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Strong solutions of stochastic equations with rank-based coefficients
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  • Published: 10 April 2012

Strong solutions of stochastic equations with rank-based coefficients

  • Tomoyuki Ichiba1,
  • Ioannis Karatzas2,3 &
  • Mykhaylo Shkolnikov4 

Probability Theory and Related Fields volume 156, pages 229–248 (2013)Cite this article

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Abstract

We study finite and countably infinite systems of stochastic differential equations, in which the drift and diffusion coefficients of each component (particle) are determined by its rank in the vector of all components of the solution. We show that strong existence and uniqueness hold until the first time three particles collide. Motivated by this result, we improve significantly the existing conditions for the absence of such triple collisions in the case of finite-dimensional systems, and provide the first condition of this type for systems with a countable infinity of particles.

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Author information

Authors and Affiliations

  1. Department of Statistics and Applied Probability, University of California, Santa Barbara, CA, 93106, USA

    Tomoyuki Ichiba

  2. INTECH Investment Management, One Palmer Square, Princeton, NJ, 08542, USA

    Ioannis Karatzas

  3. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Ioannis Karatzas

  4. Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA, 94720, USA

    Mykhaylo Shkolnikov

Authors
  1. Tomoyuki Ichiba
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  2. Ioannis Karatzas
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  3. Mykhaylo Shkolnikov
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Corresponding author

Correspondence to Mykhaylo Shkolnikov.

Additional information

This research was partially supported by NSF grants DMS-08-06211 and DMS-09-05754.

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Cite this article

Ichiba, T., Karatzas, I. & Shkolnikov, M. Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Relat. Fields 156, 229–248 (2013). https://doi.org/10.1007/s00440-012-0426-3

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  • Received: 27 September 2011

  • Accepted: 22 March 2012

  • Published: 10 April 2012

  • Issue Date: June 2013

  • DOI: https://doi.org/10.1007/s00440-012-0426-3

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Keywords

  • Stochastic differential equations
  • Strong existence
  • Strong uniqueness
  • Equations with rank-based coefficients
  • Brownian particles
  • Triple collisions

Mathematics Subject Classification

  • 60H10
  • 60J60
  • 60J65
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