Convergence rates for rank-based models with applications to portfolio theory

Ichiba, Tomoyuki; Pal, Soumik; Shkolnikov, Mykhaylo

doi:10.1007/s00440-012-0432-5

Cited by 43 publications

(50 citation statements)

References 34 publications

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“…For example, [20] considers Brownian particle systems with rank-dependent drifts, and proves ergodicity with limiting exponential distribution, for the process of spacings between ranked particles. Extending the spacing analysis, [17] proves stability of the ranked relative capitalizations, as well as long horizon estimates on the growth of certain classes of wealth processes. For particular models, such as the Atlas model of [13], the authors of [17] are able to explicitly identify the limiting density via its Laplace transform.…”

Section: Introductionmentioning

confidence: 75%

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Robust maximization of asymptotic growth

Kardaras¹,

Robertson²

2012

Ann. Appl. Probab.

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This paper addresses the question of how to invest in a robust growth-optimal way in a market where the instantaneous expected return of the underlying process is unknown. The optimal investment strategy is identified using a generalized version of the principal eigenfunction for an elliptic second-order differential operator, which depends on the covariance structure of the underlying process used for investing. The robust growth-optimal strategy can also be seen as a limit, as the terminal date goes to infinity, of optimal arbitrages in the terminology of Fernholz and Karatzas [Ann. Appl. Probab. 20 (2010) 1179-1204].Comment: Published in at http://dx.doi.org/10.1214/11-AAP802 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Introductionmentioning

confidence: 75%

“…Extending the spacing analysis, [17] proves stability of the ranked relative capitalizations, as well as long horizon estimates on the growth of certain classes of wealth processes. For particular models, such as the Atlas model of [13], the authors of [17] are able to explicitly identify the limiting density via its Laplace transform.…”

Section: Introductionmentioning

confidence: 75%

Robust maximization of asymptotic growth

Kardaras¹,

Robertson²

2012

Ann. Appl. Probab.

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“…Assume the sequence (g n ) n≥1 is bounded from below, as in (19). Then the sequence (g N j ) j≥1 is also bounded below.…”

Section: 1mentioning

confidence: 99%

Infinite systems of competing Brownian particles

Sarantsev¹

2017

Ann. Inst. H. Poincaré Probab. Statist.

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Abstract. Consider a system of infinitely many Brownian particles on the real line. At any moment, these particles can be ranked from the bottom upward. Each particle moves as a Brownian motion with drift and diffusion coefficients depending on its current rank. The gaps between consecutive particles form the (infinite-dimensional) gap process. We find a stationary distribution for the gap process. We also show that if the initial value of the gap process is stochastically larger than this stationary distribution, this process converges back to this distribution as time goes to infinity. This continues the work by Pal and Pitman (2008). Also, this includes infinite systems with asymmetric collisions, similar to the finite ones from Karatzas, Pal and Shkolnikov (2016).

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“…the survey [26]), characterizes those (γ, σ) for which the stationary distribution of Z(t) is a product of exponential random variables. Further utilizing this theory, [18] deduces verious stochastic comparison results, whereas [13] and the references therein, estimate the rate t m , m ≫ 1 of convergence in distribution of the spacing process Z(t).…”

Section: Introductionmentioning

confidence: 89%

“…Combining Proposition 1.3 with Lyapunov functions for finite atlas systems (constructed for example in [7,13]), yields the following information on convergence of the atlas m particle spacing fdd at times t m → ∞ fast enough. of finite second moment, for any fixed k ≥ 1, the joint density of (Z 1 (t m ), .…”

Section: Introductionmentioning

confidence: 99%

The infinite Atlas process: Convergence to equilibrium

Dembo¹,

Jara²,

Olla³

2019

Ann. Inst. H. Poincaré Probab. Statist.

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The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.

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Convergence rates for rank-based models with applications to portfolio theory

Cited by 43 publications

References 34 publications

Robust maximization of asymptotic growth

Robust maximization of asymptotic growth

Infinite systems of competing Brownian particles

The infinite Atlas process: Convergence to equilibrium

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