Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2014, Article ID 718948, 18 pages
http://dx.doi.org/10.1155/2014/718948
Research Article

Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs

1School of Mathematics, Shandong University, Jinan 250100, China
2School of Mathematics, Shandong Polytechnic University, Jinan 250353, China
3School of Mathematical Sciences, Shandong Normal University, Jinan 250014, China

Received 31 March 2014; Revised 23 May 2014; Accepted 18 June 2014; Published 13 July 2014

Academic Editor: Guangchen Wang

Copyright © 2014 Ruimin Xu and Tingting 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. Y. Hu and S. Peng, “Adapted solution of a backward semilinear stochastic evolution equation,” Stochastic Analysis and Applications, vol. 9, no. 4, pp. 445–459, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  2. N. I. Mahmudov and M. A. McKibben, “On backward stochastic evolution equations in Hilbert spaces and optimal control,” Nonlinear Analysis, vol. 67, no. 4, pp. 1260–1274, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M. Fuhrman and G. Tessitore, “Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control,” The Annals of Probability, vol. 30, no. 3, pp. 1397–1465, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. Fuhrman and G. Tessitore, “Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces,” Annals of Probability, vol. 32, no. 1B, pp. 607–660, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. X. Mao, “Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients,” Stochastic Processes and their Applications, vol. 58, no. 2, pp. 281–292, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. J. Lasry and P. Lions, “Mean field games,” Japanese Journal of Mathematics, vol. 2, no. 1, pp. 229–260, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. R. Buckdahn, J. Li, and S. Peng, “Mean-field backward stochastic differential equations and related partial differential equations,” Stochastic Processes and their Applications, vol. 119, no. 10, pp. 3133–3154, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. D. Andersson and B. Djehiche, “A maximum principle for SDEs of mean-field type,” Applied Mathematics and Optimization, vol. 63, no. 3, pp. 341–356, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. R. Buckdahn, B. Djehiche, J. Li, and S. Peng, “Mean-field backward stochastic differential equations: a limit approach,” The Annals of Probability, vol. 37, no. 4, pp. 1524–1565, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. R. Xu, “Mean-field backward doubly stochastic differential equations and related SPDEs,” Boundary Value Problems, vol. 2012, article 114, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. R. Buckdahn, B. Djehiche, and J. Li, “A general stochastic maximum principle for SDEs of mean-field type,” Applied Mathematics and Optimization, vol. 64, no. 2, pp. 197–216, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. J. Li, “Stochastic maximum principle in the mean-field controls,” Automatica, vol. 48, no. 2, pp. 366–373, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. G. Wang, C. Zhang, and W. Zhang, “Stochastic maximum principle for mean-field type optimal control under partial information,” IEEE Transactions on Automatic Control, vol. 59, no. 2, pp. 522–528, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. Hafayed, “A mean-field necessary and sufficient conditions for optimal singular stochastic control,” Communications in Mathematics and Statistics, vol. 1, no. 4, pp. 417–435, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  15. M. Hafayed, “A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes,” International Journal of Dynamics and Control, vol. 1, no. 4, pp. 300–315, 2013. View at Publisher · View at Google Scholar
  16. Y. Hu and S. Peng, “Maximum principle for semilinear stochastic evolution control systems,” Stochastics and Stochastics Reports, vol. 33, no. 3-4, pp. 159–180, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. Fuhrman, Y. Hu, and G. Tessitore, “Stochastic maximum principle for optimal control of SPDEs,” Comptes Rendus Mathematique, vol. 350, no. 13-14, pp. 683–688, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus