We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the order flow and only depend on rate of arrival of orders and the covariance structure of order sizes. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2157445_code1114.pdf?abstractid=1757861&mirid=1&type=2

Order book dynamics in liquid markets: limit theorems and diffusion approximations

We consider two parallel M/M/1 queues which are fed by a single Poisson arrival stream. An arrival splits into two parts, with each part joining a different queue. This is the simplest example of a fork-join model. After the individual parts receive service, they may be joined back together, though we do not consider the join part here. We study this model in the heavy traffic limit, where the service rate in either queue is only slightly larger than the arrival rate. In this limit we obtain asymptotically the joint steady-state queue length distribution. In the symmetric case, where the two servers are identical, this distribution has a very simple form. In the non-symmetric case we derive several integral representations for the distribution. We then evaluate these integrals asymptotically, which leads to simple formulas which show the basic qualitative structure of the joint distribution function.

A fork-join queueing model: Diffusion approximation, integral representations and asymptotics

The use of singular perturbation methods to obtain approximate analytic solutions to the types of diffusion equations that are used to model queueing networks is shown. As an example, a two-dimensional fork and join model is considered. Approximations to the stationary distribution of the number of customers in the system as well as the residual busy period are computed. The formulas obtained clearly show the dependence of the solution on the variables and the parameters of the model.

On the diffusion approximation to a fork and join queueing model

We consider two parallel queues. When both are non-empty, they behave as two independent M/M/1 queues. If one queue is empty the server in the other works at a different rate. We consider the heavy traffic limit, where the system is close to instability. We derive and analyze the heavy traffic diffusion approximation for this model. In particular, we obtain simple integral representations for the joint steady state density of the (scaled) queue lengths. Asymptotic and numerical properties of the solution are studied.

Heavy Traffic Analysis of Two Coupled Processors

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

Integral expression for the stationary distribution of reflected brownian motion in a wedge

This paper is concerned with the two-dimensional generalization of the single server queue “diffusion approximation”: this model consists of a stochastic process in $\mathbb{R}^ + \times \mathbb{R}^ + $ being a diffusion in the open quarter plane with absorption on the boundaries $\{ 0 \} \times \mathbb{R}^ + \cup \mathbb{R}^ + \times \{ 0 \}$ and jumps back into this open domain. On the boundary this process is a linear diffusion with absorption on $( {0,0} )$ and jumps back into the x or y axis. This model describes within the same framework such phenomena as coupling of processors or simultaneous arrivals of jobs which do not preserve the classical “product form” solutions. First we exhibit a pathwise construction of the process which leads to the existence, the uniqueness and the Markov property. From this, we derive a functional equation for the Laplace transform of the steady state distribution. The solution of this functional equation is obtained by reduction to a boundary value problem on the righ...

Francois Baccelli

Papers

Analysis of models reducible to a class of diffusion processes in the positive quarter plane