Moderate deviations for neutral functional stochastic differential equations driven by Lévy noises

Abstract

Using the weak convergence method introduced by A. Budhiraja, P. Dupuis, and A. Ganguly [Ann. Probab., 2016, 44: 1723–1775], we establish the moderate deviation principle for neutral functional stochastic differential equations driven by both Brownian motions and Poisson random measures.

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

References

  1. 1.

    Aldous D. Stopping times and tightness. Ann Probab, 1978, 6: 335–340

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bao J H, Yuan C G. Large deviations for neutral functional SDEs with jumps. Stochastics, 2015, 87: 48–70

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Budhiraja A, Dupuis P, Ganguly A. Moderate deviation principle for stochastic differential equations with jumps. Ann Probab, 2016, 44: 1723–1775

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cai Y J, Huang J H, Maroulas V. Large deviations of mean-field stochastic differential equations with jumps. Statist Probab Lett, 2015, 96: 1–9

    MathSciNet  Article  Google Scholar 

  8. 8.

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

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dembo A, Zeitouni O, Large Deviations Techniques and Applications. San Diego: Academic Press, 1989

    Google Scholar 

  10. 10.

    Dong Z, Xiong J, Zhai J L, Zhang T. A moderate deviation principle for 2-D stochastic Navier-Stokes equations driven by multiplicative Lévy noises. J Funct Anal, 2017, 272: 227–254

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dunford N, Schwartz J. Linear Operators, Part I. New York: Wiley, 1988

    Google Scholar 

  12. 12.

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

    Google Scholar 

  13. 13.

    Freidlin M I. Random perturbations of reaction-diffusion equations: the quasi-deterministic approach. Trans Amer Math Soc, 1988, 305: 665–697

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Freidlin M I, Wentzell A D. Random Perturbations of Dynamical Systems. New York: Springer, 1984

    Google Scholar 

  15. 15.

    Guillin A. Averaging principle of SDE with small diffusion: moderate deviations. Ann Probab, 2003, 31: 413–443

    MathSciNet  Article  Google Scholar 

  16. 16.

    Guillin A, Liptser R. Examples of moderate deviation principle for diffusion processes. Discrete Contin Dyn Syst Ser B, 2006, 6: 803–828

    MathSciNet  MATH  Google Scholar 

  17. 17.

    He Q, Yin G. Large deviations for multi-scale Markovian switching systems with a small diffusion. Asymptot Anal, 2014, 87: 123–145

    MathSciNet  Article  Google Scholar 

  18. 18.

    He Q, Yin G. Moderate deviations for time-varying dynamic systems driven by non-homogeneous Markov chains with two-time scales. Stochastics, 2014, 86: 527–550

    MathSciNet  Article  Google Scholar 

  19. 19.

    He Q, Yin G, Zhang Q. Large deviations for two-time-scale systems driven by non-homogeneous Markov chains and associated optimal control problems. SIAM J Control Optim, 2011, 49: 1737–1765

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  21. 21.

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

    MathSciNet  Article  Google Scholar 

  22. 22.

    Mao X. Stochastic Differential Equations and Applications. Amsterdam: Elsevier, 2007

    Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

    Peszat S. Large derivation principle for stochastic evolution equations. Probab Theory Related Fields, 1994, 98: 113–136

    MathSciNet  Article  Google Scholar 

  25. 25.

    Röckner M, Zhang T, Zhang X. Large deviations for stochastic tamed 3D Navier-Stokes equations. Appl Math Optim, 2010, 61: 267–285

    MathSciNet  Article  Google Scholar 

  26. 26.

    Sowers R. Large deviations for a reaction-diffusion equation with non-Gaussian perturbations. Ann Probab, 1992, 20: 504–537

    MathSciNet  Article  Google Scholar 

  27. 27.

    Suo Y Q, Tao J, Zhang W. Moderate deviation and central limit theorem for stochastic differential delay equations with polynomial growth. Front Math China, 2018, 13: 913–933

    MathSciNet  Article  Google Scholar 

  28. 28.

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

    MathSciNet  Article  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    MathSciNet  Article  Google Scholar 

  31. 31.

    Zhang X. Euler schemes and large deviations for stochastic Volterra equations with singular kernels. J Differential Equations, 2008, 244: 2226–2250

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11671034, 11771327, 61703001).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fubao Xi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ma, X., Xi, F. & Liu, D. Moderate deviations for neutral functional stochastic differential equations driven by Lévy noises. Front. Math. China 15, 529–554 (2020). https://doi.org/10.1007/s11464-020-0836-y

Download citation

Keywords

  • Moderate deviations
  • neutral functional stochastic differential equations
  • Poisson random measure

MSC

  • 60F10
  • 60H10