Skip to main content
SpringerLink
Log in

Parameter Estimation of Path-Dependent McKean-Vlasov Stochastic Differential Equations

  • Published:
Acta Mathematica Scientia Aims and scope Submit manuscript

Abstract

This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.

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. Ait-Sahalia Y. Nonparametric pricing of interest rate derivative securities. Econometrica, 1996, 64: 527–560

    Article  Google Scholar 

  2. Bhatt A G, Karandikar R L. Robustness of the nonlinear filter: the correlated case. Stochastic Process Appl, 2002, 97: 41–58

    Article  MathSciNet  Google Scholar 

  3. Bibby B M, Sørensen M. Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1995, 1: 17–39

    Article  MathSciNet  Google Scholar 

  4. Bishwal J P N. Large deviations inequalities for the maximum likelihood estimator and the Bayes estimators in nonlinear stochastic differential equations. Statistics & Probability Letters, 1999, 43: 207–215

    Article  MathSciNet  Google Scholar 

  5. Bossy M, Talay D. A stochastic particle method for the McKean-Vlasov and the Burgers equation. Mathematics of Computation, 1997, 66: 157–192

    Article  MathSciNet  Google Scholar 

  6. Dawson D, Vaillancourt J. Stochastic McKean-Vlasov equations. Nonlinear Differential Equations and Applications, 1995, 2: 199–229

    Article  MathSciNet  Google Scholar 

  7. Ding X, Qiao H. Euler-Maruyama approximations for stochastic McKean-Vlasov equations with non-Lipschitz coefficients. Journal of Theoretical Probability, 2021, 34: 1408–1425

    Article  MathSciNet  Google Scholar 

  8. Ding X, Qiao H, Stability for stochastic McKean-Vlasov equations with non-Lipschitz coefficients. SIAM Journal on Control and Optimization, 2021, 59: 887–905

    Article  MathSciNet  Google Scholar 

  9. Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Related Fields, 2015, 162: 707–738

    Article  MathSciNet  Google Scholar 

  10. Karatzas I, Shreve S E. Brownian motion and stochastic calculus. 2nd ed. Graduate Texts in Mathematics. Vol 113. New York: Springer-Verlag, 2005

    Google Scholar 

  11. Liptser R S, Shiryayev A N. Statistics of Random Processes: I General Theory. 2nd ed. Springer, 2001

  12. McKean H P. A Class of Markov Processes Associated with Nonlinear Parabolic Equations. Proceedings of the National Academy of Sciences, 1966, 56: 1907–1911

    Article  MathSciNet  Google Scholar 

  13. Qiao H. A Nonlinear Stochastic Evolution Equation in Hilbert space. Acta Mathematica Scientia, 2009, 29A(2): 383–391

    MathSciNet  MATH  Google Scholar 

  14. Ren P, Wu J. Least squares estimator for path-dependent McKean-Vlasov SDEs via discrete-time observations. Acta Mathematica Scientia, 2019, 39B(3): 691–716

    Article  MathSciNet  Google Scholar 

  15. Stroock D W. Lectures on Stochastic Analysis: Diffusion Theory. London Mathematical Society Students Texts, Cambridge University Press, 1987

  16. Wang F-Y. Distribution dependent SDEs for Landau type equations. Stoch Proc Appl, 2018, 128: 595–621

    Article  MathSciNet  Google Scholar 

  17. Wen J, Wang X, Mao S, Xiao X. Maximum likelihood estimation of McKean-Vlasov stochastic differential equation and its application. Applied Mathematics and Computation, 2016, 274: 237–246

    Article  MathSciNet  Google Scholar 

  18. Yoshida N. Estimation for diffusion processes from discrete observation. Journal of Multivariate Analysis, 1992, 41: 220–242

    Article  MathSciNet  Google Scholar 

  19. Zhang X. Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications. J Math Anal Appl, 2006, 316: 447–458

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second author thanks Professor Renming Song for providing her with an excellent environment in which to work at the University of Illinois at Urbana-Champaign. Both authors are grateful to the two referees, as their suggestions and comments improved the results and the presentation of this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 211189, China

    Meiqi Liu  (刘美琪) & Huijie Qiao  (乔会杰)

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Huijie Qiao  (乔会杰)

Authors
  1. Meiqi Liu  (刘美琪)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huijie Qiao  (乔会杰)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huijie Qiao  (乔会杰).

Additional information

The second author is supported by NSF of China (11001051, 11371352, 12071071) and China Scholarship Council (201906095034).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, M., Qiao, H. Parameter Estimation of Path-Dependent McKean-Vlasov Stochastic Differential Equations. Acta Math Sci 42, 876–886 (2022). https://doi.org/10.1007/s10473-022-0304-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10473-022-0304-8

Key words

2010 MR Subject Classification

Search

Navigation