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

Jianhui Huang; Jingtao Shi

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

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

Abstract

top
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.

How to cite

top

Huang, 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. [1] F. Antonelli, Backward-forward stochastic differential equations. Ann. Appl. Prob.3 (1993) 777–793. Zbl0780.60058MR1233625
  2. [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. [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. [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. [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. [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. [7] D. Cucoo and J. Cvitanic, Optimal consumption choices for a ‘large’ investor. J. Econ. Dyn. Control22 (1998) 401–436. Zbl0902.90031
  8. [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. [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. [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. [11] Y. Hu and S.G. Peng, Solution of forward-backward stochastic differential equations. Prob. Theory Relat. Fields103 (1995) 273–283. Zbl0831.60065MR1355060
  12. [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. [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. [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. [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. [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. [17] S.G. Peng, Backward stochastic differential equations and applications to optimal control. Appl. Math. Optim.27 (1993) 125–144. Zbl0769.60054MR1202528
  18. [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. [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. [20] S.G. Peng and Z. Yang, Anticipated backward stochastic differential equations. Ann. Prob.37 (2009) 877–902. Zbl1186.60053MR2537524
  21. [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. [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. [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. [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. [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. [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. [27] J.M. Yong and X.Y. Zhou, Stochastic Controls : Hamiltonian Systems and HJB Equations. Springer-Verlag, New York (1999). Zbl0943.93002MR1696772
  28. [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. [29] J.F. Zhang, The wellposedness of FBSDEs. Discrete Contin. Dyn. Syst., Ser. B 6 (2006) 927–940. Zbl1132.60315MR2223916
  30. [30] X.Y. Zhou, Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans. Autom. Control41 (1996) 1176–1179. Zbl0857.93099MR1407203

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.