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

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Highly relevant for quantitative trading in several dimensions: (1) Provides a mechanistic explanation for high-frequency volatility clustering through local super-criticality, directly informing execution algorithms and market-making strategies. (2) The identification of AMS/AMB as triggers and MLO as catalysts offers actionable microstructure insights for order routing and liquidity provision. (3) The volatility signature plot reproduction validates the model for risk management and VaR estimation at high frequencies. (4) The physical constraint framework prevents model blow-up in simulation, critical for backtesting. (5) The branching ratio and spectral radius metrics provide real-time market instability indicators. However, the model is primarily descriptive/diagnostic rather than directly predictive of returns.

The mathematical framework is sophisticated, requiring understanding of multivariate point processes, KKT optimization, and state-dependent intensity functions. The two-step estimation procedure (separable likelihood) simplifies practical implementation. The O(N) recursive computation with exponential kernels makes it feasible for millions of ticks. However, correctly identifying MLOs from tick data requires detailed trade/quote flags, and the state-dependent simulation via Ogata's thinning with physical gates adds implementation complexity. The KKT constraint handling and zero-frequency treatment require careful numerical implementation.

The paper provides detailed mathematical formulations, KKT proofs, recursive estimation algorithms, and specific parameter values. However, the Nikkei NEEDS Tick Data is a commercial dataset requiring purchase. No code repository is mentioned. The methodology is fully specified mathematically, but empirical reproduction requires access to the proprietary tick data. The burn-in period, analysis window, and kernel specifications are clearly stated.