Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations∗
Department of Applied Mathematics, The Hong Kong Polytechnic
University, Hung Hom,
Kowloon, Hong Kong,
2 School of Mathematics, Shandong University, Jinan 250100, P.R. China
Revised: 13 August 2011
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.
Mathematics Subject Classification: 93E20 / 60H10 / 34K50
Key words: Stochastic optimal control / maximum principle / stochastic differential delayed equation / anticipated backward differential equation / fully coupled forward-backward stochastic system / Clarke generalized gradient
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).
© EDP Sciences, SMAI, 2012