The Annals of Applied Probability

A functional limit theorem for limit order books with state dependent price dynamics

Christian Bayer, Ulrich Horst, and Jinniao Qiu

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We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

Article information

Ann. Appl. Probab., Volume 27, Number 5 (2017), 2753-2806.

Received: April 2015
Revised: July 2016
First available in Project Euclid: 3 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B11: Probability theory on linear topological spaces [See also 28C20] 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 91B70: Stochastic models

Limit order book functional limit theorem stochastic partial differential equation


Bayer, Christian; Horst, Ulrich; Qiu, Jinniao. A functional limit theorem for limit order books with state dependent price dynamics. Ann. Appl. Probab. 27 (2017), no. 5, 2753--2806. doi:10.1214/16-AAP1265.

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  • [1] Abergel, F. and Jedidi, A. (2015). Long-time behavior of a Hawkes process-based limit order book. SIAM J. Financial Math. 6 1026–1043.
  • [2] Biais, B., Hillion, P. and Spatt, C. (1995). An empirical analysis of the limit order book and the order flow in the Paris bourse. J. Finance 50 1655–1689.
  • [3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [4] Cebiroğlu, G. and Horst, U. (2015). Optimal order display in limit order markets with liquidity competition. J. Econom. Dynam. Control 58 81–100.
  • [5] Cont, R. and Larrard, A. D. (2012). Order book dynamics in liquid markets: Limit theorems and diffusion approximations. Available at SSRN,
  • [6] Cont, R., Stoikov, S. and Talreja, R. (2010). A stochastic model for order book dynamics. Oper. Res. 58 549–563.
  • [7] Da Prato, G. and Zabczyk, J. (2008). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge.
  • [8] Easley, D. and O’Hara, M. (1987). Price, trade size, and information in securities markets. J. Financ. Econ. 19 69–90.
  • [9] Farmer, J., Gillemot, L., Lillo, F., Mike, S. and Sen, A. (2004). What really causes large price changes? Quant. Finance 4 383–397.
  • [10] Gao, X., Dai, J. G., Dieker, A. B. and Deng, S. J. (2014). Hydrodynamic limit of order book dynamics. Preprint.
  • [11] Glosten, L. R. and Milgrom, P. R. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. J. Financ. Econ. 14 71–100.
  • [12] Hautsch, N. and Huang, R. (2012). The market impact of a limit order. J. Econom. Dynam. Control 36 501–522.
  • [13] Horst, U. and Kreher, D. (2017). A weak law of large numbers for a limit order book model with fully state dependent order dynamics. SIAM J. Financial Math. 8 314–343.
  • [14] Horst, U. and Paulsen, M. (2017). A law of large numbers for limit order books. Math. Oper. Res. To appear.
  • [15] Huang, W., Lehalle, C.-A. and Rosenbaum, M. (2015). Simulating and analyzing order book data: The queue-reactive model. J. Amer. Statist. Assoc. 110 107–122.
  • [16] Huang, W. and Rosenbaum, M. (2015). Ergodicity and diffusivity of Markovian order book models: A general framework. Available at arXiv:1505.04936v1.
  • [17] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [18] Kang, W. and Williams, R. J. (2007). An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 741–779.
  • [19] Keller-Ressel, M. and Müller, M. (2016). A Stefan-type stochastic moving boundary problem. Stoch. Partial Differ. Equ., Anal. Computat. 4 746–790.
  • [20] Kruk, Ł. (2003). Functional limit theorems for a simple auction. Math. Oper. Res. 28 716–751.
  • [21] Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6 223–240.
  • [22] Kushner, H. J. (1974). On the weak convergence of interpolated Markov chains to a diffusion. Ann. Probab. 2 40–50.
  • [23] Lachapelle, A., Lasry, J.-M., Lehalle, C.-A. and Lions, P.-L. (2016). Efficiency of the price formation process in presence of high frequency participants: A mean-field game analysis. Math. Financ. Econ. 10 223–262.
  • [24] Lakner, P., Reed, J. and Simatos, F. (2017). Scaling limit of a limit order book via the regenerative characterization of Lévy trees. Stoch. Syst. To appear.
  • [25] Lakner, P., Reed, J. and Stoikov, S. (2016). High frequency asymptotics for the limit order book. Mark. Microstruct. Liq. 2 1650004.
  • [26] Lasry, J.-M. and Lions, P.-L. (2007). Mean field games. Jpn. J. Math. 2 229–260.
  • [27] Lowther, G. (2010). Lévy’s characterization of Brownian motion. Available at
  • [28] Osterrieder, J. R. (2007). Arbitrage, market microstructure and the limit order book. Ph.D. thesis, ETH Zurich. DISS. ETH Nr 17121.
  • [29] Roşu, I. (2009). A dynamic model of the limit order book. Rev. Financ. Stud. 22 4601–4641.
  • [30] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour, XIV—1984. Lect. Notes Math. 1180 265–439. Springer, Berlin.