A Weak Law of Large Numbers for a Limit Order Book Model with Fully State Dependent Order Dynamics

Abstract

This paper studies a limit order book 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 limit order book we derive a weak law of large numbers, which states that the limit order book 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 each follow a nonlinear PDE coupled with two nonlinear ODEs that describe the best bid and ask price.