General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions

TL;DR

This paper establishes a Pontryagin-type maximum principle for optimal controls in infinite-dimensional stochastic evolution equations, addressing a longstanding problem by formulating operator-valued backward stochastic equations and defining their solutions.

Contribution

It introduces a novel formulation of operator-valued backward stochastic evolution equations and their solutions, enabling the derivation of the maximum principle in infinite dimensions.

Findings

Formulated operator-valued backward stochastic evolution equations.

Established solutions in the sense of transposition.

Proved Banach-Alaoglu-type theorems for operator sequences.

Abstract

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.