Source: Ann. Appl. Probab. Volume 15, Number 4
(2005), 2296-2330.
Atlas-type models are constant-parameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this paper we study the basic properties of this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock.
References
Bass, R. and Pardoux, E. (1987). Uniqueness of diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557--572.
Mathematical Reviews (MathSciNet):
MR917679
Fernholz, E. R. (2002). Stochastic Portfolio Theory. Springer, New York.
Fernholz, E. R. and Karatzas, I. (2005). Relative arbitrage in volatility-stabilized markets. Annals of Finance 1 149--177.
Fernholz, E. R., Karatzas, I. and Kardaras, C. (2005). Diversity and arbitrage in equity markets. Finance and Stochastics 9 1--27.
Gihman, I. I. and Skorohod, A. V. (1972). Stochastic Differential Equations. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR346904
Harrison, J. M. and Williams, R. J. (1987). Multi-dimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115--137.
Mathematical Reviews (MathSciNet):
MR877593
Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77--115.
Mathematical Reviews (MathSciNet):
MR912049
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.
Khas'minskii, R. Z. (1960). Ergodic properties of recurrent diffusion processes, and stabilization of the solution to the Cauchy problem for parabolic equations. Theory Probab. Appl. 5 179--196.
Mathematical Reviews (MathSciNet):
MR133871
Khas'minskii, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Amsterdam.
Mathematical Reviews (MathSciNet):
MR600653
Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 459--485.
Mathematical Reviews (MathSciNet):
MR894900
Williams, R. J. (1995). Semimartingale reflecting Brownian motion in the orthant. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.). IMA Vol. Math. Appl. 71 125--137. Springer, New York.
