A weak law of large numbers for a limit order book model with fully state dependent order dynamics

TL;DR

This paper establishes a weak law of large numbers for a limit order book model with state-dependent order dynamics, showing convergence to a continuous limit described by coupled PDEs and ODEs.

Contribution

It introduces a novel limit order book model with fully state-dependent dynamics and proves its convergence to a continuous-time limit with coupled PDEs and ODEs.

Findings

Convergence of the LOB model to a continuous limit as order size and tick size tend to zero.

Derivation of coupled non-linear PDEs and ODEs describing the limit dynamics.

Validation of the weak law of large numbers for the model.

Abstract

This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of large numbers, which states that the LOB model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities follow each a non-linear PDE coupled with two non-linear ODEs that describe the best bid and ask price.