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

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TLDR
In this article, rates of convergence of rank-based interacting diffusions and semimartingale reflecting Brownian motions to equilibrium were determined using Transportation Cost-Information inequalities for Markov processes.
Abstract
We determine rates of convergence of rank-based interacting diffusions and semimartingale reflecting Brownian motions to equilibrium. Bounds on fluctuations of additive functionals are obtained using Transportation Cost-Information inequalities for Markov processes. We work out various applications to the rank-based abstract equity markets used in Stochastic Portfolio Theory. For example, we produce quantitative bounds, including constants, for fluctuations of market weights and occupation times of various ranks for individual coordinates. Another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio. This produces estimates of probabilities of “beating the market”.

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Citations
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Journal ArticleDOI

Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

TL;DR: In this paper, the authors studied a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition, and derived an explicit formula for the time derivative of the flow.
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Triple and simultaneous collisions of competing Brownian particles

TL;DR: Ichiba, Karatzas, Shkolnikov as mentioned in this paper showed a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions in the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them.
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Large systems of diffusions interacting through their ranks

TL;DR: In this article, the authors study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusion tends to infinity and prove that the limiting dynamics is given by a McKean-Vlasov evolution equation.
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Reflected Brownian Motion in a Convex Polyhedral Cone: Tail Estimates for the Stationary Distribution

TL;DR: In this paper, a multidimensional obliquely reflected Brownian motion in a convex polyhedral cone is considered and sufficient conditions for existence of a stationary distribution and convergence to this distribution at an exponential rate, as time goes to infinity.
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Triple and Simultaneous Collisions of Competing Brownian Particles

TL;DR: In this article, a necessary and sufficient condition for a.s. absense of triple and simultaneous collisions was established for the case of asymmetric collisions, when the local time of collision between the particles is split unevenly between them.
References
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Book

Brownian Motion and Stochastic Calculus

TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
Book

Foundations of modern probability

TL;DR: In this article, the authors discuss the relationship between Markov Processes and Ergodic properties of Markov processes and their relation with PDEs and potential theory. But their main focus is on the convergence of random processes, measures, and sets.
Book

The concentration of measure phenomenon

TL;DR: Concentration functions and inequalities isoperimetric and functional examples Concentration and geometry Concentration in product spaces Entropy and concentration Transportation cost inequalities Sharp bounds of Gaussian and empirical processes Selected applications References Index
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The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator

Jim Pitman, +1 more
TL;DR: The two-parameter Poisson-Dirichlet distribution with a single parameter is known as the size-biased random permutation (SBNP) as discussed by the authors, which was introduced by Engen in the context of species diversity and rediscovered by Perman and the authors in the study of excursions of Bessel processes.
Journal ArticleDOI

Reflected Brownian Motion on an Orthant

TL;DR: In this paper, the authors consider a diffusion process whose state space is the nonnegative orthant, and they adopt an approach that requires a restriction on the directions of reflection, but Reiman has shown that this restriction is met by all diffusions arising as heavy traffic limits in open K-station queuing networks.