  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).
