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October 2020 A note on jump Atlas models
Clayton Barnes, Andrey Sarantsev
Braz. J. Probab. Stat. 34(4): 844-857 (October 2020). DOI: 10.1214/19-BJPS457


The market weight of a stock is its capitalization (cap) divided by the total market cap. Rank these weights from top to bottom. The capital distribution curve is a plot of weights versus ranks. For the US stock market, it is linear on a double logarithmic scale, and stable with respect to time (Stochastic Portfolio Theory (2002) Springer). This property has been captured by models with rank-dependent dynamics: Each stock’s cap logarithm is a Brownian motion with drift and diffusion coefficients depending on its current rank (Probability Theory and Related Fields 147 (2010) 123–159). However, short-term stock movements have heavy tails. One can add jumps to Brownian motions to capture this. Observed time stability follows from a long-term stability result, stated and proved here. Via simulations, we find which properties of continuous models are preserved after adding jumps.


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Clayton Barnes. Andrey Sarantsev. "A note on jump Atlas models." Braz. J. Probab. Stat. 34 (4) 844 - 857, October 2020.


Received: 1 May 2019; Accepted: 1 September 2019; Published: October 2020
First available in Project Euclid: 25 September 2020

MathSciNet: MR4153645
Digital Object Identifier: 10.1214/19-BJPS457

Keywords: capital distribution curve , Competing Brownian particles , Lévy process , stationary distribution

Rights: Copyright © 2020 Brazilian Statistical Association


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Vol.34 • No. 4 • October 2020
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