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

ESAIM: Control, Optimisation and Calculus of Variations (2012)

- Volume: 18, Issue: 4, page 1073-1096
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topHuang, Jianhui, and Shi, Jingtao. "Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1073-1096. <http://eudml.org/doc/272939>.

@article{Huang2012,

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.},

author = {Huang, Jianhui, Shi, Jingtao},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {stochastic optimal control; maximum principle; stochastic differential delayed equation; anticipated backward differential equation; fully coupled forward-backward stochastic system; Clarke generalized gradient; Clarke's generalized gradient},

language = {eng},

number = {4},

pages = {1073-1096},

publisher = {EDP-Sciences},

title = {Maximum principle for optimal control of fully coupled forward-backward stochastic differential delayed equations},

url = {http://eudml.org/doc/272939},

volume = {18},

year = {2012},

}

TY - JOUR

AU - Huang, Jianhui

AU - Shi, Jingtao

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

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2012

PB - EDP-Sciences

VL - 18

IS - 4

SP - 1073

EP - 1096

AB - 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.

LA - eng

KW - stochastic optimal control; maximum principle; stochastic differential delayed equation; anticipated backward differential equation; fully coupled forward-backward stochastic system; Clarke generalized gradient; Clarke's generalized gradient

UR - http://eudml.org/doc/272939

ER -

## References

top- [1] F. Antonelli, Backward-forward stochastic differential equations. Ann. Appl. Prob.3 (1993) 777–793. Zbl0780.60058MR1233625
- [2] F. Antonelli, E. Baruccib and M.E. Mancinoc, Asset pricing with a forward-backward stochastic differential utility. Econ. Lett.72 (2001) 151–157. Zbl0988.91039MR1840721
- [3] R. Buckdahn and Y. Hu, Hedging contingent claims for a large investor in an incomplete market. Adv. Appl. Prob.30 (1998) 239–255. Zbl0904.90009MR1618845
- [4] L. Chen and Z. Wu, Maximum principle for stochastic optimal control problem of forward-backward system with delay, in Proc. Joint 48th IEEE CDC and 28th CCC, Shanghai, P.R. China (2009) 2899–2904.
- [5] L. Chen and Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application. Automatica46 (2010) 1074–1080. Zbl1205.93163MR2877190
- [6] L. Chen and Z. Wu, A type of generalized forward-backward stochastic differential equations and applications. Chin. Ann. Math. Ser. B32 (2011) 279–292. Zbl1218.60047MR2773230
- [7] D. Cucoo and J. Cvitanic, Optimal consumption choices for a ‘large’ investor. J. Econ. Dyn. Control22 (1998) 401–436. Zbl0902.90031
- [8] J. Cvitanic and J. Ma, Hedging options for a large investor and forward-backward SDE’s. Ann. Appl. Prob.6 (1996) 370–398. Zbl0856.90011MR1398050
- [9] J. Cvitanic, X.H. Wan and J.F. Zhang, Optimal contracts in continuous-time models. J. Appl. Math. Stoch. Anal. 2006 (2006), Article ID 95203 1–27. Zbl1140.91433MR2237178
- [10] N. El Karoui, S.G. Peng and M.C. Quenez, Backward stochastic differential equations in finance. Math. Finance7 (1997) 1–71. Zbl0884.90035MR1434407
- [11] Y. Hu and S.G. Peng, Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields103 (1995) 273–283. Zbl0831.60065MR1355060
- [12] V.B. Kolmanovsky and T.L. Maizenberg, Optimal control of stochastic systems with aftereffect, in Stochastic Systems, Translated from Avtomatika i Telemekhanika. 1 (1973) 47–61. Zbl0274.93068MR459886
- [13] Q.X. Meng, A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information. Sciences in China, Mathematics 52 (2009) 1579–1588. Zbl1176.93083MR2520595
- [14] S.E.A. Mohammed, Stochastic differential equations with memory : theory, examples and applications. Stochastic Analysis and Related Topics VI. The Geido Workshop, 1996. Progress in Probability, Birkhauser (1998). Zbl0901.60030
- [15] B. Øksendal and A. Sulem, A maximum principle for optimal control of stochastic systems with delay, with applications to finance, Optimal Control and Partial Differential Equations – Innovations and Applications, edited by J.M. Menaldi, E. Rofman and A. Sulem. IOS Press, Amsterdam (2000). Zbl1054.93531
- [16] E. Pardoux and S.G. Peng, Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14 (1990) 55–61. Zbl0692.93064MR1037747
- [17] S.G. Peng, Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim.27 (1993) 125–144. Zbl0769.60054MR1202528
- [18] S.G. Peng, Backward stochastic differential equations-stochastic optimization theory and viscosity solutions of HJB equations, Topics on Stochastic Analysis (in Chinese), edited by J.A. Yan, S.G. Peng, S.Z. Fang and L.M. Wu. Science Press, Beijing (1997) 85–138.
- [19] S.G. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to the optimal control. SIAM J. Control Optim.37 (1999) 825–843. Zbl0931.60048MR1675098
- [20] S.G. Peng and Z. Yang, Anticipated backward stochastic differential equations. Ann. Prob.37 (2009) 877–902. Zbl1186.60053MR2537524
- [21] J.T. Shi and Z. Wu, The maximum principle for fully coupled forward-backward stochastic control systems. ACTA Automatica Sinica32 (2006) 161–169. MR2230926
- [22] J.T. Shi and Z. Wu, The maximum principle for partially observed optimal control of fully coupled forward-backward stochastic system. J. Optim. Theory Appl.145 (2010) 543–578. Zbl1209.49034MR2645802
- [23] G.C. Wang and Z. Wu, The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans. Autom. Control54 (2009) 1230–1242. MR2532612
- [24] N. Williams, On dynamic principal-agent problems in continuous time. Working paper (2008). Available on the website : http://www.ssc.wisc.edu/˜nwilliam/dynamic-pa1.pdf
- [25] Z. Wu, Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci.11 (1998) 249–259. Zbl0938.93066MR1651258
- [26] J.M. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions. SIAM J. Control Optim.48 (2010) 4119–4156. Zbl1202.93180MR2645476
- [27] J.M. Yong and X.Y. Zhou, Stochastic Controls : Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). Zbl0943.93002MR1696772
- [28] Z.Y. Yu, Linear-quadratic optimal control and nonzero-sum differential game of forward-backward stochastic system. Asian J. Control14 (2012) 1–13. Zbl1282.93280MR2881806
- [29] J.F. Zhang, The wellposedness of FBSDEs. Discrete Contin. Dyn. Syst., Ser. B 6 (2006) 927–940. Zbl1132.60315MR2223916
- [30] X.Y. Zhou, Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans. Autom. Control41 (1996) 1176–1179. Zbl0857.93099MR1407203

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.