Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2013, Article ID 637106, 9 pages
http://dx.doi.org/10.1155/2013/637106
Research Article

Successive Approximation of SFDEs with Finite Delay Driven by -Brownian Motion

1Glorious Sun School of Business and Management, Donghua University, 1882 West Yan-an Rood, Shanghai 200051, China
2Department of Mathematics, Donghua University, 2999 North Renmin Rood, Songjiang, Shanghai 201620, China

Received 11 November 2013; Accepted 5 December 2013

Academic Editor: Yaozhong Hu

Copyright © 2013 Litan Yan and Qinghua Zhang. 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. S. Peng, “G-expectation, G-Brownian motion and related stochastic calculus of Itô type,” in Stochastic Analysis and Applications, vol. 2 of Abel Symposia, pp. 541–567, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  2. S. Peng, “Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation,” Stochastic Processes and Their Applications, vol. 118, no. 12, pp. 2223–2253, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. Peng, “G-Brownian motion and dynamic risk measure under volatility uncertainty,” http://arxiv.org/abs/0711.2834.
  4. S. Peng, “Nonlinear expections and stochastic calculus under uncertainty,” http://arxiv.org/abs/1002.4546.
  5. F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–659, 1973. View at Google Scholar
  6. R. C. Merton, “Theory of rational option pricing,” The Bell Journal of Economics and Management Science, vol. 4, no. 1, pp. 141–183, 1973. View at Google Scholar · View at MathSciNet
  7. M. Arriojas, Y. Hu, S.-E. Mohammed, and G. Pap, “A delayed black and scholes formula,” Stochastic Analysis and Applications, vol. 25, no. 2, pp. 471–492, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. L. Denis, M. Hu, and S. Peng, “Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths,” Potential Analysis, vol. 34, no. 2, pp. 139–161, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  9. M.-S. Hu and S.-G. Peng, “On representation theorem of G-expectations and paths of G-Brownian motion,” Acta Mathematica Sinica, vol. 25, no. 3, pp. 539–546, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. M. Hu and X. Li, “Independence under the G-expectation framework,” Journal of Theoretical Probability, 2012. View at Publisher · View at Google Scholar
  11. Q. Lin, “Local time and tanaka formula for the G-Brownian motion,” Journal of Mathematical Analysis and Applications, vol. 398, no. 1, pp. 315–334, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  12. S. Peng, Y. Song, and J. Zhang, “A complete representation theorem for G-martingales,” http://arxiv.org/abs/1201.2629.
  13. Y. Song, “Some properties on G-evaluation and its applications to G-martingale decomposition,” Science China Mathematics, vol. 54, no. 2, pp. 287–300, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. Xu, H. Shang, and B. Zhang, “A Girsanov type theorem under G-framework,” Stochastic Analysis and Applications, vol. 29, no. 3, pp. 386–406, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  15. L. Yan, B. Gao, and Q. Zhang, “The fractional derivatives for G-Brownian local time”.
  16. L. Yan, X. Sun, and B. Gao, “Integral with respect to the G-Brownian local time,” http://arxiv.org/abs/1212.6353.
  17. X. Li and S. Peng, “Stopping times and related Itô's calculus with G-Brownian motion,” Stochastic Processes and Their Applications, vol. 121, no. 7, pp. 1492–1508, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  18. X. Bai and Y. Lin, “On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-lipschitz coefficients,” http://arxiv.org/abs/1002.1046v3.
  19. F. Gao, “Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion,” Stochastic Processes and Their Applications, vol. 119, no. 10, pp. 3356–3382, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Z. Chen and D. Zhang, “Exponential stability for stochastic differential equation driven by G-Brownian motion,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1906–1910, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Q. Lin, “Properties of solutions of stochastic differential equations driven by the G-Brownian motion,” http://arxiv.org/abs/1010.3158.
  22. Y. Lin, “Stochastic differential equations driven by G-Brownian motion with reflecting boundary conditions,” Electronic Journal of Probability, vol. 18, article 9, pp. 1–23, 2013. View at Publisher · View at Google Scholar
  23. Y. Ren, Q. Bi, and R. Sakthivel, “Stochastic functional differential equations with infinite delay driven by G-Brownian motion,” Mathematical Methods in the Applied Sciences, vol. 36, no. 13, pp. 1746–1759, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. Y. Ren and L. Hu, “A note on the stochastic differential equations driven by G-Brownian motion,” Statistics & Probability Letters, vol. 81, no. 5, pp. 580–585, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  25. T. Taniguchi, “Successive approximations to solutions of stochastic differential equations,” Journal of Differential Equations, vol. 96, no. 1, pp. 152–169, 1992. View at Publisher · View at Google Scholar · View at MathSciNet