Large volatility-stabilized markets

TL;DR

This paper studies large systems of interacting diffusions in financial markets, showing their empirical measures converge to a PDE solution, with an explicit stochastic representation involving squared Bessel processes.

Contribution

It provides a rigorous limit theorem for volatility-stabilized market models as the number of assets grows large, including an explicit stochastic representation of the limit.

Findings

Empirical measures converge to a PDE solution after rescaling.

Explicit stochastic representation via time-changed squared Bessel processes.

Provides insights into the asymptotic behavior of large financial systems.

Abstract

We investigate the behavior of systems of interacting diffusion processes, known as volatility-stabilized market models in the mathematical finance literature, when the number of diffusions tends to infinity. We show that, after an appropriate rescaling of the time parameter, the empirical measure of the system converges to the solution of a degenerate parabolic partial differential equation. A stochastic representation of the latter in terms of one-dimensional distributions of a time-changed squared Bessel process allows us to give an explicit description of the limit.