Transposition Method for Backward Stochastic Evolution Equations Revisited, and Its Application

TL;DR

This paper enhances the transposition method for solving backward stochastic evolution equations in infinite-dimensional spaces and applies it to derive a general Pontryagin maximum principle for stochastic control.

Contribution

It improves the transposition method to handle more general backward stochastic evolution equations and removes a key technical assumption from previous work.

Findings

Developed an improved transposition method for vector and operator-valued equations.

Derived a general Pontryagin maximum principle in infinite dimensions.

Removed a restrictive technical assumption from earlier results.

Abstract

The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In 1articular, we drop the technical assumption appeared in [Q. L\"u and X. Zhang, Springer Briefs in Mathematics,Springer, New York, 2014, Theorem 9.1].