Large systems of diffusions interacting through their ranks

Mykhaylo Shkolnikov

doi:10.1016/j.spa.2012.01.011

Large systems of diffusions interacting through their ranks

Mykhaylo Shkolnikov

Research output: Contribution to journal › Article › peer-review

38 Scopus citations

Abstract

We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.

Original language	English (US)
Pages (from-to)	1730-1747
Number of pages	18
Journal	Stochastic Processes and their Applications
Volume	122
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2012.01.011
State	Published - Apr 2012
Externally published	Yes

All Science Journal Classification (ASJC) codes

Statistics and Probability
Modeling and Simulation
Applied Mathematics

Keywords

Capital distributions
Diffusion processes
McKean-Vlasov equation
Particle method
Porous medium equation
Rank-based market models

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10.1016/j.spa.2012.01.011

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abstract = "We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.",

keywords = "Capital distributions, Diffusion processes, McKean-Vlasov equation, Particle method, Porous medium equation, Rank-based market models",

author = "Mykhaylo Shkolnikov",

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AU - Shkolnikov, Mykhaylo

N1 - Funding Information: The author’s research was supported in part by NSF grant DMS-0806211 .

PY - 2012/4

Y1 - 2012/4

N2 - We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.

AB - We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.

KW - Capital distributions

KW - Diffusion processes

KW - McKean-Vlasov equation

KW - Particle method

KW - Porous medium equation

KW - Rank-based market models

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JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 4

ER -

Large systems of diffusions interacting through their ranks

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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