Second-Order Approximation of Limit Order Books in a Single-Scale Regime

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.