Sensitivity of functionals of McKean-Vlasov SDE's with respect to the initial distribution
Filippo de Feo, Salvatore Federico, Fausto Gozzi, Nizar Touzi
TL;DR
This work addresses how distributional perturbations of the initial law affect distributionally robust criteria for systems driven by McKean–Vlasov SDEs. By importing Bartl–Drapeau–Obloj–Wiesel's origin-sensitivity framework into the infinite-dimensional MKV setting and employing the Buckdahn–Li–Peng tangent/adjoint machinery, the authors derive an explicit derivative formula for the DRO value at zero perturbation, expressed via the adjoint of the MKV flow and the Lions/Wasserstein derivatives of the terminal functional φ. They establish differentiability of the MKV solution with respect to the initial datum through a carefully constructed auxiliary process and MF-linearization, obtaining uniform bounds and representation formulas for the derivative and its adjoint. The paper culminates with a systemic-risk example (variance of the log-monetary reserve) illustrating the practical computation of the sensitivity and linking the general theory to a concrete mean-field contagion setting. Overall, the results provide a rigorous, operational method to quantify model risk in mean-field dynamics under distributional shifts, with potential applications in finance and systemic-risk assessment.
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.