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Abstract and Applied Analysis
Volume 2012, Article ID 537376, 29 pages
http://dx.doi.org/10.1155/2012/537376
Research Article

A Maximum Principle for Controlled Time-Symmetric Forward-Backward Doubly Stochastic Differential Equation with Initial-Terminal Sate Constraints

1Institute for Financial Studies and Institute of Mathematics, Shandong University, Shandong, Jinan 250100, China
2Institute of mathematics, Shandong University, Shandong, Jinan 250100, China

Received 2 October 2012; Accepted 15 November 2012

Academic Editor: Jen-Chih Yao

Copyright © 2012 Shaolin Ji et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. El Karoui, S. Peng, and M. C. Quenez, “Backward stochastic differential equations in finance,” Mathematical Finance, vol. 7, no. 1, pp. 1–71, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. É. Pardoux and S. G. Peng, “Adapted solution of a backward stochastic differential equation,” Systems & Control Letters, vol. 14, no. 1, pp. 55–61, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. F. Antonelli, “Backward-forward stochastic differential equations,” The Annals of Applied Probability, vol. 3, no. 3, pp. 777–793, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. Y. Hu and S. Peng, “Solution of forward-backward stochastic differential equations,” Probability Theory and Related Fields, vol. 103, no. 2, pp. 273–283, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Ma, P. Protter, and J. M. Yong, “Solving forward-backward stochastic differential equations explicitly—a four step scheme,” Probability Theory and Related Fields, vol. 98, no. 3, pp. 339–359, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, vol. 1702 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1999.
  7. J. Yong, “Finding adapted solutions of forward-backward stochastic differential equations: method of continuation,” Probability Theory and Related Fields, vol. 107, no. 4, pp. 537–572, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. Y. Hu, “N-person differential games governed by semilinear stochastic evolution systems,” Applied Mathematics and Optimization, vol. 24, no. 3, pp. 257–271, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. J. Yong and X. Y. Zhou, Stochastic Controls Hamiltonian Systems and HJB Equations, vol. 43, Springer, New York, NY, USA, 1999.
  10. R. Buckdahn and Y. Hu, “Hedging contingent claims for a large investor in an incomplete market,” Advances in Applied Probability, vol. 30, no. 1, pp. 239–255, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. Cvitanić and J. Ma, “Hedging options for a large investor and forward-backward SDE's,” The Annals of Applied Probability, vol. 6, no. 2, pp. 370–398, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Hu, “Potential kernels associated with a filtration and forward-backward SDEs,” Potential Analysis, vol. 10, no. 2, pp. 103–118, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. P. Briand and Y. Hu, “Probabilistic approach to singular perturbations of semilinear and quasilinear parabolic PDEs,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 35, no. 7, pp. 815–831, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. R. Buckdahn and Y. Hu, “Probabilistic approach to homogenizations of systems of quasilinear parabolic PDEs with periodic structures,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 32, no. 5, pp. 609–619, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. E. Pardoux and S. Peng, “Backward doubly stochastic differential equations and systems of quasilinear SPDEs,” Probability Theory and Related Fields, vol. 98, no. 2, pp. 209–227, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. S. Peng and Y. Shi, “A type of time-symmetric forward-backward stochastic differential equations,” Comptes Rendus Mathématique, vol. 336, no. 9, pp. 773–778, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Y. Han, S. Peng, and Z. Wu, “Maximum principle for backward doubly stochastic control systems with applications,” SIAM Journal on Control and Optimization, vol. 48, no. 7, pp. 4224–4241, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. A. Bensoussan, Stochastic Control by Functional Analysis Methods, vol. 11, North-Holland, Amsterdam, The Netherlands, 1982.
  19. S. Ji, “Dual method for continuous-time Markowitz's problems with nonlinear wealth equations,” Journal of Mathematical Analysis and Applications, vol. 366, no. 1, pp. 90–100, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Ji and S. Peng, “Terminal perturbation method for the backward approach to continuous time mean-variance portfolio selection,” Stochastic Processes and their Applications, vol. 118, no. 6, pp. 952–967, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. S. Ji and X. Y. Zhou, “A maximum principle for stochastic optimal control with terminal state constraints, and its applications,” Communications in Information and Systems, vol. 6, no. 4, pp. 321–337, 2006. View at Google Scholar · View at Zentralblatt MATH
  22. S. Ji and X. Y. Zhou, “A generalized Neyman-Pearson lemma for g-probabilities,” Probability Theory and Related Fields, vol. 148, no. 3-4, pp. 645–669, 2010. View at Publisher · View at Google Scholar
  23. T. R. Bielecki, H. Jin, S. R. Pliska, and X. Y. Zhou, “Continuous-time mean-variance portfolio selection with bankruptcy prohibition,” Mathematical Finance, vol. 15, no. 2, pp. 213–244, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. N. El Karoui, S. Peng, and M. C. Quenez, “A dynamic maximum principle for the optimization of recursive utilities under constraints,” The Annals of Applied Probability, vol. 11, no. 3, pp. 664–693, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. J. Yong, “Forward-backward stochastic differential equations with mixed initial-terminal conditions,” Transactions of the American Mathematical Society, vol. 362, no. 2, pp. 1047–1096, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. J. Yong, “Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions,” SIAM Journal on Control and Optimization, vol. 48, no. 6, pp. 4119–4156, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. D. Nualart and É. Pardoux, “Stochastic calculus with anticipating integrands,” Probability Theory and Related Fields, vol. 78, no. 4, pp. 535–581, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. I. Ekeland, “On the variational principle,” Journal of Mathematical Analysis and Applications, vol. 47, pp. 324–353, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH