ONE-DIMENSIONAL BROWNIAN PARTICLE SYSTEMS WITH RANK DEPENDENT DRIFTS

By Soumik Pal, Jim Pitman. 2007

139 citations recorded by the source corpus.

We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For nitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We also study one particular countably innite system, where only the minimum Brownian particle gets a constant upward drift, and prove that independent and identically distributed exponential spacings remain stationary under the dynamics of such a process. Some related conjectures in this direction have also been discussed.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ef85fc7c3fd5c37adddf76f4b0f31d450b86c186

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Citations in the corpus

Convergence rates for rank-based models with applications to portfolio theory (2011, methodology): “We recall from Theorem 8 in [32] the following facts.”
HYBRID ATLAS MODELS (2009, methodology): “With γi = 0, ρi,j = 0, 1 ≤ i, j ≤ n and σ 2 1 = · · · = σ 2 n , we recover the case studied by Banner, Fernholz and Karatzas [3] and Pitman and Pal [22].”
On collisions of Brownian particles (2008, background): “Recent work on interacting particle systems by Pal & Pitman [19] clarifies the long-range behavior of the spacings between the arranged Brownian particles under the equal variance condition: σ̃1 = · · · = σ̃n ; the setting of systems with countably many particle is also studied there, and related wo”