Consider a multidimensional diffusion model where the drift and the diffusion coefficients for individual coordinates are functions of the relative sizes of their current value compared to the others. Two such models were introduced by Fernholz and Karatzas as models for equity markets to reflect some well-known empirically observed facts. In the first model, called ‘Rank-based’, the time-dynamics is determined by the ordering in which the coordinate values can be arranged at any point of time. In the other, named the ‘Volatility-stabilized’, the parameters are functions of the ratio of the current value to the total sum over all coordinates. We show some remarkable properties of these models, in particular, phase transitions and infinite divisibility. Relationships with existing models of queueing, dynamic spin glasses, and statistical genetics will be discussed. Part of the material is based on separate joint work with Sourav Chatterjee and Jim Pitman.

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Last updated: 19 Jul 2024