Necessary conditions for optimality for stochastic evolution equations

TL;DR

This paper establishes necessary optimality conditions for stochastic evolution control problems in Hilbert spaces using the maximum principle and backward stochastic evolution equations, allowing coefficients to depend on control variables.

Contribution

It introduces a maximum principle for stochastic evolution equations with control-dependent coefficients using the semigroup approach.

Findings

Derived necessary conditions for optimality in stochastic evolution control.

Extended the maximum principle to include control-dependent coefficients.

Utilized backward stochastic evolution equations for the adjoint system.

Abstract

This paper is concerned with providing the maximum principle for a control problem governed by a stochastic evolution system on a separable Hilbert space. In particular, necessary conditions for optimality for this stochastic optimal control problem are derived by using the adjoint backward stochastic evolution equation. Moreover, all coefficients appearing in this system are allowed to depend on the control variable. We achieve our results through the semigroup approach.