Paper: arXiv 2407.04354

Authors: Johannes Muhle-Karbe, Eyal Neuman, Yonatan Shadmi

Abstract

Maglaras, Moallemi, and Zheng (2021) have introduced a flexible queueing model for fragmented limit-order markets, whose fluid limit remains remarkably tractable. In the present study we prove that, in the limit of small and frequent orders, the discrete system indeed converges to the fluid limit, which is characterized by a system of coupled nonlinear ODEs with singular coefficients at the origin. Moreover, we establish that the fluid system is asymptotically stable for an arbitrary number of limit order books in that, over time, it converges to the stationary equilibrium state studied by Maglaras et al. (2021).

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical proof for the fluid limit convergence and asymptotic stability of fragmented limit-order markets using coupled nonlinear ODEs. However, it is entirely theoretical with no empirical data analysis or backtesting, placing it squarely in the Lab Rats quadrant.

Research Flowchart

  flowchart TD
    A[Research Goal: Prove convergence & stability of fragmented limit-order market fluid limit] --> B[Input: Discrete Queueing Model for Fragmented LOBs]
    B --> C[Limit Scaling: Apply Small and Frequent Orders Regime]
    C --> D[Fluid Limit Proof: Prove convergence to coupled nonlinear ODEs with singular coefficients]
    D --> E[Asymptotics Analysis: Evaluate global stability for arbitrary number of LOBs]
    E --> F[Key Outcome: Verified convergence to unique stationary equilibrium]

Cited References (24)

  1. A Cross‐Border Market Model with Limited Transmission Capacities (2022) — link
  2. Trading in Fragmented Markets (2019) — link
  3. Limit Order Books, Diffusion Approximations and Reflected SPDEs: From Microscopic to Macroscopic Models (2018) — link
  4. Optimal make–take fees for market making regulation (2018) — link
  5. Competing on Speed (2018) — link
  6. A diffusion approximation for limit order book models (2016) — link
  7. A Markov Model of a Limit Order Book: Thresholds, Recurrence, and Trading Strategies (2015) — link
  8. HYDRODYNAMIC LIMIT OF ORDER-BOOK DYNAMICS (2014) — link
and 16 more
  1. Queueing Dynamics and State Space Collapse in Fragmented Limit Order Book Markets (2014) — link
  2. Market Liquidity: Theory, Evidence, and Policy (2013) — link
  3. Optimal order placement in limit order markets (2012) — link
  4. Price Dynamics in a Markovian Limit Order Market (2011) — link
  5. Eigenvalues of rank-one updated matrices with some applications (2007) — link
  6. Scheduling Flexible Servers with Convex Delay Costs: Heavy-Traffic Optimality of the Generalized cµ-Rule (2004) — link
  7. Strong approximations for Markovian service networks (1998) — link
  8. Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse (1998) — link
  9. State space collapse with application to heavy traffic limits for multiclass queueing networks (1998) — link
  10. Strong approximation theorems for density dependent Markov chains (1978) — link
  11. Random Models of Dynamical Systems Introduction to SDE’s TD 3 : Stochastic differential equations and diffusion processes (2021)
  12. Regulation NMS: Final Rules and Amendments to Joint Industry Plans (2005)
  13. Differential Equations, Dynamical Systems, and an Introduction to Chaos (2003) — link
  14. Heavy TraÆ Limits for Some Queueing Networks (2001)
  15. Review: P. Billingsley, Convergence of probability measures (1971) — link
  16. Limits of limit-order books

0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.