ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2012 - Volume 18, Issue 04  

Research Article

Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations ∗

Jianhui Huanga1 and Jingtao Shia2

a1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China. majhuang@inet.polyu.edu.hk

a2 School of Mathematics, Shandong University, Jinan 250100, P.R. China; shijingtao@sdu.edu.cn

Abstract

This paper deals with the optimal control problem in which the controlled system is described by a fully coupled anticipated forward-backward stochastic differential delayed equation. The maximum principle for this problem is obtained under the assumption that the diffusion coefficient does not contain the control variables and the control domain is not necessarily convex. Both the necessary and sufficient conditions of optimality are proved. As illustrating examples, two kinds of linear quadratic control problems are discussed and both optimal controls are derived explicitly.

(Received August 12 2010)

(Revised August 13 2011)

(Online publication January 16 2012)

Key Words:

  • Stochastic optimal control;
  • maximum principle;
  • stochastic differential delayed equation;
  • anticipated backward differential equation;
  • fully coupled forward-backward stochastic system;
  • Clarke generalized gradient

Mathematics Subject Classification:

  • 93E20;
  • 60H10;
  • 34K50

Footnotes

∗  Jianhui Huang was supported by Departmental General Research Fund (No. A-SA69) and RGC Earmarked Grants of The Hong Kong Polytechnic University (Nos. 500909, 501010). Jingtao Shi was supported by China Postdoctoral Science Foundation Funded Project (No. 20100481278), Postdoctoral Innovation Foundation Funded Project of Shandong Province (No. 201002026), National Natural Sciences Foundations of China (No. 11126209) and Shandong Province (No. ZR2011AQ012), Research Award Fund for Outstanding Young and Middle-aged Scientists of Shandong Province (No. BS2011SF010), and Independent Innovation Foundation of Shandong University (IIFSDU, No. 2010TS060).