PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

doi:10.1007/s10473-022-0304-8

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 876-886.doi: 10.1007/s10473-022-0304-8

PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

刘美琪¹, 乔会杰^1,2

1. Department of Mathematics, Southeast University, Nanjing, 211189, China;
2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

收稿日期:2020-10-13 修回日期:2021-06-14 发布日期:2022-06-24
通讯作者: Huijie QIAO,E-mail:hjqiaogean@seu.edu.cn E-mail:hjqiaogean@seu.edu.cn
基金资助:
The second author is supported by NSF of China (11001051, 11371352, 12071071) and China Scholarship Council (201906095034).

PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

Meiqi LIU¹, Huijie QIAO^1,2

1. Department of Mathematics, Southeast University, Nanjing, 211189, China;
2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

Received:2020-10-13 Revised:2021-06-14 Published:2022-06-24
Contact: Huijie QIAO,E-mail:hjqiaogean@seu.edu.cn E-mail:hjqiaogean@seu.edu.cn
Supported by:
The second author is supported by NSF of China (11001051, 11371352, 12071071) and China Scholarship Council (201906095034).

摘要/Abstract

摘要： 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.

关键词: Path-dependent McKean-Vlasov stochastic differential equations, maximum likelihood estimation, the strong consistency, numerical simulation

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.

Key words: Path-dependent McKean-Vlasov stochastic differential equations, maximum likelihood estimation, the strong consistency, numerical simulation

中图分类号:

60H10

刘美琪, 乔会杰. PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(3): 876-886.

Meiqi LIU, Huijie QIAO. PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2022, 42(3): 876-886.

参考文献

[1] Ait-Sahalia Y. Nonparametric pricing of interest rate derivative securities. Econometrica, 1996, 64:527-560
[2] Bhatt A G, Karandikar R L. Robustness of the nonlinear filter:the correlated case. Stochastic Process Appl, 2002, 97:41-58
[3] Bibby B M, Sørensen M. Martingale estimation functions for discretely observed diffusion processes. Bernoulli, 1995, 1:17-39
[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
[5] Bossy M, Talay D. A stochastic particle method for the McKean-Vlasov and the Burgers equation. Mathematics of Computation, 1997, 66:157-192
[6] Dawson D, Vaillancourt J. Stochastic McKean-Vlasov equations. Nonlinear Differential Equations and Applications, 1995, 2:199-229
[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
[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
[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
[10] Karatzas I, Shreve S E. Brownian motion and stochastic calculus. 2nd ed. Graduate Texts in Mathematics. Vol 113. New York:Springer-Verlag, 2005
[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
[13] Qiao H. A Nonlinear Stochastic Evolution Equation in Hilbert space. Acta Mathematica Scientia, 2009, 29A(2):383-391
[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
[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
[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
[18] Yoshida N. Estimation for diffusion processes from discrete observation. Journal of Multivariate Analysis, 1992, 41:220-242
[19] Zhang X. Euler-Maruyama approximations for SDEs with non-Lipschitz coefficients and applications. J Math Anal Appl, 2006, 316:447-458

PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

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