Paper: arXiv 2511.18117

Authors: Levon Mahseredjian

Abstract

This paper develops a theoretical mesoscopic model of the limit order book driven by multivariate Hawkes processes, designed to capture temporal self-excitation and the spatial propagation of order flow across price levels. In contrast to classical zero-intelligence or Poisson based queueing models, the proposed framework introduces mathematically defined migration events between neighbouring price levels, whose intensities are themselves governed by the underlying Hawkes structure. This provides a principled stochastic mechanism for modeling interactions between order arrivals, cancellations, and liquidity movement across adjacent queues. Starting from a microscopic specification of Hawkes driven order flow, we derive a diffusion approximation which yields a reflected mesoscopic stochastic differential equation (SDE) system for queue volumes. The limiting generator is obtained through a Taylor expansion of the microscopic generator, demonstrating how temporal excitation together with spatial migration determine the drift and diffusion structure of the limit order book in the mesoscopic regime. The resulting model extends existing diffusion limits by incorporating correlated excitations and price level to price level liquidity movement within a unified Hawkes based formulation. By establishing this diffusive limit, the paper provides a mathematically consistent bridge between high frequency event based models and macroscopic stochastic descriptions of market microstructure. The work is entirely theoretical and lays a foundation for future analytical and numerical developments without relying on empirical calibration.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical treatment of limit order book dynamics using a functional central limit theorem on multivariate Hawkes processes with spatial liquidity migration. However, it is entirely theoretical with zero empirical validation, data analysis, or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Bridge High-Frequency Order Book Dynamics to Macroscopic Models] --> B[Microscopic Model Input: Multivariate Hawkes Processes & Spatial Liquidity Migration]
    B --> C[Derive Microscopic Generator for Order Queue Volumes]
    C --> D[Apply Taylor Expansion to Microscopic Generator]
    D --> E[Diffusion Approximation & Asymptotic Limit Analysis]
    E --> F[Outcome: Reflected Mesoscopic SDE System capturing Drift and Diffusion Structures]

Cited References (14)

  1. Limit Order Books, Diffusion Approximations and Reflected SPDEs: From Microscopic to Macroscopic Models (2018) — link
  2. Price Dynamics in a Markovian Limit Order Market (2011) — link
  3. A Mathematical Approach to Order Book Modelling (2010) — link
  4. Markov Processes: Characterization and Convergence (2005)
  5. Stochastic differential equations with reflecting boundary conditions (1984) — link
  6. “A diffusion limit for a Hawkes-type limit order book model,” (2019)
  7. “A continuum limit for order book models (2018)
  8. “Modelling order flow dynamics using Hawkes processes,” (2017)
and 6 more
  1. “A generalized method for multivariate Hawkes processes,” (2015)
  2. “Queue-reactive Hawkes processes for limit order books,” (2015)
  3. Limit Order Markets: A Survey (2007)
  4. Clustering of order arrivals , price impact and trade path optimisation (2006)
  5. ii) Regularity of baselines. The baseline functions for arrivals and can-cellations satisfy global Lipschitz and linear growth bounds in the queue state
  6. (iii) Moment bounds. The initial states satisfy sup n E ∥ X kn (0) ∥ 2 < ∞ for k ∈ { b, a }

0 of 14 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.