## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 3 (2018), 1491-1535.

### 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 $J^{\mathrm{c}}_{-}<J^{\mathrm{c}}_{+}$ such that buy orders below $J^{\mathrm{c}}_{-}$ and sell orders above $J^{\mathrm{c}}_{+}$ stay in the order book forever while all other orders are eventually matched. We extend Luckock’s model by adding market orders, that is, 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 $J^{\mathrm{c}}_{-}$ and $J^{\mathrm{c}}_{+}$ of Luckock.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 3 (2018), 1491-1535.

**Dates**

Received: May 2016

Revised: April 2017

First available in Project Euclid: 1 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1527840025

**Digital Object Identifier**

doi:10.1214/17-AAP1336

**Mathematical Reviews number (MathSciNet)**

MR3809470

**Zentralblatt MATH identifier**

06919731

**Subjects**

Primary: 82C27: Dynamic critical phenomena

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C26: Dynamic and nonequilibrium phase transitions (general) 60J05: Discrete-time Markov processes on general state spaces

**Keywords**

Continuous double auction order book rank-based Markov chain self-organized criticality Stigler–Luckock model market microstructure

#### Citation

Swart, Jan M. Rigorous results for the Stigler–Luckock model for the evolution of an order book. Ann. Appl. Probab. 28 (2018), no. 3, 1491--1535. doi:10.1214/17-AAP1336. https://projecteuclid.org/euclid.aoap/1527840025