## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 5 (2017), 2753-2806.

### A functional limit theorem for limit order books with state dependent price dynamics

Christian Bayer, Ulrich Horst, and Jinniao Qiu

#### Abstract

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 5 (2017), 2753-2806.

**Dates**

Received: April 2015

Revised: July 2016

First available in Project Euclid: 3 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1509696034

**Digital Object Identifier**

doi:10.1214/16-AAP1265

**Mathematical Reviews number (MathSciNet)**

MR3719946

**Zentralblatt MATH identifier**

1379.60036

**Subjects**

Primary: 60B11: Probability theory on linear topological spaces [See also 28C20] 90B22: Queues and service [See also 60K25, 68M20]

Secondary: 91B70: Stochastic models

**Keywords**

Limit order book functional limit theorem stochastic partial differential equation

#### Citation

Bayer, Christian; Horst, Ulrich; Qiu, Jinniao. A functional limit theorem for limit order books with state dependent price dynamics. Ann. Appl. Probab. 27 (2017), no. 5, 2753--2806. doi:10.1214/16-AAP1265. https://projecteuclid.org/euclid.aoap/1509696034