Fluid-Limits of Fragmented Limit-Order Markets

TL;DR

This paper proves that in fragmented limit-order markets, the discrete queueing system converges to a tractable fluid limit described by nonlinear ODEs, which is stable and converges to equilibrium over time.

Contribution

It establishes the rigorous convergence of the discrete model to the fluid limit and proves the stability of the fluid system for any number of order books.

Findings

Discrete system converges to fluid limit of nonlinear ODEs

Fluid system is asymptotically stable

Convergence to stationary equilibrium over time

Abstract

Maglaras, Moallemi, and Zheng (2021) have introduced a flexible queueing model for fragmented limit-order markets, whose fluid limit remains remarkably tractable. In the present study we prove that, in the limit of small and frequent orders, the discrete system indeed converges to the fluid limit, which is characterized by a system of coupled nonlinear ODEs with singular coefficients at the origin. Moreover, we establish that the fluid system is asymptotically stable for an arbitrary number of limit order books in that, over time, it converges to the stationary equilibrium state studied by Maglaras et al. (2021).