Partially observed linear quadratic control problem with delay via backward separation method

Abstract

Summary In this paper, we study a partially observed linear quadratic optimal control problem derived by stochastic differential delay equations. Combining backward separation method with stochastic filtering, we obtain optimal feedback regulators in some special cases. Some filtering results for anticipated backward stochastic differential equations are also developed by expressing the solutions of the anticipated backward stochastic differential equations as some Itô's processes. Copyright © 2016 John Wiley & Sons, Ltd.

A Partially Observed Nonzero‐Sum Stochastic Differential Game with Delays and its Application to Finance

“…Otherwise, the filtering problem will become very difficult, and the optimal filtering results for both the state and the adjoint processes will not be obtained due to the delayed and advanced time durations. One can refer to Section V in [16].…”

“…Wang et al [15] applied the direct method, the approximation method and the Malliavin derivative method to concern with maximum principles for the partially observed forward-backward stochastic system. Wu and Shu [16,17] investigated both linear-quadratic and nonlinear optimal control problems for partially observed stochastic delay systems. Xiong et al [18] considered a partially observed nonzero-sum SDG of forward-backward stochastic differential equations, and obtained the necessary and sufficient maximum principle.…”

“… applied the direct method, the approximation method and the Malliavin derivative method to concern with maximum principles for the partially observed forward‐backward stochastic system. Wu and Shu investigated both linear‐quadratic and nonlinear optimal control problems for partially observed stochastic delay systems. Xiong et al.…”

Linear Quadratic Optimal Control Problems of Delayed Backward Stochastic Differential Equations

“…Stochastic differential delayed equations (SDDEs, for short) are nice tools to described the dynamics of some natural and social phenomena (see Mohammed [15,16]). Since then, massive research on related topics has become a desirable and serious endeavor among researchers in stochastic optimal control, differential games, mathematical finance and so on (see [17,21,2,18,32,3,9,10,6,28,34,30,12]). The forward SDDEs characterize the dynamic changes of state processes with given initial state trajectories.…”

On linear quadratic optimal control of discrete‐time complex‐valued linear systems

“…The LQR problem has been extended to several different situations. For example, LQR control with time delay was investigated in the works of Liang and Zhang and Wu and Shu; the LQR problem for stochastic systems was studied in the works of Liu et al and Wu and Zhuang; and LQR for stochastic time‐delay systems was solved in the work of Li et al In the work of Wu et al, the LQR problem was solved for the so‐called antilinear system (which is a special case of the complex‐valued linear systems), and a so‐called anti‐Riccati equation–based solution was established.…”