A C1-Itô’s Formula for Flows of Semimartingale Distributions

We provide an Itô’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 -Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 2006) to this context. As...

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Bibliographic Details
Published in:Applied mathematics & optimization 2024-08, Vol.90 (1), p.26
Main Authors: Bouchard, Bruno, Tan, Xiaolu, Wang, Jixin
Format: Article
Language:English
Subjects:
Applied mathematics
Calculus of Variations and Optimal Control
Optimization
Control
Decomposition
Martingales
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical and Computational Physics
Optimal control
Optimization
Simulation
Systems Theory
Theoretical
Viscosity
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Summary:We provide an Itô’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 -Itô’s formula in Gozzi and Russo (Stoch Process Appl 116(11):1563–1583, 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.
ISSN:0095-4616
1432-0606
DOI:10.1007/s00245-024-10165-y