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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 545–585 | Cite as

Comparison Techniques for Competing Brownian Particles

  • Andrey SarantsevEmail author
Article
  • 12 Downloads

Abstract

Consider a finite system of Brownian particles on the real line. Each particle has drift and diffusion coefficients depending on its current rank relative to other particles, as in Karatzas et al. (Ann I H Poincare-PR 52(1):323–354, 2016). We prove some comparison results for these systems. As an example, we show that if we remove a few particles from the top, then the gaps between adjacent particles become stochastically larger, the local times of collision between adjacent particles become stochastically smaller, and the remaining particles shift upward, in the sense of stochastic ordering.

Keywords

Reflected Brownian motion Competing Brownian particles Asymmetric collisions Skorohod problem Stochastic comparison 

Mathematics Subject Classification (2010)

Primary 60K35 Secondary 60J60 60J65 60H10 91B26 

Notes

Acknowledgements

I would like to thank Ioannis Karatzas, Soumik Pal, Xinwei Feng, Amir Dembo, and Vladas Sidoravicius for help and useful discussion. This research was partially supported by NSF grants DMS 1007563, DMS 1308340, DMS 1405210, and DMS 1409434.

References

  1. 1.
    Banner, A.D., Fernholz, E.R., Ichiba, T., Karatzas, I., Papathanakos, V.: Hybrid atlas models. Ann. Appl. Probab. 21(2), 609–644 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banner, A.D., Fernholz, E.R., Karatzas, I.: Atlas models of equity markets. Ann. Appl. Probab. 15(4), 2296–2330 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Banner, A.D., Ghomrasni, R.: Local times of ranked continuous semimartingales. Stoch. Process. Appl. 118(7), 1244–1253 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruggeman, C., Sarantsev, A.: Multiple collisions in systems of competing Brownian particles. Bernoulli. 24(1), 156–201 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chatterjee, S., Pal, S.: A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Rel. Fields 147(1–2), 123–159 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dembo, A., Shkolnikov, M., Varadhan, S.R.S., Zeitouni, O.: Large deviations for diffusions interacting through their ranks. Comm. Pure Appl. Math. 69(7), 1259–1313 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dembo, A., Tsai, L.-C.: Equilibrium fluctuation of the atlas model. Ann. Probab. 45(6B), 4529–4560 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Fernholz, E.R., Ichiba, T., Karatzas, I.: Two Brownian particles with rank-based characteristics and skew-elastic collisions. Stoch. Process. Appl. 123(8), 2999–3026 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haddad, J.-P., Mazumdar, R.R., Piera, F.J.: Pathwise comparison results for stochastic fluid networks. Queueing Syst. 66(2), 155–168 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Harrison, J.M.: The heavy traffic approximation for single server queues in series. J. Appl. Probab. 10, 613–629 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Harrison, J.M.: The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Probab. 10(4), 886–905 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Harrison, J.M., Reiman, I.M.: Reflected Brownian motion on an orthant. Ann. Probab. 9(2), 302–308 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Harrison, J.M., Williams, J.M.: Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22(2), 77–115 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ichiba, T.: Topics in Multi-dimensional Diffusion Theory: Attainability, Reflection, Ergodicity and Rankings. ProQuest LLC, Ann Arbor (2009). Thesis (Ph.D.)—Columbia UniversityGoogle Scholar
  15. 15.
    Ichiba, T., Karatzas, I.: On collisions of Brownian particles. Ann. Appl. Probab. 20(3), 951–977 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ichiba, T., Karatzas, I., Shkolnikov, M.: Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Rel. Fields 156(1–2), 229–248 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ichiba, T., Pal, S., Shkolnikov, M.: Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Rel. Fields 1–34, (2012)Google Scholar
  18. 18.
    Jourdain, B.: Diffusion processes associated with nonlinear evolution equations for signed measures. Methodol. Comput. Appl. Probab. 2(1), 69–91 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Jourdain, B., Malrieu, F.: Propagation of chaos and Poincare inequalities for a system of particles interacting through their cdf. Ann. Appl. Probab. 18(5), 1706–1736 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jourdain, B., Reygner, J.: Capital distribution and portfolio performance in the mean-field Atlas model. Ann. Finance. 11(2), 151–198 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jourdain, B., Reygner, J.: Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation. Stoch. Partial Differ. Equ. Anal. Comput. 1(3), 455–506 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Jourdain, B., Reygner, J.: The small noise limit of order-based diffusion processes. Electron. J. Probab. 19(29), 1–36 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Karatzas, I., Pal, S., Shkolnikov, M.: Systems of Brownian particles with asymmetric collisions. Ann. I H Poincare-PR. 52(1), 323–354 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Karatzas, I., Sarantsev, A.: Diverse market models of competing Brownian particles with splits and mergers (2014). Available at arXiv:1404.0748
  25. 25.
    Kella, O., Ramasubramanian, S.: Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant. Math. Oper. Res. 37(2), 301–312 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kella, O., Whitt, W.: Stability and structural properties of stochastic storage networks. J. Appl. Probab. 33(4), 1169–1180 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pal, S., Pitman, J.: One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18(6), 2179–2207 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pal, S., Shkolnikov, M.: Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24(4), 1482–1508 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ramasubramanian, S.: A subsidy-surplus model and the Skorokhod problem in an orthant. Math. Oper. Res. 25(3), 509–538 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Reygner, J.: Chaoticity of the stationary distribution of rank-based interacting diffusions. Electron. Commun. Prob. 20(60), 1–20 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sarantsev, A.: Infinite systems of competing Brownian particles. Ann. I H Poincare-PR. 53(4), 2279–2315 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sarantsev, A.: Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20(29), 1–28 (2015)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Sarantsev, A.: Two-sided infinite systems of competing Brownian particles. ESAIM Probab. Statist 21, 317–349 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Shkolnikov, M.: Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21(5), 1911–1932 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Shkolnikov, M.: Large systems of diffusions interacting through their ranks. Stoch. Process. Appl. 122(4), 1730–1747 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Williams, R.J.: Semimartingale reflecting Brownian motions in the orthant. In: Stochastic Networks, volume 71 of IMA Vol. Mathematics and Applications, pp. 125–137. Springer, New York (1995)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of NevadaRenoUSA

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