Scaling Limit of a Limit Order Book Model via the Regenerative Characterization of Lévy Trees

Abstract

We consider the following Markovian dynamic on point processes: at constant rate and with equal probability, either the rightmost atom of the current configuration is removed, or a new atom is added at a random distance from the rightmost atom. Interpreting atoms as limit buy orders, this process was introduced by Lakner et al. [Lakner et al. (2016) High frequency asymptotics for the limit order book. Mark. Microstructure Liq. 2:1650004 [83 pages]] to model a one-sided limit order book.

We consider this model in the regime where the total number of orders converges to a reflected Brownian motion, and complement the results of Lakner et al. [Lakner P, Reed J, Stoikov S (2016) High frequency asymptotics for the limit order book. Mark. Microstructure Liq. 2:1650004 [83 pages]] by showing that, in the case where the mean displacement at which a new order is added is positive, the measure-valued process describing the whole limit order book converges to a simple functional of this reflected Brownian motion. Our results make it possible to derive useful and explicit approximations on various quantities of interest such as the depth or the total value of the book.

Our approach leverages an unexpected connection with Lévy trees. More precisely, the cornerstone of our approach is the regenerative characterization of Lévy trees due to Weill [Weill M (2007) Regenerative real trees. Ann. Probab. 35:2091–2121. MR2353384 (2008j:60205)], which provides an elegant proof strategy which we unfold.