Rigorous results for the Stigler-Luckock model for the evolution of an order book

Abstract: In 1964, G.J. Stigler introduced a stochastic model for the evolution of an order book on a stock market. This model was independently rediscovered and generalized by H. Luckock in 2003. In his formulation, traders place buy and sell limit orders of unit size according to independent Poisson processes with possibly different intensities. Newly arriving buy (sell) orders are either immediately matched to the best available matching sell (buy) order or stay in the order book until a matching order arrives. Assuming stationarity, Luckock showed that the distribution functions of the best buy and sell order in the order book solve a differential equation, from which he was able to calculate the position of two prices $J^{\rm c}-<J^{\rm c}+$ such that buy orders below $J^{\rm c}-$ and sell orders above $J^{\rm c}+$ stay in the order book forever while all other orders are eventually matched. We extend Luckock's model by adding market orders, i.e., with a certain rate traders arrive at the market that take the best available buy or sell offer in the order book, if there is one, and do nothing otherwise. We give necessary and sufficient conditions for such an extended model to be positive recurrent and show how these conditions are related to the prices $J^{\rm c}-$ and $J^{\rm c}+$ of Luckock.