Skip to main content
SpringerLink
Log in

Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

Employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bao J, Wang F, Yuan C. Ergodicity for neutral type SDEs with infinite memory. arXiv: 1805.03431

  2. Bao J, Yuan C. Convergence rate of EM scheme for SDDEs. Proc Amer Math Soc, 2013, 141(9): 3231–3243

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao J, Yuan C. Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87(1): 48–70

    Article  MathSciNet  MATH  Google Scholar 

  4. Budhiraja A, Chen, J, Dupuis P. Large deviations for stochastic partial differential equations driven by Poisson random measure. Stochastic Process Appl, 2013, 123(2): 523–560

    Article  MathSciNet  MATH  Google Scholar 

  5. Budhiraja A, Dupuis P. A variational representation for positive functionals of infinite dimensional Brownian motion. Probab Math Statist, 2000, 20(1): 39–61

    MathSciNet  MATH  Google Scholar 

  6. Budhiraja A, Dupuis P, Fischer M. Large deviation properties of weakly interacting processes via weak convergence methods. Ann Probab, 2012, 40(1): 74–102

    Article  MathSciNet  MATH  Google Scholar 

  7. Budhiraja A, Dupuis P, Ganguly A. Moderate deviation principles for stochastic differential equations with jumps. Ann Probab, 2016, 44(3): 1723–1775

    Article  MathSciNet  MATH  Google Scholar 

  8. Budhiraja A, Dupuis P, Maroulas V. Large deviations for infinite dimensional stochastic dynamical systems. Ann Probab, 2008, 36(4): 1390–1420

    Article  MathSciNet  MATH  Google Scholar 

  9. Budhiraja A, Dupuis P, Maroulas V. Variational representations for continuous time processes. Ann Inst Henri Poincaré Probab Stat, 2011, 47(3): 725–747

    Article  MathSciNet  MATH  Google Scholar 

  10. Budhiraja A, Nyquist P. Large deviations for multidimensional state-dependent shotnoise processes. J Appl Probab, 2015, 52(4): 1097–1114

    Article  MathSciNet  MATH  Google Scholar 

  11. Cardon-Webber C. Large deviations for Burger's-type SPDE. Stochastic Process Appl, 1999, 84(1): 53–70

    Article  MathSciNet  MATH  Google Scholar 

  12. Cerrai S, Röckner M. Large deviations for stochastic reaction diffusion systems with multiplicative noise and non-Lipschitz reaction term. Ann Probab, 2004, 32(18): 1100–1139

    MathSciNet  MATH  Google Scholar 

  13. Chow P. Large deviation problem for some parabolic Itô equations. Comm Pure Appl Math, 1992, 45(1): 97–120

    Article  MathSciNet  MATH  Google Scholar 

  14. Cortissoz J. On the Skorokhod representation theorem. Proc Amer Math Soc, 2007, 135(12): 3995–4007

    Article  MathSciNet  MATH  Google Scholar 

  15. Dembo A, Zeitouni O. Large Deviations Techniques and Applications. 2nd ed. Applications of Mathematics, Vol 38. Berlin: Springer-Verlag, 1998

    Google Scholar 

  16. Duan J, Millet A. Large deviations for the Boussinesq equations under random in uences. Stochastic Process Appl, 2009, 119(6): 2052–2081

    Article  MathSciNet  MATH  Google Scholar 

  17. Dupuis P, Ellis R. A Weak Convergence Approach to the Theory of Large Deviations. New York: Wiley, 1997

    Book  MATH  Google Scholar 

  18. Fatheddin P. Central limit theorem for a class of SPDEs. J Appl Probab, 2015, 52(3): 786–796

    Article  MathSciNet  MATH  Google Scholar 

  19. Freidlin M. Random perturbations of reaction-diffusion equations: the quasideterministic approximation. Trans Amer Math Soc, 1988, 305(2): 665–697

    MathSciNet  MATH  Google Scholar 

  20. Freidlin M, Wentzell A. Random Perturbation of Dynamical Systems. Grundlehren Math Wiss, Vol 260. New York: Springer, 1984

    Google Scholar 

  21. Horbacz K. The central limit theorem for random dynamical systems. J Stat Phys, 2016, 164(6): 1261–1291

    Article  MathSciNet  MATH  Google Scholar 

  22. Kallianpur G, Xiong J. Large deviations for a class of stochastic partial differential equations. Ann Probab, 1996, 24(1): 320–345

    Article  MathSciNet  MATH  Google Scholar 

  23. Karatzas I, Shreve S. Brownian Motion and Stochastic Calculus. 2nd ed. Grad Texts in Math, Vol 113. New York: Springer-Verlag, 1991

    Google Scholar 

  24. Klebaner F C. Introduction to Stochastic Calculus with Applications. 2nd ed. London: Imperial College Press, 2005

    Book  MATH  Google Scholar 

  25. Li Y, Wang R, Zhang S. Moderate deviations for a stochastic heat equation with spatially correlated noise. Acta Appl Math, 2015, 139(1): 59–80

    Article  MathSciNet  MATH  Google Scholar 

  26. Liu W. Large deviations for stochastic evolution equations with small multiplicative noise. Appl Math Optim, 2010, 61(1): 27–56

    Article  MathSciNet  MATH  Google Scholar 

  27. Ma X, Xi F. Moderate deviations for neutral stochastic differential delay equations with jumps. Statist Probab Lett, 2017, 126: 97–107

    Article  MathSciNet  MATH  Google Scholar 

  28. Mao X, Matina J. Khasminskii-type theorems for stochastic differential delay equations. Stoch Anal Appl, 2005, 23(5): 1045–1069

    Article  MathSciNet  MATH  Google Scholar 

  29. Maroulas V. Uniform large deviations for infinite dimensional stochastic systems with jumps. Mathematika, 2011, 57(1): 175–192

    Article  MathSciNet  MATH  Google Scholar 

  30. Mohammed S E A. Stochastic Functional Differential Equations. Boston: Longman Scientific & Technical, 1984

    MATH  Google Scholar 

  31. Ren J, Zhang X. Large deviations for homeomorphism ows of non-Lipschitz SDE. Bull Sci Math, 2005, 129(8): 643–655

    Article  MathSciNet  Google Scholar 

  32. Sritharan S, Sundar P. Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stochastic Process Appl, 2006, 116(11): 1636–1659

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang R, Zhai J, Zhang T. A moderate deviation principle for 2-D stochastic Navier-Stokes equations. J Differential Equations, 2015, 258(10): 3363–3390

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang R, Zhang T. Moderate deviations for stochastic reaction-diffusion equations with multiplicative noise. Potential Anal, 2015, 42(1): 99–113

    Article  MathSciNet  MATH  Google Scholar 

  35. Xiong J. Large deviations for diffusion processes in duals of nuclear spaces. Appl Math Optim, 1996, 34(1): 1–27

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhai J, Zhang T. Large deviations for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. Bernoulli, 2015, 21(4): 2351–2392

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and corrections. This work was supported in part by the National Natural Science Foundation of China (Grant No. 11401592), the Natural Science Foundation of Hunan Province (No. 13JJ5043), and the Mathematics and Interdisciplinary Sciences Project of Central South University.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Yongqiang Suo & Wei Zhang

  2. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, UK

    Yongqiang Suo

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Jin Tao

Authors
  1. Yongqiang Suo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jin Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Suo, Y., Tao, J. & Zhang, W. Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth. Front. Math. China 13, 913–933 (2018). https://doi.org/10.1007/s11464-018-0710-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-018-0710-3

Keywords

MSC

Search

Navigation