A $C^1$-It\^o's formula for flows of semimartingale distributions

TL;DR

This paper extends Itô's formula to $C^1$-functionals of flows of semimartingale distributions, enabling new analysis of McKean-Vlasov control problems with minimal regularity assumptions.

Contribution

It introduces a $C^1$-Itô's formula for flows of distributions of semimartingales, extending previous results and applying it to McKean-Vlasov control problems with a novel verification theorem.

Findings

Established a $C^1$-Itô's formula for distribution flows.

Applied the formula to McKean-Vlasov control problems.

Proved a verification theorem with minimal regularity requirements.

Abstract

We provide an It\^o's formula for $C^1$-functionals of flows of conditional marginal distributions of continuous semimartingales. This is based on the notion of weak Dirichlet process, and extends the $C^1$-It\^o's formula in Gozzi and Russo (2006) to this context. As the first application, we study a class of McKean-Vlasov optimal control problems, and establish a verification theorem which only requires $C^1$-regularity of its value function, which is equivalently the (viscosity) solution of the associated HJB master equation. It goes together with a novel duality result.