A general maximum principle for optimal control of stochastic differential delay systems

TL;DR

This paper establishes a comprehensive maximum principle for stochastic optimal control problems involving systems with delays in both the coefficients and the control domain, using innovative transformation and adjoint equation techniques.

Contribution

It introduces a new method transforming delayed variational equations into delay-free Volterra integral equations and develops novel adjoint equations for delayed stochastic control systems.

Findings

Derived a general maximum principle for stochastic delay systems.

Proposed a new approach transforming delayed variational equations.

Expressed adjoint equations in compact anticipated backward stochastic differential equations.

Abstract

In this paper, we solve an open problem and obtain a general maximum principle for a stochastic optimal control problem where the control domain is an arbitrary non-empty set and all the coefficients (especially the diffusion term and the terminal cost) contain the control and state delay. In order to overcome the difficulty of dealing with the cross term of state and its delay in the variational inequality, we propose a new method: transform a delayed variational equation into a Volterra integral equation without delay, and introduce novel first-order, second-order adjoint equations via the backward stochastic Volterra integral equation theory. Finally we express these two kinds of adjoint equations in more compact anticipated backward stochastic differential equation types for several special yet typical control systems.