Home Trajectory fitting estimation for stochastic differential equations driven by fractional Brownian motion
Article
Licensed
Unlicensed Requires Authentication

Trajectory fitting estimation for stochastic differential equations driven by fractional Brownian motion

  • Hector Araya ORCID logo EMAIL logo and John Barrera
Published/Copyright: October 27, 2023

Abstract

We consider the problem of drift parameter estimation in a stochastic differential equation driven by fractional Brownian motion with Hurst parameter H ( 1 2 , 1 ) and small diffusion. The technique that we used is the trajectory fitting method. Strong consistency and asymptotic distribution of the estimator are established as a small diffusion coefficient goes to zero.

MSC 2020: 60G22; 62M09; 60H3

Communicated by Stanislav Molchanov


Award Identifier / Grant number: 11230051

Award Identifier / Grant number: ECOS210037(C21E07)

Award Identifier / Grant number: AMSUD210023

Funding statement: The first author was partially supported by FONDECYT 11230051, Proyecto ECOS210037(C21E07), Mathamsud AMSUD210023. The work of the second author was supported by a scholarship granted by the Postgraduate and Programs Direction of Universidad Técnica Federico Santa María.

References

[1] K. Bertin, S. Torres and C. A. Tudor, Drift parameter estimation in fractional diffusions driven by perturbed random walks, Statist. Probab. Lett. 81 (2011), no. 2, 243–249. 10.1016/j.spl.2010.10.003Search in Google Scholar

[2] K. Bertin, S. Torres and C. A. Tudor, Maximum-likelihood estimators and random walks in long memory models, Statistics 45 (2011), no. 4, 361–374. 10.1080/02331881003768750Search in Google Scholar

[3] A. Beskos, J. Dureau and K. Kalogeropoulos, Bayesian inference for partially observed stochastic differential equations driven by fractional Brownian motion, Biometrika 102 (2015), no. 4, 809–827. 10.1093/biomet/asv051Search in Google Scholar

[4] K. Es-Sebaiy, R. Belfadli and Y. Ouknine, Parameter estimation for fractional Ornstein–Uhlenbeck processes: Non-ergodic case, Front. Environ. Sci. Eng. China 1 (2011), no. 1, 1–16. Search in Google Scholar

[5] M. Gubinelli, Controlling rough paths, J. Funct. Anal. 216 (2004), no. 1, 86–140. 10.1016/j.jfa.2004.01.002Search in Google Scholar

[6] Y. Hu and D. Nualart, Parameter estimation for fractional Ornstein–Uhlenbeck processes, Statist. Probab. Lett. 80 (2010), no. 11–12, 1030–1038. 10.1016/j.spl.2010.02.018Search in Google Scholar

[7] Y. Hu, D. Nualart, W. Xiao and W. Zhang, Exact maximum likelihood estimator for drift fractional Brownian motion at discrete observation, Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 5, 1851–1859. 10.1016/S0252-9602(11)60365-2Search in Google Scholar

[8] Y. Hu, D. Nualart and H. Zhou, Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter, Stat. Inference Stoch. Process. 22 (2019), no. 1, 111–142. 10.1007/s11203-017-9168-2Search in Google Scholar

[9] K. Kubilius, V. Skorniakov and D. Melichov, Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift, J. Stat. Comput. Simul. 86 (2016), no. 10, 1954–1969. 10.1080/00949655.2015.1095301Search in Google Scholar

[10] K. Kubilius, Y. Mishura and K. Ralchenko, Parameter Estimation in Fractional Diffusion Models, Bocconi & Springer Ser. 8, Springer, Cham, 2017. 10.1007/978-3-319-71030-3Search in Google Scholar

[11] Y. Kutoyants, Minimum-distance parameter estimation for diffusion-type observations, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 8, 637–642. Search in Google Scholar

[12] J. A. León and S. Tindel, Malliavin calculus for fractional delay equations, J. Theoret. Probab. 25 (2012), no. 3, 854–889. 10.1007/s10959-011-0349-4Search in Google Scholar

[13] Y. Mishura and G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics 80 (2008), no. 5, 489–511. 10.1080/17442500802024892Search in Google Scholar

[14] I. Nourdin and T. T. D. Tran, Statistical inference for Vasicek-type model driven by Hermite processes, Stochastic Process. Appl. 129 (2019), no. 10, 3774–3791. 10.1016/j.spa.2018.10.005Search in Google Scholar

[15] D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. Search in Google Scholar

[16] B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes: Kendall’s Library of Statistics 8, Wiley, New York, 2010. Search in Google Scholar

[17] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, 1987. Search in Google Scholar

[18] B. Saussereau, Nonparametric inference for fractional diffusion, Bernoulli 20 (2014), no. 2, 878–918. 10.3150/13-BEJ509Search in Google Scholar

[19] C. A. Tudor and F. G. Viens, Statistical aspects of the fractional stochastic calculus, Ann. Statist. 35 (2007), no. 3, 1183–1212. 10.1214/009053606000001541Search in Google Scholar

[20] L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), no. 1, 251–282. 10.1007/BF02401743Search in Google Scholar

[21] M. Zähle, Integration with respect to fractal functions and stochastic calculus. I, Probab. Theory Related Fields 111 (1998), no. 3, 333–374. 10.1007/s004400050171Search in Google Scholar

Received: 2022-10-20
Accepted: 2023-03-20
Published Online: 2023-10-27
Published in Print: 2023-12-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/rose-2023-2018/html
Scroll to top button