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

Published Online:https://doi.org/10.1287/stsy.2017.0005

References

  • Abergel F, Jedidi A (2013) A mathematical approach to order book modeling. Int. J. Theor. Appl. Finance 16:1350025. MR3085985.Google Scholar
  • Addario-Berry L, Broutin N (2011) Total progeny in killed branching random walk. Probab. Theory Related Fields 151:265–295. MR2834719.Google Scholar
  • Aldous D (1993) The continuum random tree III. Ann. Probab. 21:248–289. MR1207226.Google Scholar
  • Aldous D (2017) Waves in a spatial queue. Stoch. Syst. 7:197–236.LinkGoogle Scholar
  • Altman E, Levy H (1994) Queueing in space. Adv. Appl. Probab. 26:1095–1116. MR1303878.Google Scholar
  • Bertoin J (1996) Lévy processes. Cambridge Tracts in Mathematics, Vol. 121 (Cambridge University Press, Cambridge, UK). MR1406564.Google Scholar
  • Biais B, Hillion P, Spatt C (1995) An empirical analysis of the limit order book and the order flow in the paris bourse. J. Finance 50:1655–1689.Google Scholar
  • Biggins JD, Lubachevsky BD, Shwartz A, Weiss A (1991) A branching random walk with a barrier. Ann. Appl. Probab. 1:573–581. MR1129775.Google Scholar
  • Billingsley P (1999) Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, Second ed. (John Wiley & Sons, New York). MR1700749.Google Scholar
  • Blanchet J, Chen X (2013) Continuous-time modeling of bid-ask spread and price dynamics in limit order books. arXiv:1310.1103.Google Scholar
  • Bouman N, Borst SC, van Leeuwaarden JSH (2011) Stability of spatial wireless systems with random admissible-set scheduling. Proc. VALUETOOLS ’11 (ICST (Institute for Computer Sciences, Social-Informatics and Telecommunications Engineering), ICST, Brussels, Belgium, Belgium), 57–65.Google Scholar
  • Bramson MD (1978) Maximal displacement of branching Brownian motion. Comm. Pure Appl. Math. 31:531–581. MR0494541.Google Scholar
  • Bramson M (1998) State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30:89–148. MR1663763.Google Scholar
  • Brown M, Peköz EA, Ross SM (2010) Some results for skip-free random walk. Probab. Engrg. Inform. Sci. 24:491–507. MR2725345.Google Scholar
  • Brunet E, Derrida B (1997) Shift in the velocity of a front due to a cutoff. Phys. Rev. E 56:2597–2604.Google Scholar
  • Coffman EG Jr, Gilbert EN (1987) Polling and greedy servers on a line. Queueing Syst. 2:115–145. MR905435.Google Scholar
  • Cont R, de Larrard A Order book dynamics in liquid markets: Limit theorems and diffusion approximations. arXiv:1202.6412.Google Scholar
  • Cont R, de Larrard A (2013) Price dynamics in a Markovian limit order market. SIAM J. Financial Math. 4:1–25. MR3032934.Google Scholar
  • Cont R, Kukanov A, Stoikov S (2014) The price impact of order book events. J. Financial Econometrics 12:47.Google Scholar
  • Cont R, Stoikov S, Talreja R (2010) A stochastic model for order book dynamics. Oper. Res. 58(3):549–563. MR2680564.LinkGoogle Scholar
  • Ding J, Zeitouni O (2014) Extreme values for two-dimensional discrete Gaussian free field. Ann. Probab. 42:1480–1515. MR3262484.Google Scholar
  • Duquesne T, Le Gall J-F (2002) Random trees, Lévy processes and spatial branching processes. Astérisque 281:vi+147. MR1954248.Google Scholar
  • Durrett R, Kesten H, Waymire E (1991) On weighted heights of random trees. J. Theoret. Probab. 4:223–237. MR1088403.Google Scholar
  • Foss S, Rolla LT, Sidoravicius V (2015) Greedy walk on the real line. Ann. Probab. 43:1399–1418. MR3342666.Google Scholar
  • Foucault T, Kadan O, Kandel E (2005) Limit order book as a market for liquidity. Rev. Financial Stud. 18:1171.Google Scholar
  • Garèche A, Disdier G, Kockelkoren J, Bouchaud JP (2013) Fokker-Planck description for the queue dynamics of large tick stocks. Phys. Rev. E 88:032809.Google Scholar
  • Getoor RK (1979) Excursions of a Markov process. Ann. Probab. 7:244–266. MR525052.Google Scholar
  • Gould MD, Porter MA, Williams S, McDonald M, Fenn DJ, Howison SD (2013) Limit order books. Quant. Finance 13:1709–1742. MR3175940.Google Scholar
  • Gromoll HC (2004) Diffusion approximation for a processor sharing queue in heavy traffic. Ann. Appl. Probab. 14:555–611. MR2052895.Google Scholar
  • Horst U, Paulsen M (2017) A law of large numbers for limit order books. Math. Oper. Res. 42(4):1280–1312.LinkGoogle Scholar
  • Hull JC (2018) Options, Futures, and Other Derivatives, 10th ed. (Pearson, New York).Google Scholar
  • Jacod J, Shiryaev AN (2003) Limit Theorems for Stochastic Processes. Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Second ed., Vol. 288 (Springer, Berlin). MR1943877.Google Scholar
  • Janson S (2006) Random cutting and records in deterministic and random trees. Random Structures Algorithms 29:139–179. MR2245498.Google Scholar
  • Kallenberg O (2002) Foundations of modern probability. Probability and Its Applications (New York), Second ed. (Springer, New York). MR1876169.Google Scholar
  • Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Graduate Texts in Mathematics, Second ed., Vol. 113 (Springer, New York). MR1121940.Google Scholar
  • Kelly F, Yudovina E (2017) A Markov model of a limit order book: Thresholds, recurrence, and trading strategies. Math. Oper. Res., ePub ahead of print July 21, 2017, https://doi.org/10.1287/moor.2017.0857.Google Scholar
  • Kesten H (1994) A limit theorem for weighted branching process trees. The Dynkin Festschrift. Progr. Probab., Vol. 34 (Birkäuser, Boston), 153–166. MR1311717.Google Scholar
  • Kirilenko A, Sowers RB, Meng X (2013) A multiscale model of high-frequency trading. Algorithmic Finance 2:59–98.Google Scholar
  • Lakner P, Reed J, Stoikov S (2016) High frequency asymptotics for the limit order book. Mark. Microstructure Liq. 2:1650004.Google Scholar
  • Lambert A, Simatos F (2015) Asymptotic behavior of local times of compound poisson processes with drift in the infinite variance case. J. Theoret. Probab. 28:41–91. MR3320960.Google Scholar
  • Lambert A, Simatos F, Zwart B (2013) Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and Processor-Sharing queues. Ann. Appl. Probab. 23:2357–2381.Google Scholar
  • Le Gall J-F, Miermont G (2012) Scaling limits of random trees and planar maps. Probability and Statistical Physics in Two and More Dimensions. Clay Math. Proc., Vol. 15 (Amer. Math. Soc., Providence, RI), 155–211. MR3025391.Google Scholar
  • Limic V (2000) On the behavior of LIFO preemptive resume queues in heavy traffic. Electron. Comm. Probab. 5:13–27 (electronic). MR1736721.Google Scholar
  • Limic V (2001) A LIFO queue in heavy traffic. Ann. Appl. Probab. 11:301–331. MR1843048.Google Scholar
  • Luckock H (2003) A steady-state model of the continuous double auction. Quant. Finance 3:385–404.Google Scholar
  • Núñez-Queija R (2001) Note on the GI/GI/1 queue with LCFS-PR observed at arbitrary times. Probab. Engrg. Inform. Sci. 15:179–187. MR1828573.Google Scholar
  • Osterrieder JR (2007) Arbitrage, the limit order book and market microstructure aspects in financial market models. PhD thesis, ETH, Zurich. MR2715947.Google Scholar
  • Pardoux E, Wakolbinger A (2011) From exploration paths to mass excursions—Variations on a theme of Ray and Knight. Blath J, Imkeller P, Roelly S, eds. Surveys in Stochastic Processes. EMS Series of Congress Reports, 87–106.Google Scholar
  • Reiman MI (1984) Some diffusion approximations with state space collapse. Modelling and Performance Evaluation Methodology (Paris, 1983). Lecture Notes in Control and Inform. Sci., Vol. 60 (Springer, Berlin), 209–240. MR893658.Google Scholar
  • Roelly-Coppoletta S (1986) A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17:43–65. MR878553.Google Scholar
  • Simatos F (2014) Coupling limit order books and branching random walks. J. Appl. Probab. 51:625–639. MR3256216.Google Scholar
  • Swart JM (2016) Rigorous results for the Stigler-Luckock model for the evolution of an order book. arXiv:1605.01551.Google Scholar
  • van de Ven PM, Borst SC, Ying L (2013) Inefficiency of MaxWeight scheduling in spatial wireless networks. Comput. Commun. 36:1350–1359.Google Scholar
  • Weill M (2007) Regenerative real trees. Ann. Probab. 35:2091–2121. MR2353384.Google Scholar
  • Williams RJ (1998) Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Syst. 30:27–88. MR1663759.Google Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.