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

Bruno Bouchard; Xiaolu Tan; Jixin Wang

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arXiv · 2307.07165

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

Bruno Bouchard·Xiaolu Tan·Jixin Wang

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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.

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BibTeX RIS

Bruno Bouchard, Xiaolu Tan, Jixin Wang. 2023-07-14. A $C^1$-It\^o's formula for flows of semimartingale distributions. https://arxiv.org/abs/2307.07165

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