Abstract
Using a recently introduced representation of the second order adjoint state as the solution of a function-valued backward stochastic partial differential equation (SPDE), we calculate the viscosity super- and subdifferential of the value function evaluated along an optimal trajectory for controlled semilinear SPDEs. This establishes the well-known connection between Pontryagin’s maximum principle and dynamic programming within the framework of viscosity solutions. As a corollary, we derive that the correction term in the stochastic Hamiltonian arising in nonsmooth stochastic control problems is nonpositive. These results directly lead us to a stochastic verification theorem for fully nonlinear Hamilton–Jacobi–Bellman equations in the framework of viscosity solutions.
Funding Statement
This work was funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 910 “Control of self-organizing nonlinear systems: Theoretical methods and concepts of application,” Project (A10) “Control of stochastic mean-field equations with applications to brain networks,” while the second author was affiliated with Technische Universität Berlin.
Acknowledgments
The authors would like to express their gratitude to Andrzej Święch for his helpful comments on a first version of this manuscript. Furthermore, the authors would like to thank the anonymous reviewers for their constructive feedback, which helped to improve the manuscript.
Citation
Wilhelm Stannat. Lukas Wessels. "Necessary and sufficient conditions for optimal control of semilinear stochastic partial differential equations." Ann. Appl. Probab. 34 (3) 3251 - 3287, June 2024. https://doi.org/10.1214/23-AAP2038
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