- Published 2013 in J. Applied Mathematics

In the classical case, many random phenomena are described by stochastic differential equations (SDEs), such as the evolution of the stock prices. However, there also exist many phenomena which are characteristic of past dependence; that is, their present value depends not only on the present situation but also 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. See [1–4] and the references therein. A stochastic control system whose state function is described by the solution of an SDDE is called a delayed stochastic system. This kind of stochastic control problem appears widely in different research fields; see, for example, [3, 5]. It is worth pointing out that the delayed responses make it more difficult to deal with the system, not only for the infinite dimensional problem, but also for the absence of Itô’s formula to deal with the delayed part of the trajectory. One fundamental research direction for stochastic optimal control problems is to establish necessary optimality conditions—Pontryagin maximum principle. By the duality between linear SDEs and backward stochastic differential equations (BSDEs), stochastic maximum principle for forward, backward, and forward-backward systems has been studied by many authors, including Peng [6, 7], Wu [8, 9], Xu [10], and Yong [11]. Recently, Peng and Yang [12] introduced a new type of BSDEs called anticipated BSDEs of the following form:

@article{Zhang2013MaximumPF,
title={Maximum Principle for Delayed Stochastic Linear-Quadratic Control Problem with State Constraint},
author={Feng Zhang},
journal={J. Applied Mathematics},
year={2013},
volume={2013},
pages={964765:1-964765:10}
}