分布不确定下随机微分方程参数最小二乘估计

Abstract

Abstract:

Under distribution uncertainty, on the basis of discrete observation data we investigate the consistency of the least squares estimator (LSE) of the parameter for the stochastic differential equation (SDE) where the noise are characterized by G-Brownian motion. In order to obtain our main result of consistency of parameter estimation, we provide some lemmas by the theory of stochastic calculus of sublinear expectation. The result shows that under some regularity conditions, the least squares estimator is strong consistent uniformly on the prior set. An illustrative example is discussed.

Key words: Stochastic differential equation disturbed by G-Brownian motiion (G-SDE), Sublinear expectation, Least squares estimator, Exponential martingale inequality for capacity, Strong consistency

CLC Number:

O211.6

Chen Fei,Weiyin Fei. Consistency of Least Squares Estimation to the Parameter for Stochastic Differential Equations Under Distribution Uncertainty[J].Acta mathematica scientia,Series A, 2019, 39(6): 1499-1513.

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References 30

1	Artzner P , Delbaen F , Eber J M , Heath D . Coherent measures of risk. Math Finance, 1999, 9 (3): 203- 228
2	Bishwal J P N. Parameter Estimation in Stochastic Differential Equations. Berlin Heidelberg: Springer-Verlag, 2008
3	Chen Z . Strong laws of large numbers for sub-linear expectations. Science China Mathematics, 2016, 59 (5): 945- 954 doi: 10.1007/s11425-015-5095-0
4	Chen Z , Epstein L . Ambiguity, risk and asset returns in continuous time. Econometrica, 2002, 70 (4): 1403- 1443 doi: 10.1111/1468-0262.00337
5	Cheridito P , Kawaguchi H , Maejima M . Fractional Ornstein-Uhlenbeck processes. Electron J Probab, 2003, 8: 1- 14 doi: 10.1214/ECP.v8-1064
6	Delbaen F , Peng S , Rosazza Gianin E . Representation of the penalty term of dynamic concave utilities. Finance Stoch, 2010, 14: 449- 472 doi: 10.1007/s00780-009-0119-7
7	Ellsberg D . Risk, ambiguity, and the Savage axioms. Quarterly Journal of Economics, 1961, 75: 643- 669 doi: 10.2307/1884324
8	Epstein L , Ji S . Ambiguity volatility and asset pricing in continuous time. Journal of Mathematical Economics, 2014, 50: 269- 282 doi: 10.1016/j.jmateco.2013.09.005
9	Fan J , Peng H . Nonconcave penalized likelihood with a diverging number of parameters. Annals of Statistics, 2004, 32: 928- 961 doi: 10.1214/009053604000000256
10	Fei C , Fei W Y , Yan L T . Existence and stability of solutions to highly nonlinear stochastic differential delay equations driven by G-Brownian motion. Appl Math J Chinese Univ, 2019, 34B (2): 184- 204
11	Fei W Y . Optimal portfolio choice based on α-MEU under ambiguity. Stochastic Models, 2009, 25: 455- 482 doi: 10.1080/15326340903088826
12	Fei W Y, Fei C. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion. 2013, arXiv: 1309.0209v1
13	Fei W Y, Fei C. On exponential stability for stochastic differential equations disturbed by G-Brownian motion. 2013, arXiv: 1311.7311v1
14	Gao F Q . Pathwise properties and homeomorphic flows for stochastic differential equations driven by G-Brownian motion. Stochastic Processes and their Applications, 2009, 119: 3356- 3382 doi: 10.1016/j.spa.2009.05.010
15	Hu M , Ji S , Peng S , Song Y . Backward stochastic differential equations driven by G-Brownian motion. Stochastic Processes and their Applications, 2014, 124: 759- 784 doi: 10.1016/j.spa.2013.09.010
16	Hu Y , Nualart D . Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statistics and Probability Letters, 2010, 80: 1030- 1038 doi: 10.1016/j.spl.2010.02.018
17	Huber P J. Robust Statistics. New York: John Wiley and Sons, 1981
18	Kasonga R . The consistemcy of a non-linear least squares estimation from diffusion processes. Stochastic Processes and their Applications, 1988, 30: 263- 275 doi: 10.1016/0304-4149(88)90088-9
19	Knight F H. Risk, Uncertainty and Profit. Boston: Houghton Mifflin, 1921
20	Kutoyants Yu A. Statistical Inference for Ergodic Diffusion Processes. London: Springer-Verlag, 2004
21	Lin L , Dong P , Song Y , Zhu L . Upper expectation parametric regression. Statistica Sinica, 2017, 27 (3): 1265- 1280
22	Lin L , Shi Y , Wang X , Yang S . k-sample upper expectation linear regression-modeling, identifiability, estimation and prediction. Journal of Statistical Planning and Inference, 2016, 170: 15- 26 doi: 10.1016/j.jspi.2015.09.002
23	Lin Q . Some properties of stochastic differential equations driven by G-Brownian motion. Acta Math Sin (Engl Ser), 2013, 29: 923- 942 doi: 10.1007/s10114-013-0701-y
24	Lin Y . Stochastic differential eqations driven by G-Brownian motion with reflecting boundary. Electron J Probab, 2013, 18 (9): 1- 23
25	Luo P , Wang F . Stochastic differential equations driven by G-Brownian motion and ordinary differential equations. Stochastic Processes and their Applications, 2014, 124: 3869- 3885 doi: 10.1016/j.spa.2014.07.004
26	Mao X. Stochastic Differential Equations and Their Applications, 2nd Edition. Chichester: Horwood Pub, 2007
27	Peng S. Nonlinear expectations and stochastic calculus under uncertainty. 2010, arXiv: 1002.4546v1
28	Prakasa Rao B L S. Statistical Inference for Diffusion Type Processes. London: Arnold, 1999
29	Shen G J , Yin X W , Yan L T . Least squares estimation for Ornstein-Uhlenbeck processes driven by the weighted fractional Brownian motion. Acta Mathematica Scientia, 2016, 36B (2): 394- 408
30	Sun C F , Ji S L . The least squares estimator of random variables under sublinear expectations. Journal of Mathematical Analysis and Applications, 2017, 451 (2): 906- 923 doi: 10.1016/j.jmaa.2017.02.020

$n $	$1 \times 10^4$	$ 2 \times 10^4$	$3 \times 10^4$	$4 \times 10^4$	$5 \times 10^4$
$\bar{\hat{\theta}}_{n, T}(10, 2^9) $	1.0313	1.0193	1.0144	1.0089	1.0057
$ \underline{\hat{\theta}}_{n, T}(10, 2^9) $	1.0104	1.0027	1.0014	0.9972	0.9978
$\bar{\hat{\theta}}_{n, T}(10, 2^9)- \underline{\hat{\theta}}_{n, T}(10, 2^9) $	0.0209	0.0166	0.0130	0.0117	0.0079

$J$	$2^{3}$	$2^4$	$2^5$	$2^6$	$2^7$
$\bar{\hat{\theta}}_{5 \times 10^3, T}(10, J)$	1.1108	1.0955	1.0903	1.0780	1.0672
$ \underline{\hat{\theta}}_{5 \times 10^3, T}(10, J) $	0.9718	0.9713	0.9879	0.9888	0.9995
$\bar{\hat{\theta}}_{5 \times 10^3, T}(10, J)-\underline{\hat{\theta}}_{5 \times 10^3, T}(10, J)$	0.1390	0.1241	0.1023	0.0892	0.0677

[1]	Wang Jixia, Xiao Qingxian. Local Estimation of the Diffusion Coefficient for Time-Inhomogeneous Diffusion Models [J]. Acta mathematica scientia,Series A, 2015, 35(2): 332-344.
[2]	TAO Bao. Asymptotic Properties of the Negative Extreme-value Index Estimator [J]. Acta mathematica scientia,Series A, 2014, 34(3): 611-618.
[3]	YIN Chang-Ming, LI Yong-Ming, WANG Peng-Yan. Strong Convergence Rates of the Maximum Quasi-Likelihood Estimator in Generalized Linear Models [J]. Acta mathematica scientia,Series A, 2009, 29(4): 1058-1064.
[4]	Li Guoliang; Liu Luqin. Strong Consistency of a Class of Estimators in Partial Linear Model under Martingale Difference Sequence [J]. Acta mathematica scientia,Series A, 2007, 27(5): 788-801.
[5]	Ding Jieli;Chen Xiru. Strong Consistency of the Maximum Likelihood Estimator in Generalized Linear Models [J]. Acta mathematica scientia,Series A, 2006, 26(2): 168-173.
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Consistency of Least Squares Estimation to the Parameter for Stochastic Differential Equations Under Distribution Uncertainty

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