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