Maximum Principle for Delayed Stochastic Linear-Quadratic Control Problem with State Constraint

Abstract

This paper is concerned with one kind of delayed stochastic linear-quadratic optimal control problems with state constraints. The control domain is not necessarily convex and the control variable does not enter the diffusion coefficient. Necessary conditions in the form of maximum principle as well as sufficient conditions are established.

Maximum principle for delayed stochastic mean-field control problem with state constraint

“…These results will play an important role in exploring the maximum principle of Problem 3.1. One should note that our control domain U is bounded, while the case U is unbounded can be treated via the bounded case with a convergence technique as mentioned in Zhang [23].…”

“…Therefore current research is divided into two kinds of methods to develop the stochastic maximum principle for delayed systems. One involves adjoint equation; it is given by the anticipated BSDE which was introduced by Peng and Yang [21]; see Chen and Wu [19,20], Yu [22] and Zhang [23]. Another method is to derive a system of three-coupled adjoint equations, which consists of two BSDEs and one backwards ordinary stochastic equation; see Øksendal and Sulem [18].…”

“…Many studies of the real problems show that there are state constraints in stochastic optimal control problems. And then we need Ekeland's variational principle to deal with the problem with state constraint; see [24,25] and [23]. To the best of the authors' knowledge, there are few results about the delayed stochastic optimal control problem.…”

“…To the best of the authors' knowledge, there are few results about the delayed stochastic optimal control problem. Reference [23] discussed this kind of problems, but only for the linear-quadratic (LQ) case. The aim of this article is to study the stochastic optimal control problem of mean-field system with delay and state constraints.…”