Skip to main content
SpringerLink
Log in

Comparison theorem for distribution-dependent neutral SFDEs

  • Published:
Journal of Evolution Equations Aims and scope Submit manuscript

Abstract

In this paper, the existence and uniqueness of strong solutions to distribution-dependent neutral SFDEs are proved. We give the conditions such that the order preservation of these equations holds. Moreover, we show these conditions are also necessary when the coefficients are continuous. Under sufficient conditions, the result extends the one in the distribution-independent case, and the necessity of these conditions is new even in distribution-independent case.

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. X. Bai, J. Jiang, Comparison theorems for neutral stochastic functional differential equations, J. Differ. Equ. 260(2016), 7250–7277.

    Article  MathSciNet  Google Scholar 

  2. J. Bao, C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math. 116(2011), 119–132.

    Article  MathSciNet  Google Scholar 

  3. R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications I, Berlin: Springer, 2018.

    Book  Google Scholar 

  4. M.-F. Chen, F.-Y. Wang, On order-preservation and positive correlations for multidimensional diffusion processes, Prob. Theory. Relat. Fields 95(1993), 421–428.

    Article  MathSciNet  Google Scholar 

  5. L. Gal’cuk, M. Davis, A note on a comparison theorem for equations with different diffusions, Stochastics 6(1982), 147–149.

    Article  MathSciNet  Google Scholar 

  6. X. Huang, M. Röckner, F.-Y. Wang, Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs, Discret. Contin. Dyn. Syst. 39(2019), 3017–3035.

    Article  MathSciNet  Google Scholar 

  7. X. Huang, F.-Y. Wang, Order-preservation for multidimensional stochastic functional differential equations with jumps, J. Evol. Equ. 14(2014), 445–460.

    Article  MathSciNet  Google Scholar 

  8. N. Ikeda, S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math. 14(1977), 619–633.

    MathSciNet  MATH  Google Scholar 

  9. T. Kamae, U.Krengel, G. L. O’Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab. 5(1977), 899–912.

    Article  MathSciNet  Google Scholar 

  10. X. Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics 37(1991), 49–59.

    MathSciNet  MATH  Google Scholar 

  11. G. L. O’Brien, A new comparison theorem for solution of stochastic differential equations, Stochastics 3(1980), 245–249.

    Article  MathSciNet  Google Scholar 

  12. S. Peng, Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab. 37(2009), 877–902.

    Article  MathSciNet  Google Scholar 

  13. S. Peng, X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stoch. Proc. Appl. 116(2006), 370–380.

    Article  MathSciNet  Google Scholar 

  14. A.V. Skorohod, Studies in the theory of random process, Boston: Addison-Wesley, 1965.

    Google Scholar 

  15. J.-M. Wang, Stochastic comparison for Lévy-type processes, J. Theor. Probab. 26(2013), 997–1019.

    Article  Google Scholar 

  16. F.-Y. Wang, The stochastic order and critical phenomena for superprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9(2006), 107–128.

    Article  MathSciNet  Google Scholar 

  17. F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl. 128(2018), 595-621.

    Article  MathSciNet  Google Scholar 

  18. T. Yamada, On comparison theorem for solutions of stochastic differential equations and its applications, J. Math. Kyoto Univ. 13(1973), 495–512.

    MathSciNet  Google Scholar 

  19. Z. Yang, X. Mao, C. Yuan, Comparison theorem of one-dimensional stochastic hybrid systems, Syst. Control Lett. 57(2008), 56–63.

    Article  MathSciNet  Google Scholar 

  20. X. Zhu, On the comparison theorem for multi-dimensional stochastic differential equations with jumps (in Chinese), Sci. Sin. Math. 42(2012), 303–311.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the associated editor and referees for their helpful comments and suggestions.

Author information

Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Xing Huang

  2. Department of Mathematics, Swansea University, Swansea, SA1 8EN, UK

    Chenggui Yuan

Authors
  1. Xing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chenggui Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Huang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by NNSFC (11801406).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, X., Yuan, C. Comparison theorem for distribution-dependent neutral SFDEs. J. Evol. Equ. 21, 653–670 (2021). https://doi.org/10.1007/s00028-020-00595-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00028-020-00595-w

Keywords

Mathematics Subject Classification

Search

Navigation