Communications in Mathematical Sciences

Volume 20 (2022)

Number 1

Least squares estimation for delay McKean–Vlasov stochastic differential equations and interacting particle systems

Pages: 265 – 296

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n1.a8

Authors

Min Zhu (College of Railway Transportation, Hunan University of Technology, Zhuzhou, Hunan, China)

Yanyan Hu (School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China)

Abstract

The aim of this paper is to solve the problem of parameter estimation for delay McKean–Vlasov stochastic differential equations (SDEs) with the coefficients exhibiting super-linear growth in the state component. Specifically, we propose a least squares estimator for an unknown parameter in the drift of a delay McKean–Vlasov SDEs with a small noise dispersion parameter by making use of time-discretized interacting particle systems and prove the weak convergence between the estimator and the true value, under suitable conditions. To achieve our main purposes on weak convergence, we give the approximation of the distribution of delay McKean–Vlasov SDEs at the discrete points and take advantage of calculating skills on the space of probability measures with finite order moments. Moreover, the asymptotic distribution of least squares estimator is derived via the properties of solutions for the corresponding interacting particle systems.

Keywords

McKean–Vlasov SDE, interacting particle systems, discrete observation, least squares method, consistency of LSE, asymptotic distribution

2010 Mathematics Subject Classification

60G52, 60J75, 62F12, 62M05

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This paper was supported by the National Natural Science Foundation of China (No. 11901188), and by the Scientific Research Funds of Hunan Provincial Education Department of China (No. 19B156).

Received 19 January 2021

Received revised 21 June 2021

Accepted 21 June 2021

Published 10 December 2021