The Annals of Applied Probability

Diverse market models of competing Brownian particles with splits and mergers

Ioannis Karatzas and Andrey Sarantsev

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349–374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplosive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1329-1361.

Received: April 2014
Revised: February 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65]
Secondary: 91B26: Market models (auctions, bargaining, bidding, selling, etc.)

Competing Brownian particles splits mergers diverse markets arbitrage opportunity portfolio


Karatzas, Ioannis; Sarantsev, Andrey. Diverse market models of competing Brownian particles with splits and mergers. Ann. Appl. Probab. 26 (2016), no. 3, 1329--1361. doi:10.1214/15-AAP1118.

Export citation


  • [1] Audrino, F., Fernholz, E. R. and Ferretti, R. G. (2007). A forecasting model for stock market diversity. Ann. Finance 3 213–240.
  • [2] Banner, A. D., Fernholz, E. R. and Karatzas, I. (2005). Atlas models of equity markets. Ann. Appl. Probab. 15 2296–2330.
  • [3] Banner, A. D. and Ghomrasni, R. (2008). Local times of ranked continuous semimartingales. Stochastic Process. Appl. 118 1244–1253.
  • [4] Bass, R. F. and Pardoux, É. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557–572.
  • [5] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
  • [6] Chatterjee, S. and Pal, S. (2010). A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 123–159.
  • [7] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Modelling and Applied Probability 38. Springer, Berlin.
  • [8] Fernholz, E. R. (2002). Stochastic Portfolio Theory. Applications of Mathematics (New York) 48. Springer, New York.
  • [9] Fernholz, E. R., Ichiba, T. and Karatzas, I. (2013). Two Brownian particles with rank-based characteristics and skew-elastic collisions. Stochastic Process. Appl. 123 2999–3026.
  • [10] Fernholz, E. R., Ichiba, T. and Karatzas, I. (2013). A second-order stock market model. Ann. Finance 9 439–454.
  • [11] Fernholz, E. R. and Karatzas, I. (2005). Relative arbitrage in volatility-stabilized markets. Ann. Finance 1 149–177.
  • [12] Fernholz, E. R. and Karatzas, I. (2009). Stochastic portfolio Theory: An Overview. Handb. Numer. Anal. 15 89–167.
  • [13] Fernholz, E. R., Karatzas, I. and Kardaras, C. (2005). Diversity and relative arbitrage in equity markets. Finance Stoch. 9 1–27.
  • [14] Ichiba, T. (2009). Topics in multi-dimensional diffusion theory: Attainability, reflection, ergodicity and rankings. Ph.D. thesis, Columbia University, ProQuest LLC, Ann Arbor, MI.
  • [15] Ichiba, T. and Karatzas, I. (2010). On collisions of Brownian particles. Ann. Appl. Probab. 20 951–977.
  • [16] Ichiba, T., Karatzas, I. and Shkolnikov, M. (2013). Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156 229–248.
  • [17] Ichiba, T., Pal, S. and Shkolnikov, M. (2013). Convergence rates for rank-based models with applications to portfolio theory. Probab. Theory Related Fields 156 415–448.
  • [18] Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I. and Fernholz, E. R. (2011). Hybrid atlas models. Ann. Appl. Probab. 21 609–644.
  • [19] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [20] Karatzas, I., Pal, S. and Shkolnikov, M. (2016). Systems of Brownian particles with asymmetric collisions. Ann. Inst. Henri Poincaré Probab. Stat. 52 323–354.
  • [21] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [22] Kardaras, C. (2008). Balance, growth and diversity of financial markets. Ann. Finance 4 369–397.
  • [23] Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
  • [24] Osterrieder, J. R. and Rheinlander, T. (2006). Arbitrage opportunities in diverse markets via a non-equivalent measure change. Ann. Finance 2 287–301.
  • [25] Pal, S. and Pitman, J. (2008). One-dimensional Brownian particle systems with rank-dependent drifts. Ann. Appl. Probab. 18 2179–2207.
  • [26] Pal, S. and Shkolnikov, M. (2014). Concentration of measure for Brownian particle systems interacting through their ranks. Ann. Appl. Probab. 24 1482–1508.
  • [27] Ruf, J. and Runggaldier, W. (2013). A systematic approach to constructing market models with arbitrage. Preprint. Available at arXiv:1309.1988.
  • [28] Sarantsev, A. (2014). On a class of diverse market models. Ann. Finance 10 291–314.
  • [29] Sarantsev, A. (2015). Infinite systems of competing Brownian particles. Preprint. Available at arXiv:1403.4229.
  • [30] Sarantsev, A. (2015). Multiple collisions in systems of competing Brownian particles. Preprint. Available at arXiv:1309.2621.
  • [31] Sarantsev, A. (2015). Triple and simultaneous collisions of competing Brownian particles. Electron. J. Probab. 20 1–28.
  • [32] Shkolnikov, M. (2011). Competing particle systems evolving by interacting Lévy processes. Ann. Appl. Probab. 21 1911–1932.
  • [33] Strong, W. (2014). Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension. Finance Stoch. 18 487–514.
  • [34] Strong, W. and Fouque, J.-P. (2011). Diversity and arbitrage in a regulatory breakup model. Ann. Finance 7 349–374.