Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2013, Article ID 132923, 7 pages
http://dx.doi.org/10.1155/2013/132923
Research Article

Existence and Uniqueness of Solution for a Class of Stochastic Differential Equations

1Department of Mathematics, Guangdong University of Education, Guangzhou 510310, China
2School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, China
3Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 2 August 2013; Accepted 21 August 2013

Academic Editors: K. Ammari and S. Kim

Copyright © 2013 Junfei Cao 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. D. Xu, Z. Yang, and Y. Huang, “Existence-uniqueness and continuation theorems for stochastic functional differential equations,” Journal of Differential Equations, vol. 245, no. 6, pp. 1681–1703, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. T. Caraballo, I. D. Chueshov, P. Marín-Rubio, and J. Real, “Existence and asymptotic behaviour for stochastic heat equations with multiplicative noise in materials with memory,” Discrete and Continuous Dynamical Systems, vol. 18, no. 2-3, pp. 253–270, 2007. View at Google Scholar · View at Scopus
  3. J. Cao, Q. Yang, and Z. Huang, “On almost periodic mild solutions for stochastic functional differential equations,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 275–286, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss, and J. Valero, “Non-autonomous and random attractors for delay random semilinear equations without uniqueness,” Discrete and Continuous Dynamical Systems, vol. 21, no. 2, pp. 415–443, 2008. View at Google Scholar · View at Scopus
  5. J. Cao, Q. Yang, and Z. Huang, “Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations,” Stochastics, vol. 83, no. 3, pp. 259–275, 2011. View at Publisher · View at Google Scholar · View at Scopus
  6. H. Keller, “Attractors and bifurcations of the stochastic Lorenz system,” Tech. Rep. 389, Institut für Dynamische Systeme, Universität Bremen, 1996. View at Google Scholar
  7. T. Caraballo, J. Real, and T. Taniguchi, “The exponential stability of neutral stochastic delay partial differential equations,” Discrete and Continuous Dynamical Systems, vol. 18, no. 2-3, pp. 295–313, 2007. View at Google Scholar · View at Scopus
  8. J. Cao, Q. Yang, Z. Huang, and Q. Liu, “Asymptotically almost periodic solutions of stochastic functional differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1499–1511, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. I. Chueshov and B. Schmalfuss, “Qualitative behavior of a class of stochastic parabolic PDES with dynamical boundary conditions,” Discrete and Continuous Dynamical Systems, vol. 18, no. 2-3, pp. 315–338, 2007. View at Google Scholar · View at Scopus
  10. F. Wei and K. Wang, “The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 516–531, 2007. View at Google Scholar
  11. Z. Fan, M. Liu, and W. Cao, “Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1142–1159, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. H. Schurz, “Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 334–345, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Zhou and M. Xue, “The existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay,” Journal of Applied Mathematics, vol. 1, pp. 95–105, 2008. View at Google Scholar
  14. X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publication, Chichester, UK, 1997.
  15. G. Jiang and X. Wang, “Regularity property of solution to two-parameter stochastic volterra equation with non-lipschitz coefficients,” Acta Mathematica Sinica, vol. 33, pp. 872–882, 2013. View at Google Scholar
  16. W. Fei, “Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition,” Nonlinear Analysis: Theory, Methods & Applications, vol. 76, pp. 202–214, 2013. View at Google Scholar
  17. T. Caraballo, F. Morillas, and J. Valero, “Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities,” Journal of Differential Equations, vol. 253, no. 2, pp. 667–693, 2012. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Wu, W. Wang, and J. Ren, “Anticipated backward stochastic differential equations with non-Lipschitz coefficients,” Statistics and Probability Letters, vol. 82, no. 3, pp. 672–682, 2012. View at Publisher · View at Google Scholar · View at Scopus
  19. F. Jiang and Y. Shen, “A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional differential equations with non-Lipschitz coefficients,” Computers and Mathematics with Applications, vol. 61, no. 6, pp. 1590–1594, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. T. Taniguchi, “The existence of energy solutions to 2-dimensional non-Lipschitz stochastic Navier-Stokes equations in unbounded domains,” Journal of Differential Equations, vol. 251, no. 12, pp. 3329–3362, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Fan and L. Jiang, “Finite and infinite time interval BSDEs with non-Lipschitz coefficients,” Statistics and Probability Letters, vol. 80, no. 11-12, pp. 962–968, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. J. Bao and Z. Hou, “Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients,” Computers and Mathematics with Applications, vol. 59, no. 1, pp. 207–214, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. J. Ren, J. Wu, and X. Zhang, “Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations,” Bulletin des Sciences Mathematiques, vol. 134, no. 4, pp. 391–404, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. T. Taniguchi, “The existence and uniqueness of energy solutions to local non-Lipschitz stochastic evolution equations,” Journal of Mathematical Analysis and Applications, vol. 360, no. 1, pp. 245–253, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. Y. Wang and Z. Huang, “Backward stochastic differential equations with non-Lipschitz coefficients,” Statistics & Probability Letters, vol. 79, pp. 1438–1443, 2009. View at Google Scholar
  26. Q. Lin, “A class of backward doubly stochastic differential equations with non-Lipschitz coefficients,” Statistics and Probability Letters, vol. 79, no. 20, pp. 2223–2229, 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. B. Xie, “The stochastic parabolic partial differential equation with non-Lipschitz coefficients on the unbounded domain,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 705–718, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. W. Fei, “Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients,” Information Sciences, vol. 177, no. 20, pp. 4329–4337, 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. J. Ren and X. Zhang, “Continuity modulus of stochastic homeomorphism flows for SDEs with non-Lipschitz coefficients,” Journal of Functional Analysis, vol. 241, no. 2, pp. 439–456, 2006. View at Publisher · View at Google Scholar · View at Scopus
  30. X. Zhang, “Homeomorphic flows for multi-dimensional SDEs with non-Lipschitz coefficients,” Stochastic Processes and Their Applications, vol. 115, no. 3, pp. 435–448, 2005. View at Publisher · View at Google Scholar · View at Scopus
  31. I. Gihman and A. Skorohod, Stochastic Differentialgleichungen, Akademie, Berlin, Germany, 1971.
  32. S. Wang, M. Wu, and Z. Jia, Inequality of Matrix, Science Press, 2006.
  33. A. Wentzell, Theorie Zufälliger Prozesse, Birkhäuser, Basel, Switzerland, 1979.
  34. E. Lorenz, “Deterministic nonperiodic flows,” Journal of the Atmospheric Sciences, vol. 12, pp. 139–163, 1963. View at Google Scholar
  35. H. Peitgen, H. J. Jürgens, and D. Saupe, Chaos and Fractals, Springer, New York, NY, USA, 1992.
  36. L. Arnold, Random Dynamical Systems, Preliminary Version, 1994.
  37. B. Schmallfuß, “The random attractor of the stochastic Lorenz system,” Zeitschrift für Angewandte Mathematik und Physik, vol. 48, pp. 951–975, 1997. View at Google Scholar
  38. L. Arnold, Random Differentialgleichungen, R. Oldenbourg, München, Germany, 1973.