For a broad class of law-dependent stochastic partial differential equations, the paper derives the cost gradient, a Pontryagin maximum principle, and existence of a deterministic optimal control.
An introductory approach to duality in optimal stochastic control
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Pontryagin Maximum Principle for McKean-Vlasov Stochastic Reaction-Diffusion Equations
For a broad class of law-dependent stochastic partial differential equations, the paper derives the cost gradient, a Pontryagin maximum principle, and existence of a deterministic optimal control.