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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. Animportant class of market models is based on systems of competing Brownianparticles: each company has a capitalization whose logarithm behaves as aBrownian motion with drift and diffusion coefficients depending on its currentrank. We study such models with a fluctuating number of companies: If at somemoment the share of the total market capitalization of a company reaches afixed level, then the company is split into two parts of random size. Companiesare also allowed to merge, when an exponential clock rings. We find conditionsunder which this system is nonexplosive (i.e., the number of companies remainsfinite at all times) and diverse, yet does not admit arbitrage opportunities.

Diverse market models of competing Brownian particles with splits and mergers