with concrete market micro-structure phenomenon like volatility signature plots.

Research Flowchart

  flowchart TD
    A[Research Goal: Model LOB Dynamics with ExsdHawkes] --> B[Data: 3 Months Tick Data (Mitsubishi UFJ)]
    B --> C[Methodology: Extended State-Dependent Hawkes Process]
    C --> D[Computational Step: Separable MLE via KKT Conditions]
    D --> E[Simulation: Enforce Physical Constraints/State Pausing]
    E --> F[Findings: Reproduction of Volatility Signature Plots]
    F --> G[Outcome: MLOs Identified as Catalyst for Instability]

Cited References (17)

1. Limit order books (2016)
2. Hawkes processes in finance (2015)
3. Statistical modeling of high-frequency financial data (2011)
4. An introduction to the theory of point processes: volume I: elementary theory and methods (2003)
5. High-dimensional Hawkes processes for limit order books: modelling, empirical analysis and numerical calibration (2018)
6. Demonstration simplifiee d’un theoreme de Knight (1971)
7. State-dependent Hawkes processes and their application to limit order book modelling (2022)
8. Statistical models for earthquake occurrences and residual analysis for point processes (1988)

0 of 17 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.