The Annals of Applied Probability

Diverse market models of competing Brownian particles with splits and mergers

Ioannis Karatzas and Andrey Sarantsev

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Abstract

We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349–374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplosive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1329-1361.

Dates
Received: April 2014
Revised: February 2015
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1465905005

Digital Object Identifier
doi:10.1214/15-AAP1118

Mathematical Reviews number (MathSciNet)
MR3513592

Zentralblatt MATH identifier
1347.91229

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65]
Secondary: 91B26: Market models (auctions, bargaining, bidding, selling, etc.)

Keywords
Competing Brownian particles splits mergers diverse markets arbitrage opportunity portfolio

Citation

Karatzas, Ioannis; Sarantsev, Andrey. Diverse market models of competing Brownian particles with splits and mergers. Ann. Appl. Probab. 26 (2016), no. 3, 1329--1361. doi:10.1214/15-AAP1118. https://projecteuclid.org/euclid.aoap/1465905005


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