Necessary and sufficient optimality conditions for relaxed and strict control problems of backward systems

TL;DR

This paper develops necessary and sufficient optimality conditions for relaxed and strict control problems in nonlinear backward stochastic systems, addressing non-convex control sets with a novel approach.

Contribution

It introduces a new method to derive optimality conditions for both relaxed and strict controls in backward stochastic differential equations, including non-convex control sets.

Findings

Established necessary and sufficient conditions for relaxed controls.

Extended results to strict control problems as a special case.

Provided a new approach applicable to nonlinear backward systems.

Abstract

We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear backward stochastic differential equation. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are measure-valued processes. The second is a particular case of the first and relates to strict control problems.