Linear Quadratic Optimal Control Problems of Delayed Backward Stochastic Differential Equations

TL;DR

This paper addresses linear quadratic optimal control for delayed backward stochastic differential equations, deriving explicit feedback controls and introducing new delayed Riccati equations with solvability analysis.

Contribution

It introduces a novel class of delayed Riccati equations and provides explicit optimal control representations for delayed backward stochastic systems.

Findings

Explicit optimal control as linear feedback of past and future states

Introduction of a new class of delayed Riccati equations

Proof of unique solvability of the Riccati equations

Abstract

This paper is concerned with a linear quadratic optimal control problem of delayed backward stochastic differential equations. An explicit representation is derived for the optimal control, which is a linear feedback of the entire past history and the future state trajectory in a short period of time. This is one of the major distinctive features of the delayed backward stochastic linear quadratic optimal control problem. To obtain the optimal feedback, a new class of delayed Riccati equations is introduced and the unique solvability of their solutions are discussed in detail.