NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMAL CONTROL OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

• 成果类型: Article
• 署名作者: Stannat, Wilhelm; Wessels, Lukas
• 署名单位: Technical University of Berlin; University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP2038
• 发表日期: 2024
• 页码: 3251-3287
• 关键词: verification theorems; MAXIMUM PRINCIPLE; HJB equations; FRAMEWORK

摘要

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.