Strong solutions of stochastic equations with rank-based coefficients

Tomoyuki Ichiba, Ioannis Karatzas, Mykhaylo Shkolnikov

Research output: Contribution to journalArticlepeer-review

36 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)229-248
Number of pages20
JournalProbability Theory and Related Fields
Issue number1-2
StatePublished - Jun 2013
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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


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