Polynomial McKean-Vlasov SDEs

TL;DR

This paper introduces a novel class of polynomial McKean-Vlasov SDEs with state-dependent coefficients, providing new existence, uniqueness, and moment computation results that extend current stochastic analysis methods.

Contribution

It develops a new framework for polynomial McKean-Vlasov SDEs, enabling solution existence, uniqueness, and moment calculations via non-linear ODEs, which were previously inaccessible.

Findings

Established existence and uniqueness of solutions for the new class of SDEs.

Derived non-linear ODEs for computing moments of these SDEs.

Provided new results on global solutions to certain ODEs.

Abstract

We study a new class of McKean-Vlasov stochastic differential equations (SDEs), possibly with common noise, applying the theory of time-inhomogeneous polynomial processes. The drift and volatility coefficients of these SDEs depend on the state variables themselves as well as their conditional moments in a way that mimics the standard polynomial structure. Our approach leads to new results on the existence and uniqueness of solutions to such conditional McKean-Vlasov SDEs which are, to the best of our knowledge, not obtainable using standard methods. Moreover, we show in the case without common noise that the moments of these McKean-Vlasov SDEs can be computed by non-linear ODEs. As a by-product, this also yields new results on the existence and uniqueness of global solutions to certain ODEs.