Sensitivity of functionals of McKean-Vlasov SDE's with respect to the initial distribution

TL;DR

This paper investigates how the initial distribution affects functionals of McKean-Vlasov SDEs, using advanced gradient techniques to analyze sensitivity in a mean field stochastic framework.

Contribution

It adapts finite-dimensional methods to infinite-dimensional settings, providing new insights into the gradient of solutions of McKean-Vlasov SDEs with respect to initial distributions.

Findings

Derived the infinite-dimensional gradient of McKean-Vlasov SDE solutions.

Extended existing properties of the gradient process.

Provided a framework for distributional sensitivity analysis.

Abstract

We examine the sensitivity at the origin of the distributional robust optimization problem in the context of a model generated by a mean field stochastic differential equation. We adapt the finite dimensional argument developed by Bartl, Drapeau, Obloj \& Wiesel to our framework involving the infinite dimensional gradient of the solution of the mean field SDE with respect to its initial data. We revisit the derivation of this gradient process as previously introduced by Buckdahn, Li \& Peng, and we complement the existing properties so as to satisfy the requirement of our main result.