Stochastic maximum principle for equations with delay: the non-convex case

TL;DR

This paper extends the stochastic Pontryagin maximum principle to controlled delay equations with non-convex control sets, using first and second order adjoint BSDEs to derive necessary optimality conditions.

Contribution

It develops a general stochastic maximum principle for delay equations with non-convex controls, including second order adjoint equations and handling delays in both state and control.

Findings

Derived necessary conditions for optimality in delay equations.

Formulated maximum principle using first and second order adjoint BSDEs.

Addressed control problems with delays in state and control.

Abstract

In this paper we develop necessary conditions for optimality, in the form of the stochastic Pontryagin maximum principle, for controlled equations with pointwise delay in the state and with control dependent noise, in the general case of controls $u \in U$ with $U$ not necessarily convex. The maximum principle is formulated by means of first and second order adjoint BSDEs. We also outline how to deal with control problems with pointwise delay both in the state and in the control.