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.
Highlights
Several mathematical models of the limit order book have been proposed in recent years, ranging from stylized models such as the Stigler-Luckock model to more complex models such as those proposed by Cont et al (2010) or Garèche et al (2013)
The discrete model that we study is a variant of the limit order book model proposed by Lakner et al (2016)
Link with Lévy Trees: Detailed Discussion The following lemma is at the heart of our approach to prove Theorem 2.1
Summary
Context The limit order book is a financial trading mechanism that facilitates the buying and selling of securities by market participants. It keeps track of orders made by traders, which makes it possible to fulfill them in the future. Several mathematical models of the limit order book have been proposed in recent years, ranging from stylized models such as the Stigler-Luckock model (see Kelly and Yudovina 2017, Luckock 2003, Swart 2016) to more complex models such as those proposed by Cont et al (2010) or Garèche et al (2013). These models may be categorized as being either discrete and closely adhering to the inherent quantized nature of the limit order book, or as being continuous in order to better capture the high frequency regime in which the order book typically evolves
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