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

Feng Zhang

doi:10.1155/2013/964765

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.

Highlights

Many random phenomena are described by stochastic differential equations (SDEs), such as the evolution of the stock prices
There exist many phenomena which are characteristic of past dependence; that is, their present value depends on the present situation and on the past history. Such models may be identified as stochastic differential delay equations (SDDEs)
SDDEs have a wide range of applications in physics, biology, engineering, economics, and finance

Summary

Introduction

Many random phenomena are described by stochastic differential equations (SDEs), such as the evolution of the stock prices. A duality between linear SDDEs and anticipated BSDEs was established in [12], which gave a new way to study the maximum principle for delayed stochastic control problems. It is worth pointing out that when we apply Ekeland’s variational principle to deal with the case when there are state constraints, we need the continuity of the state process X(⋅) and the lower semicontinuity of a penalty functional Jρ(V(⋅)) in the control variable V(⋅), which is impossible to prove when the control domain is unbounded To overcome this difficulty, we adopt a convergence technique inspired by Tang and Li [16].

Formulation of the Problem

Maximum Principle in the Case When U Is Bounded in R

Maximum Principle for General Control Domain U