2012/04/15 by Qi Lü, Xu Zhang, Lü, Qi +1 · 3 citations
Economics, Econometrics and Finance · Engineering · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1204.3275
openalex publication_date 2012/04/15 · openalex created_date 2022/12/13 · openalex updated_date 2026/07/28
The main purpose of this paper is to give a solution to a long-standing unsolved problem in stochastic control theory, i.e., to establish the Pontryagin-type maximum principle for optimal controls of general infinite dimensional nonlinear stochastic evolution equations. Both drift and diffusion terms can contain the control variables, and the control domains are allowed to be nonconvex. The key to reach it is to provide a suitable formulation of operator-valued backward stochastic evolution equations (BSEEs for short), as well as a way to define their solutions. Besides, both vector-valued and operator-valued BSEEs, with solutions in the sense of transposition, are studied. As a crucial preliminary, some weakly sequential Banach-Alaoglu-type theorems are established for uniformly bounded linear operators between Banach spaces.