Nonlinear McKean-Vlasov diffusions under the weak Hormander condition with quantile-dependent coefficients

TL;DR

This paper establishes strong existence and uniqueness for a class of degenerate McKean-Vlasov stochastic differential equations with quantile-dependent coefficients under a weak Hörmander condition, using density bounds and a Feynman-Kac approach.

Contribution

It introduces a novel method to prove existence and uniqueness for degenerate, quantile-dependent McKean-Vlasov equations under weak Hörmander conditions.

Findings

Proved strong existence and uniqueness under weak Hörmander condition.

Developed a new approach using density bounds for time-inhomogeneous diffusions.

Applied Feynman-Kac formula to construct contraction maps for the system.

Abstract

In this paper, the strong existence and uniqueness for a degenerate finite system of quantile-dependent McKean-Vlasov stochastic differential equations are obtained under a weak H\"{o}rmander condition. The approach relies on the apriori bounds for the density of the solution to time inhomogeneous diffusions. The time inhomogeneous Feynman-Fac formula is used to construct a contraction map for this degenerate system.