Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Systems of Brownian particles with asymmetric collisions

Ioannis Karatzas, Soumik Pal, and Mykhaylo Shkolnikov

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Abstract

We study systems of Brownian particles on the real line which interact by splitting the local times of collisions among themselves in an asymmetric manner. We prove strong existence and uniqueness of such processes and identify them with the collections of ordered processes in a Brownian particle system, in which the drift coëfficients, the diffusion coëfficients, and the collision local times for the individual particles are assigned according to their ranks. These Brownian systems can be viewed as generalizations of those arising in first-order models for equity markets in the context of stochastic portfolio theory, and are able to correct for several shortcomings of such models while being equally amenable to computations. We also show that, in addition to being of interest in their own right, such systems of Brownian particles arise as universal scaling limits of systems of jump processes on the integer lattice with local interactions. A key step in the proof is the analysis of a generalization of Skorokhod maps which include “local times” at the intersection of faces of the nonnegative orthant. The result extends the convergence of the totally asymmetric simple exclusion process (TASEP) to its continuous analogue. Finally, we identify those among the Brownian particle systems which have a probabilistic structure of determinantal type.

Résumé

Nous étudions des systèmes de particules browniennes sur l’axe réel qui interagissent de façon asymétrique en fonction de leurs temps locaux de collision. Nous prouvons l’existence et l’unicité au sens fort de tels processus et les identifions avec un système de particules constitué d’une famille de processus browniens ordonnés où les coefficients de dérive et de diffusion, ainsi que les temps locaux de collision entre les particules, dépendent de leurs rangs. Ces systèmes browniens peuvent être compris comme des généralisations de processus stochastiques issus des marchés boursiers et de la gestion de portefeuilles. Nous pouvons pallier certaines lacunes de ces modèles tout en conservant la possibilité d’effectuer des calculs explicites. Nous montrons aussi, qu’en plus de leur intérêt intrinsèque, de tels systèmes de particules browniennes apparaissent comme des limites universelles de processus de sauts sur un réseau avec des interactions locales. Une étape clef dans la preuve est l’analyse d’une généralisation des applications de Skorokhod qui inclut les temps locaux à l’intersection des faces de l’orthant positif. Le résultat généralise la convergence du processus d’exclusion simple totalement asymétrique (TASEP) vers sa version continue. Finalement, nous identifions parmi les systèmes de particules browniennes ceux qui possèdent une structure probabiliste déterminantale.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 323-354.

Dates
Received: 30 August 2013
Revised: 11 September 2014
Accepted: 19 September 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089271

Digital Object Identifier
doi:10.1214/14-AIHP646

Mathematical Reviews number (MathSciNet)
MR3449305

Zentralblatt MATH identifier
1333.60206

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H10: Stochastic ordinary differential equations [See also 34F05] 91B26: Market models (auctions, bargaining, bidding, selling, etc.)

Keywords
Determinantal processes Interacting particle systems Invariance principles Reflected Brownian motions Skorokhod maps Stochastic Portfolio Theory Strong solutions of stochastic differential equations Triple collisions

Citation

Karatzas, Ioannis; Pal, Soumik; Shkolnikov, Mykhaylo. Systems of Brownian particles with asymmetric collisions. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 323--354. doi:10.1214/14-AIHP646. https://projecteuclid.org/euclid.aihp/1452089271


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