Well-posedness of state-dependent rank-based interacting systems

TL;DR

This paper investigates the mathematical conditions under which certain complex stochastic systems with rank-based interactions are well-defined and unique, especially when coefficients are discontinuous or vanish, with implications for financial models.

Contribution

It establishes strong and weak well-posedness results for state-dependent rank-based SDEs, including cases with discontinuous and vanishing diffusion coefficients.

Findings

Strong well-posedness for planar systems with rank-dependent drift.

Weak well-posedness for high-dimensional systems with elliptic diffusion.

Positivity of solutions when diffusion coefficients vanish at zero.

Abstract

We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong well-posedness for a planar system with rank-dependent drift coefficients, and non-rank-dependent and non-uniformly elliptic diffusion coefficients. We then state weak well-posedness for two classes of high-dimensional rank-based interacting SDEs with elliptic diffusion coefficients. Finally, we address the positivity of solutions in the case where the diffusion coefficients vanish at zero.