Paper: arXiv 2308.00805

Authors: Ulrich Horst, Dörte Kreher, Konstantins Starovoitovs

Abstract

We establish a first and second-order approximation for an infinite dimensional limit order book model (LOB) in a single (‘‘critical’’) scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first-order and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a standard coupled ODE-PDE system, the second-order approximation is non-standard and described in terms of an infinite-dimensional stochastic evolution equation driven by a cylindrical Brownian motion. The driving noise processes exhibit a non-trivial correlation in terms of the model parameters. We prove that the evolution equation has a unique solution and that the sequence of standardized LOB models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a simplified version of our model to market data and show that the model accurately captures correlations between price and volume fluctuations.

Complexity vs Empirical Score

  • Math Complexity: 0.0/10
  • Empirical Rigor: 0.0/10
  • Quadrant: Philosophers — conceptual discussion, limited math and data

Why this score: Scoring failed: Could not find valid JSON in LLM response: line 1 column 1 (char 0)

Research Flowchart

  flowchart TD
    A[Research Goal: Derive 1st & 2nd order approximations for infinite-dimensional LOB] --> B[Input Data: Empirical LOB market data & critical scaling assumptions]
    B --> C[Methodology: Formulate coupled ODE-PDE & stochastic evolution equations]
    C --> D[Analysis: Solve martingale problem driven by cylindrical Brownian motion]
    D --> E[Model Calibration: Fit simplified LOB model to equity market data]
    E --> F[Key Findings: Unique solution found; weak convergence proved; captures price-volume correlations]