August 2023 Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension
Andrea Cosso, Fausto Gozzi, Idris Kharroubi, Huyên Pham, Mauro Rosestolato
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Ann. Appl. Probab. 33(4): 2863-2918 (August 2023). DOI: 10.1214/22-AAP1880

Abstract

We study the optimal control of path-dependent McKean–Vlasov equations valued in Hilbert spaces motivated by non-Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions (Lions (Audio Conference, 2006–2012)), and prove a related functional Itô formula in the spirit of Dupire ((2009), Functional Itô Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS) and Wu and Zhang (Ann. Appl. Probab. 30 (2020) 936–986). The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.

Citation

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Andrea Cosso. Fausto Gozzi. Idris Kharroubi. Huyên Pham. Mauro Rosestolato. "Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension." Ann. Appl. Probab. 33 (4) 2863 - 2918, August 2023. https://doi.org/10.1214/22-AAP1880

Information

Received: 1 January 2021; Revised: 1 December 2021; Published: August 2023
First available in Project Euclid: 10 July 2023

MathSciNet: MR4612657
zbMATH: 07720494
Digital Object Identifier: 10.1214/22-AAP1880

Subjects:
Primary: 49L25 , 60K35 , 93E20

Keywords: dynamic programming principle , functional Itô calculus , Master Bellman equation , Path-dependent McKean–Vlasov SDEs in Hilbert space , pathwise measure derivative , viscosity solutions

Rights: Copyright © 2023 Institute of Mathematical Statistics

Vol.33 • No. 4 • August 2023
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