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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This research was partially supported by NSF grants DMS-08-06211 and DMS-09-05754.
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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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DOI: https://doi.org/10.1007/s00440-012-0426-3
Keywords
- Stochastic differential equations
- Strong existence
- Strong uniqueness
- Equations with rank-based coefficients
- Brownian particles
- Triple collisions