Scaling Limit of a Limit Order Book Model via the Regenerative Characterization of Lévy Trees
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.  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
... tion, and complement the results of Lakner et al.  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. The cornerstone of our approach is the regenerative characterization of Lévy trees proved in Weill  , which provides an elegant and intuitive proof strategy which we unfold. Moreover, the proofs rely on new results of independent interest on branching random walks with a barrier. Link with Lévy trees. The most demanding part of the proof of our main result is to show that the price process converges to a reflected Brownian motion. To prove this, we exploit an unexpected connection between our model and Lévy trees. Informally speaking, Lévy trees are the continuous scaling limits of Galton Watson trees, and both Lévy trees and Galton Watson trees are coded by so-called contour functions. The definition of a Galton Watson tree translates immediately to some regenerative property, say (R'), satisfied by its contour function: informally, successive excursions above a certain level are i.i.d., see Section 2 for more details. In the discrete setting this regenerative property is easily seen to actually characterize contour functions of Galton Watson trees. Weill  extended this characterization to the continuous setting, i.e., showed that any continuous stochastic process satisfying some regenerative property (R) (the analog of (R') for continuous processes) must be the contour function of some Lévy tree. The above mentioned result lies at the heart of the proof of our main result. Indeed, it follows by simple inspection that, in our model, the price process satisfies a regenerative property very close to (R') and that, in the asymptotic regime that we are interested in, this difference should be washed out in the limit. This suggests an elegant way to study the asymptotic behavior of the price process, namely by showing that any accumulation point of the price process must satisfy the continuous regenerative property (R). We will then know thanks to  that any accumulation point must be the contour function of a Lévy tree, thereby drastically reducing the possible accumulation points. A few additional arguments will then make it possible to deduce that, among this class of stochastic processes, the limit process must actually be a reflected Brownian motion and that the whole measure-valued process converges to a simple functional thereof. Although this proof strategy is very attractive at a conceptual level, many technical details need to be taken care of along the way and Section 5 of the paper is dedicated to working these details out. A key argument which is used repeatedly in the proofs is the coupling established in Simatos  between the model studied here and a branching random walk. Leveraging this coupling leads us to derive new results of independent interest on branching random walks with a barrier, which are gathered in Appendix A. As an aside, it is interesting to note that the discrete regenerative property (R') is satisfied by a classical queueing system, namely the Last-In-First-Out (LIFO) queue, also called Last-Come-First-Served (LCFS). This property was for instance exploited in Núñez-Queija  to study its stationary behavior. Combined with the above reasoning it provides a new interpretation for the results by Limic [27, 28] , where it is proved that the scaling limit of the LIFO queue is the height process of a Lévy tree. Organization of the paper. Section 2 introduces basic notation, presents our main result (Theorem 2.1) and discusses more formally the connection with Lévy trees. Before proceeding to the proof of Theorem 2.1 in Section 5, we introduce in Section 3 the coupling of Simatos , and additional notation together with preliminary results in Section 4. We conclude the paper by discussing in Section 6 possible extensions of our results. Finally, as mentioned above, some results of independent interest on branching random walks, which we need along the way, are gathered and proved in Appendix A. Acknowledgements. F. Simatos would like to thank N. Broutin for useful discussions about branching random walks that lead to the results of Appendix A.