April  2022, 21(4): 1505-1536. doi: 10.3934/cpaa.2022027

Least squares estimation for distribution-dependent stochastic differential delay equations

1. 

School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

2. 

Hunan University of Technology, Zhuzhou 412007, Hunan, China

* Corresponding author

Received  July 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

Fund Project: The second author is partially supported by the National Natural Science Foundation of China (No. 12071031). The third author is partially supported by the National Natural Science Foundation of China (No. 11901188) and the Scientific Research Funds of Hunan Provincial Education Department of China (19B156)

The parametric estimation of drift parameter for distribution - dependent stochastic differential delay equations with a small diffusion is presented. The principle technique of our investigation is to construct an appropriate contrast function and carry out a limiting type of argument to show the consistency and convergence rate of the least squares estimator of the drift parameter via interacting particle systems. In addition, two examples are constructed to demonstrate the effectiveness of our work.

Citation: Yanyan Hu, Fubao Xi, Min Zhu. Least squares estimation for distribution-dependent stochastic differential delay equations. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1505-1536. doi: 10.3934/cpaa.2022027
References:
[1]

J. Bao and X. Huang, Approximations of McKean-Vlasov stochastic differential equations with irregular coefficients, J. Theor. Probab., (2021), 29 pp. doi: 10.1007/s10959-021-01082-9.

[2]

R. BuckdahnJ. Li and J. Ma, A mean-field stochastic control problem with partial observations, Ann. Appl. Probab., 27 (2017), 3201-3245.  doi: 10.1214/17-AAP1280.

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I. Mean Field FBSDEs, Control, and Games, 1$^{nd}$ edition, Springer, Cham, 2018. doi: 0.1007/978-3-319-58920-6.

[4]

R. BuckdahnJ. Li and J. Ma, Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), 7135-7173.  doi: 10.1090/tran/7576.

[5]

Y. Hu and H. Long, Least squares estimator for Ornstein-Uhlenbeck processes driven by $\alpha$-stable motions, Stochastic Process. Appl., 119 (2009), 2465-2480.  doi: 10.1016/j.spa.2008.12.006.

[6]

X. Huang, Path-Distribution Dependent SDEs with Singular Coefficients, Electron. J. Probab., 26 (2021), 1-21.  doi: 10.1214/21-EJP630.

[7]

X. HuangM. Röckner and F. Wang, Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst., 39 (2019), 3017-3035.  doi: 10.3934/dcds.2019125.

[8]

X. HuangM. Röckner and F. Wang, Comparison theorem for distribution-dependent neutral SFDEs, J. Evol. Equ., 21 (2021), 653-670.  doi: 10.1007/s00028-020-00595-w.

[9]

M. Kac, Foundations of kinetic theory, Proc. Third Berkeley Symp. on Math. Stat. and Prob., 3 (1956), 171-197.  doi: 10.1525/9780520350694-012.

[10]

M. Kac, Probability and Related Topics in the Physical Sciences, 1$^{nd}$ edition, Interscience Publishers, New York, 1960. doi: 10.2307/2309209.

[11]

R. A. Kasonga, The consistency of a non-linear least squares estimator from diffusion processes, Stochastic Process. Appl., 2 (1988), 263-275.  doi: 10.1143/PTPS.69.101.

[12]

H. LongC. Ma and Y. Shimizu, Least squares estimators for stochastic differential equations driven by small Lévy noises, Stochastic Process. Appl., 127 (2017), 1475-1495.  doi: 10.1016/J.SPA.2016.08.006.

[13]

H. LongY. Shimizu and W. Sun, Least squares estimators for discretely observed stochastic processes driven by small Lévy noises, J. Multivariate Anal., 116 (2013), 422-439.  doi: 10.1016/j.jmva.2013.01.012.

[14]

C. Ma, A note on 'Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises', Statist. Probab. Lett., 80 (2010), 1528-1531.  doi: 10.1016/J.SPL.2010.06.006.

[15]

M. Kac, Stochastic differential equations and applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.47.

[16]

P. Ren and F. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differ. Equ., 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.

[17]

P. Ren and J. Wu, Least squares estimation for path-distribution dependent stochastic differential equations, Appl. Math. Comput., 410 (2021), 126457.  doi: 10.1016/j.amc.2021.126457.

[18]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Appl. Math. Comput., 2 (2021), 1131-1158.  doi: 10.3150/20-BEJ1268.

[19]

A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 3nd edition, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511802256.

[20]

F. Wang, Distribution-dependent SDEs for Landau type equations, Stochastic Process. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.

[21]

N. Yoshida, Asymptotic expansion for statistics related to small diffusions, J. Japan Statist. Soc., 22 (1992), 139-159.  doi: 10.11329/jjss1970.22.139.

show all references

References:
[1]

J. Bao and X. Huang, Approximations of McKean-Vlasov stochastic differential equations with irregular coefficients, J. Theor. Probab., (2021), 29 pp. doi: 10.1007/s10959-021-01082-9.

[2]

R. BuckdahnJ. Li and J. Ma, A mean-field stochastic control problem with partial observations, Ann. Appl. Probab., 27 (2017), 3201-3245.  doi: 10.1214/17-AAP1280.

[3]

R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I. Mean Field FBSDEs, Control, and Games, 1$^{nd}$ edition, Springer, Cham, 2018. doi: 0.1007/978-3-319-58920-6.

[4]

R. BuckdahnJ. Li and J. Ma, Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), 7135-7173.  doi: 10.1090/tran/7576.

[5]

Y. Hu and H. Long, Least squares estimator for Ornstein-Uhlenbeck processes driven by $\alpha$-stable motions, Stochastic Process. Appl., 119 (2009), 2465-2480.  doi: 10.1016/j.spa.2008.12.006.

[6]

X. Huang, Path-Distribution Dependent SDEs with Singular Coefficients, Electron. J. Probab., 26 (2021), 1-21.  doi: 10.1214/21-EJP630.

[7]

X. HuangM. Röckner and F. Wang, Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst., 39 (2019), 3017-3035.  doi: 10.3934/dcds.2019125.

[8]

X. HuangM. Röckner and F. Wang, Comparison theorem for distribution-dependent neutral SFDEs, J. Evol. Equ., 21 (2021), 653-670.  doi: 10.1007/s00028-020-00595-w.

[9]

M. Kac, Foundations of kinetic theory, Proc. Third Berkeley Symp. on Math. Stat. and Prob., 3 (1956), 171-197.  doi: 10.1525/9780520350694-012.

[10]

M. Kac, Probability and Related Topics in the Physical Sciences, 1$^{nd}$ edition, Interscience Publishers, New York, 1960. doi: 10.2307/2309209.

[11]

R. A. Kasonga, The consistency of a non-linear least squares estimator from diffusion processes, Stochastic Process. Appl., 2 (1988), 263-275.  doi: 10.1143/PTPS.69.101.

[12]

H. LongC. Ma and Y. Shimizu, Least squares estimators for stochastic differential equations driven by small Lévy noises, Stochastic Process. Appl., 127 (2017), 1475-1495.  doi: 10.1016/J.SPA.2016.08.006.

[13]

H. LongY. Shimizu and W. Sun, Least squares estimators for discretely observed stochastic processes driven by small Lévy noises, J. Multivariate Anal., 116 (2013), 422-439.  doi: 10.1016/j.jmva.2013.01.012.

[14]

C. Ma, A note on 'Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises', Statist. Probab. Lett., 80 (2010), 1528-1531.  doi: 10.1016/J.SPL.2010.06.006.

[15]

M. Kac, Stochastic differential equations and applications, 2$^{nd}$ edition, Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.47.

[16]

P. Ren and F. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differ. Equ., 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.

[17]

P. Ren and J. Wu, Least squares estimation for path-distribution dependent stochastic differential equations, Appl. Math. Comput., 410 (2021), 126457.  doi: 10.1016/j.amc.2021.126457.

[18]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Appl. Math. Comput., 2 (2021), 1131-1158.  doi: 10.3150/20-BEJ1268.

[19]

A. W. van der Vaart, Asymptotic statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 3nd edition, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511802256.

[20]

F. Wang, Distribution-dependent SDEs for Landau type equations, Stochastic Process. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.

[21]

N. Yoshida, Asymptotic expansion for statistics related to small diffusions, J. Japan Statist. Soc., 22 (1992), 139-159.  doi: 10.11329/jjss1970.22.139.

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