Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 892304, 13 pages
http://dx.doi.org/10.1155/2015/892304
Research Article

Maximum Principle for Optimal Control Problems of Forward-Backward Regime-Switching Systems Involving Impulse Controls

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

Received 15 April 2014; Accepted 28 August 2014

Academic Editor: Guangchen Wang

Copyright © 2015 Shujun Wang and Zhen Wu. 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. L. Pontryagin, V. Boltyanskti, R. Gamkrelidze, and E. Mischenko, The Mathematical Theory of Optimal Control Processes, John Wiley & Sons, New York, NY, USA, 1962.
  2. J. Bismut, “An introductory approach to duality in optimal stochastic control,” SIAM Review, vol. 20, no. 1, pp. 62–78, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. G. Peng, “A general stochastic maximum principle for optimal control problems,” SIAM Journal on Control and Optimization, vol. 28, no. 4, pp. 966–979, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. E. Pardoux and S. Peng, “Adapted solution of a backward stochastic differential equation,” Systems and Control Letters, vol. 14, no. 1, pp. 55–61, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. D. Duffie and L. G. Epstein, “Stochastic differential utility,” Econometrica, vol. 60, no. 2, pp. 353–394, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  6. 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 MathSciNet · View at Scopus
  7. S. Peng, “Backward stochastic differential equations and applications to optimal control,” Applied Mathematics and Optimization, vol. 27, no. 2, pp. 125–144, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. W. S. Xu, “Stochastic maximum principle for optimal control problem of forward and backward system,” Journal of the Australian Mathematical Society B, vol. 37, no. 2, pp. 172–185, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  9. Z. Wu, “Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems,” Systems Science and Mathematical Sciences, vol. 11, no. 3, pp. 249–259, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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 · View at MathSciNet
  11. G. Wang and Z. Wu, “The maximum principles for stochastic recursive optimal control problems under partial information,” IEEE Transactions on Automatic Control, vol. 54, no. 6, pp. 1230–1242, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. G. Wang and Z. Yu, “A Pontryagin's maximum principle for non-zero sum differential games of BSDEs with applications,” IEEE Transactions on Automatic Control, vol. 55, no. 7, pp. 1742–1747, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. Z. Wu, “A general maximum principle for optimal control of forward-backward stochastic systems,” Automatica, vol. 49, no. 5, pp. 1473–1480, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. S. Crépey, “About the pricing equations in finance,” in Paris-Princeton Lectures on Mathematical Finance 2010, vol. 2003 of Lecture Notes in Mathematics, pp. 63–203, Springer, Berlin, Germany, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. S. Crépey and A. Matoussi, “Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison,” The Annals of Applied Probability, vol. 18, no. 5, pp. 2041–2069, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. Donnelly, “Sufficient stochastic maximum principle in a regime-switching diffusion model,” Applied Mathematics and Optimization, vol. 64, no. 2, pp. 155–169, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. R. Tao and Z. Wu, “Maximum principle for optimal control problems of forward-backward regime-switching system and applications,” Systems & Control Letters, vol. 61, no. 9, pp. 911–917, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. R. Tao, Z. Wu, and Q. Zhang, “BSDEs with regime switching: weak convergence and applications,” Journal of Mathematical Analysis and Applications, vol. 407, no. 1, pp. 97–111, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. M. H. A. Davis and A. R. Norman, “Portfolio selection with transaction costs,” Mathematics of Operations Research, vol. 15, no. 4, pp. 676–713, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Øksendal and A. Sulem, “Optimal consumption and portfolio with both fixed and proportional transaction costs,” SIAM Journal on Control and Optimization, vol. 40, no. 6, pp. 1765–1790, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. A. Cadenillas and F. Zapatero, “Classical and impulse stochastic control of the exchange rate using interest rates and reserves,” Mathematical Finance, vol. 10, no. 2, pp. 141–156, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. Jeanblanc-Picqué, “Impulse control method and exchange rate,” Mathematical Finance, vol. 3, pp. 161–177, 1993. View at Google Scholar
  23. R. Korn, “Some applications of impulse control in mathematical finance,” Mathematical Methods of Operations Research, vol. 50, no. 3, pp. 493–518, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic, Dordrecht, The Netherlands, 2003.
  25. Z. Wu and F. Zhang, “Stochastic maximum principle for optimal control problems of forward-backward systems involving impulse controls,” IEEE Transactions on Automatic Control, vol. 56, no. 6, pp. 1401–1406, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Z. Wu and F. Zhang, “Maximum principle for stochastic recursive optimal control problems involving impulse controls,” Abstract and Applied Analysis, vol. 2012, Article ID 709682, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, vol. 43 of Applications of Mathematics, Springer, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  28. I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet