Mean field stochastic differential equations with a diffusion coefficient with irregular distributional dependence
Jani Nykänen
TL;DR
This paper studies one-dimensional mean-field SDEs with a diffusion coefficient that depends discontinuously on the law via a threshold function of the distribution. By applying a Lamperti-type transformation, the authors convert the original distribution-dependent SDE into a Gaussian SDE with deterministic coefficients, enabling rigorous analysis of existence and uniqueness of strong solutions. They establish conditions under which a unique strong solution exists on a maximal interval, and identify scenarios that lead to global solvability or finite-time explosion, as well as cases where multiple solutions or nonexistence can occur. The transformation and monotonicity framework provide a tractable approach to regime-switching diffusion coefficients driven by the law of the process, with potential applications to regimes in finance and other fields where volatility responds discontinuously to the evolving distribution.
Abstract
We study mean field stochastic differential equations with a diffusion coefficient that depends on the distribution function of the unknown process in a discontinuous manner, which is a type of distribution dependent regime switching. To determine the distribution function we show that under certain conditions these equations can be transformed into SDEs with deterministic coefficients using a Lamperti-type transformation. We prove an existence and uniqueness result and consider cases when the uniqueness may fail or a solution exists only for a finite time.