Statistical inference for ergodic diffusion with Markovian switching

Yuzhong Cheng; Hiroki Masuda

doi:10.3934/dcdsb.2025056

Article Contents

2025, Volume 30, Issue 10: 3910-3940. Doi: 10.3934/dcdsb.2025056

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Statistical inference for ergodic diffusion with Markovian switching

Yuzhong Cheng^1, and
Hiroki Masuda^2, ,

1.
Joint Graduate School of Mathematics for Innovation, Kyushu University, 744 Motooka Fukuoka, Japan
2.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo, Japan

^*Corresponding author: Hiroki Masuda
^*Corresponding author: Hiroki Masuda

Received: September 1, 2024

Early access: April 14, 2025

Published online: October 1, 2025

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Abstract

This study explores a Gaussian quasi-likelihood approach for estimating parameters of diffusion processes with Markovian regime switching. Assuming the ergodicity under high-frequency sampling, we will show the asymptotic normality of the unknown parameters contained in the drift and diffusion coefficients and present a consistent explicit estimator for the generator of the Markov chain. Simulation experiments are conducted to illustrate the theoretical results obtained.

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Mathematics Subject Classification: Primary: 62M05, 60H10, 60J60; Secondary: 60K37.

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Figure 1. Sample paths of SDE solution $ X $ and associated Markov chain $ Z $

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Figure 2. Boxplots of the estimators for $ h = 0.01 $ with four schemes:(i) $ T = 100,n = 10000 $; (ii) $ T = 300,n = 30000 $; (iii) $ T = 500,n = 50000 $; (iv) $ T = 1000, n = 100000 $. The red dashed line indicates the true value of the parameters

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Figure 3. Boxplots of the estimators for $ h = 0.001 $ with four schemes:(i) $ T = 100,n = 100000 $; (ii) $ T = 300,n = 300000 $; (iii) $ T = 500,n = 500000 $; (iv) $ T = 1000, n = 1000000 $. The red dashed line indicates the true value of the parameters

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Figure 4. Histograms of the standardized estimators for $ h = 0.01 $. The red curve indicates the standard normal density

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Figure 5. Histograms of the standardized estimators for $ h = 0.001 $. The red curve indicates the standard normal density

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Table 1. The mean and the standard deviation (Std. Dev.) of the estimators with true values $ \theta = (1,2,0.1,0.2) $ and $ h = 0.01 $

	$ T=100, n=10000 $		$ T=300, n=30000 $		$ T=500, n=50000 $		$ T=1000, n=100000 $
Estimator	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
$ \hat{\beta}_{1,n} $	1.130	1.018	1.027	0.125	1.018	0.088	0.999	0.072
$ \hat{\beta}_{2,n} $	1.546	1.192	1.913	0.452	2.002	0.374	1.981	0.092
$ \hat{\sigma}_{1,n} $	0.367	0.422	0.141	0.189	0.104	0.999	0.100	0.000
$ \hat{\sigma}_{2,n} $	0.169	0.045	0.194	0.021	0.198	0.072	0.198	0.001

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Table 2. The mean and the standard deviation (Std. Dev.) of the estimators with true values $ \theta = (1,2,0.1,0.2) $ and $ h = 0.001 $

	$ T=100, n=100000 $		$ T=300, n=300000 $		$ T=500,n=500000 $		$ T=1000, n=1000000 $
Estimator	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
$ \hat{\beta}_{1,n} $	1.042	0.187	1.018	0.122	1.011	0.103	0.999	0.065
$ \hat{\beta}_{2,n} $	1.372	1.157	1.879	0.526	1.970	0.266	2.006	0.118
$ \hat{\sigma}_{1,n} $	0.459	0.460	0.153	0.213	0.110	0.102	0.100	0.000
$ \hat{\sigma}_{2,n} $	0.161	0.049	0.194	0.024	0.199	0.010	0.200	0.000

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Table 3. The mean and the standard deviation (Std. Dev.) of the estimators with true values $ Q = \begin{pmatrix} -0.01 & 0.01 \\ 0.01 & -0.01 \\ \end{pmatrix} $ and $ h = 0.01 $

	$ T=100, n=10000 $		$ T=300, n=30000 $		$ T=500, n=50000 $		$ T=1000, n=100000 $
Estimator	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
$ \hat{q}_{11}^{(n)} $	-0.0095	0.0172	-0.0099	0.0097	-0.0103	0.0071	-0.0109	0.0057
$ \hat{q}_{12}^{(n)} $	0.0095	0.0172	0.0099	0.0097	0.0103	0.0071	0.0109	0.0057
$ \hat{q}_{21}^{(n)} $	0.0664	0.1463	0.0280	0.0645	0.0180	0.0387	0.0121	0.0061
$ \hat{q}_{22}^{(n)} $	-0.0664	0.1463	-0.0280	0.0645	-0.0180	0.0387	-0.0121	0.0061

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Table 4. The mean and the standard deviation (Std. Dev.) of the estimators with true values $ Q = \begin{pmatrix} -0.01 & 0.01 \\ 0.01 & -0.01 \\ \end{pmatrix} $ and $ h = 0.001 $

	$ T=100, n=100000 $		$ T=300, n=300000 $		$ T=500, n=500000 $		$ T=1000, n=1000000 $
Estimator	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.	Mean	Std. Dev.
$ \hat{q}_{11}^{(n)} $	-0.0133	0.0273	-0.0098	0.0091	-0.0107	0.0123	-0.0102	0.0057
$ \hat{q}_{12}^{(n)} $	0.0133	0.0273	0.0098	0.0091	0.0107	0.0123	0.0102	0.0057
$ \hat{q}_{21}^{(n)} $	0.0643	0.1783	0.0270	0.0343	0.0163	0.0247	0.0126	0.0058
$ \hat{q}_{22}^{(n)} $	-0.0643	0.1783	-0.0270	0.0343	-0.0163	0.0247	-0.0126	0.0058

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