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Second order approximations for limit order books


In this paper, we derive a second order approximation for an infinite-dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g. the volume standing at the top of the book). We study the fluctuations of the price and volume process relative to their first order approximation given in ODE–PDE form under two different scaling regimes. In the first case, we suppose that price changes are really rare, yielding a constant first order approximation for the price. This leads to a measure-valued SDE driven by an infinite-dimensional Brownian motion in the second order approximation of the volume process. In the second case, we use a slower rescaling rate, which leads to a non-degenerate first order approximation and gives a PDE with random coefficients in the second order approximation for the volume process. Our results can be used to derive confidence intervals for models of optimal portfolio liquidation under market impact.

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  1. In [12], the claim was proved for \(p^{(n)}\equiv p\) not depending on . However, the proof in [12] can easily be extended to more general \(p^{(n)}\) as long as \(p^{(n)}\) converges uniformly to some \(p\).

  2. By ‘weak’ solution, we mean ‘weak’ in the PDE sense, i.e., we consider \(Z^{u}\) as a distribution-valued process, being an element of \(H^{-3}\). The exact solution concept is explicitly defined in Theorem 6.9 below.


  1. Abergel, F., Jedidi, A.: A mathematical approach to order book modeling. Int. J. Theor. Appl. Finance 16(5), 1350025 (2013)

    Article  MathSciNet  Google Scholar 

  2. Abergel, F., Jedidi, A.: Long-time behavior of a Hawkes process-based limit order book. SIAM J. Financ. Math. 6, 1026–1043 (2015)

    Article  MathSciNet  Google Scholar 

  3. Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Finance 10, 143–157 (2010)

    Article  MathSciNet  Google Scholar 

  4. Bacry, E., Muzy, J.F.: Hawkes model for price and trades high-frequency dynamics. Quant. Finance 14(7), 1147–1166 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bayer, C., Horst, U., Qiu, J.: A functional limit theorem for limit order books with state dependent price dynamics. Ann. Appl. Probab. 27, 2753–2806 (2017)

    Article  MathSciNet  Google Scholar 

  6. Cont, R., de Larrard, A.: Order book dynamics in liquid markets: limit theorems and diffusion approximations (2012). Working paper. Available online at

  7. Cont, R., de Larrard, A.: Price dynamics in a Markovian limit order market. SIAM J. Financ. Math. 4, 1–25 (2013)

    Article  MathSciNet  Google Scholar 

  8. Farmer, J., Gillemot, L., Lillo, F., Mike, S., Sen, A.: What really causes large price changes? Quant. Finance 4, 383–397 (2004)

    Article  Google Scholar 

  9. Gao, X., Deng, S.J.: Hydrodynamic limit of order book dynamics. Probab. Eng. Inf. Sci. 32, 96–125 (2016)

    Article  MathSciNet  Google Scholar 

  10. Guo, X., Ruan, Z., Zhu, L.: Dynamics of order positions and related queues in a limit order book (2015). Working paper. Available online at

  11. Horst, U., Kreher, D.: A diffusion approximation for limit order book models (2017). Working paper. Available online at

  12. Horst, U., Kreher, D.: A weak law of large numbers for a limit order book model with state dependent order dynamics. SIAM J. Financ. Math. 8, 314–343 (2017)

    Article  MathSciNet  Google Scholar 

  13. Horst, U., Paulsen, M.: A law of large numbers for limit order books. Math. Oper. Res. 42, 1280–1312 (2017)

    Article  MathSciNet  Google Scholar 

  14. Horst, U., Wei, X.: A scaling limit for limit order books driven by Hawkes random measures (2017). Working paper. Available online at

  15. Huang, W., Rosenbaum, M.: Ergodicity and diffusivity of Markovian order book models: a general framework. SIAM J. Financ. Math. 8, 874–900 (2017)

    Article  MathSciNet  Google Scholar 

  16. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2002)

    MATH  Google Scholar 

  17. Jaisson, T., Rosenbaum, M.: Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25, 600–631 (2015)

    Article  MathSciNet  Google Scholar 

  18. Kang, H.W., Kurtz, T.G., Popovic, L.: Central limit theorems and diffusion approximations for multiscale Markov chain models. Ann. Appl. Probab. 24, 721–759 (2014)

    Article  MathSciNet  Google Scholar 

  19. Keller-Ressel, M., Müller, M.: A Stefan-type stochastic moving boundary problem. Stoch. Partial Differ. Equ., Anal. Computat. 4, 746–790 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Lakner, P., Reed, J., Simatos, F.: Scaling limit of a limit order book model via the regenerative characterization of Lévy trees. Stochastic Systems 7, 342–373 (2017)

    Article  MathSciNet  Google Scholar 

  21. Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Letta, G., Pratelli, M. (eds.) Probability and Analysis, Lect. Sess. C.I.M.E., Varenna/Italy, 1985. Lect. Notes Math., vol. 1206, pp. 167–241 (1986)

    Chapter  Google Scholar 

  22. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.-L. (ed.) École D’Été de Probabilités de Saint Flour XIV—1984. Lect. Notes Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)

    Chapter  Google Scholar 

  23. Zheng, B., Roueff, F., Abergel, F.: Modelling bid and ask prices using constrained Hawkes processes: ergodicity and scaling limit. SIAM J. Financial Math. 5, 99–136 (2014)

    Article  MathSciNet  Google Scholar 

  24. Zheng, Z.: Stochastic Stefan problems: Existence, Uniqueness and Modeling of Market Limit Orders. PhD thesis, Graduate College of the University of Illinois at Urbana-Champaign (2012). Available online at

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Correspondence to Ulrich Horst.

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Financial support through the CRC TRR 190 is gratefully acknowledged.

Appendix A: A technical estimate for \(L^{2}\)-martingales

Appendix A: A technical estimate for \(L^{2}\)-martingales

Lemma A.1

([21, Theorem 6.1])

There exists a constant \(C > 0\) such that for all martingale differences \((X_{i})\) with values in , we have

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Horst, U., Kreher, D. Second order approximations for limit order books. Finance Stoch 22, 827–877 (2018).

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