OPTIMAL CONTROL OF PATH-DEPENDENT MCKEAN-VLASOV SDES IN INFINITE-DIMENSION

• 成果类型: Article
• 署名作者: Cosso, Andrea; Gozzi, Fausto; Kharroubi, Idris; Pham, Huyen; Rosestolato, Mauro
• 署名单位: University of Milan; Luiss Guido Carli University; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Universite Paris Cite; Universite Paris Cite; University of Salento
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1880
• 发表日期: 2023
• 页码: 2863-2918
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; hamilton-jacobi equations; Optimal portfolio choice; kolmogorov equations; VISCOSITY SOLUTIONS; Bellman equations; hilbert-spaces; regular solutions; ii verification; labor income

摘要

We study the optimal control of path-dependent McKean-Vlasov equa-tions valued in Hilbert spaces motivated by non-Markovian mean-field mod-els 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 con-tinuous 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 Ito formula in the spirit of Dupire ((2009), Functional Ito 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 equa-tion notably in the special case when there is no dependence on the law of the control.