PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS

doi:10.1016/S0252-9602(10)60067-7

摘要/Abstract

摘要：

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small α-stable noises, observed at n regularly spaced time points t_i=i/n, i=1, …, n on [0,1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator(LSE) when a small dispersion parameter ε→0 and n → ∝simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.

关键词: Asymptotic distribution of LSE, consistency of LSE, discrete observations, least squares method, parameter estimation, small α-stable noises, stable distribution, stochastic differential equations

Abstract:

Key words: Asymptotic distribution of LSE, consistency of LSE, discrete observations, least squares method, parameter estimation, small α-stable noises, stable distribution, stochastic differential equations

中图分类号:

60G52

龙红卫. PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS[J]. 数学物理学报(英文版), 2010, 30(3): 645-663.

Long Hongwei. PARAMETER ESTIMATION FOR A CLASS OF STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SMALL STABLE NOISES FROM DISCRETE OBSERVATIONS[J]. Acta mathematica scientia,Series B, 2010, 30(3): 645-663.

参考文献

[1] Brockwell P J, Davis R A, Yang Y. Estimation for non-negative Lévy-driven Ornstein-Uhlenbeck processes. J Appl Probab, 2007, 44: 977--989

[2] Ditlevsen P D. Observation of α-stable noise induced millennial climate changes from an ice-core record.
Geophys Res Lett, 1999, 26: 1441--1444

[3] Ditlevsen P D. Anomalous jumping in a double-well potential. Physical Review E, 1999, 60: 172--179

[4] Fristedt B. Sample functions of stochastic processes with stationary, independent increments//Ney P, Port S.
Advances in Probability and Related Topics. Vol 3. New York: Marcel Dekker, 1974: 241--396

[5] Genon-Catalot V. Maximum contrast estimation for diffusion processes from discrete observations. Statistics, 1990, 21: 99--116

[6] Gloter A, S{\o}rensen M. Estimation for stochastic differential equations with a small diffusion coefficient. Stochastic Process Appl, 2009, 119: 679--699

[7] Hu Y, Long H. Parameter estimation for Ornstein-Uhlenbeck processes driven by α-stable Lévy motions. Communications on Stochastic Analysis, 2007, 1: 175--192

[8] Hu Y, Long H. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. Stochastic Process Appl, 2009, 119: 2465--2480

[9] Janicki A, Weron A. Simulation and Chaotic Behavior of α-stable Stochastic Processes. New York: Marcel Dekker, 1994

[10] Joulin A. On maximal inequalities for stable stochastic integrals. Potential Anal, 2007, 26: 57--78

[11] Kallenberg O. On the existence and path properties of stochastic integrals. Ann Probab, 1975, 3: 262--280

[12] Kallenberg O. Some time change representations of stable integrals, via predictable transformations of local martingales. Stochastic Process Appl, 1992, 40: 199--223

[13] Kunitomo N, Takahashi A. The asymptotic expansion approach to the valuation of interest rate contingent claims. Math Finance, 2001, 11: 117--151

[14] Kutoyants Yu A. Parameter Estimation for Stochastic Processes. Berlin: Heldermann, 1984

[15] Kutoyants Yu A. Identification of Dynamical Systems with Small Noise. Dordrecht: Kluwer, 1994

[16] Laredo C F. A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process. Ann Statist, 1990, 18: 1158--1171

[17] Long H. Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises. Statistics and Probability Letters, 2009, 79: 2076--2085

[18] Masuda H. Simple estimators for non-linear Markovian trend from sampled data: I. ergodic cases [Preprint]. Fukuoka: Kyushu University, 2005

[19] Rosinski J, Woyczynski W A. Moment inequalities for real and vector p-stable stochastic integrals//Probab- ility in Banach Spaces V, Lecture Notes in Math, Vol 1153: 369--386. Berlin: Springer, 1985

[20] Rosinski J, Woyczynski W A. On Ito stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann Probab, 1986, 14: 271--286

[21] Samorodnitsky G, Taqqu M S. Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman & Hall, 1994

[22] Sato K I. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999

[23] Shimizu Y. M-estimation for discretely observed ergodic diffusion processes with infinite jumps. Stat Inference Stoch Process, 2006, 9: 179--225

[24] Shimizu Y, Yoshida N. Estimation of parameters for diffusion processes with jumps from discrete observations. Stat Inference Stoch Process, 2006, 9: 227--277

[25] Sørensen M. Small dispersion asymptotics for diffusion martingale estimating functions [Preprint]. Copenhagen: University of Copenhagen, 2000

[26] Sørensen M, Uchida M. Small diffusion asymptotics for discretely sampled stochastic differential equations. Bernoulli, 2003, 9: 1051--1069

[27] Spiliopoulos K. Methods of moments estimation of Ornstein-Uhlenbeck processes driven by general
Lévy process [Preprint]. College Park: University of Maryland, 2008

[28] Takahashi A. An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets, 1999, 6: 115--151

[29] Takahashi A, Yoshida N. An asymptotic expansion scheme for optimal investment problems. Stat Inference Stoch Process, 2004, 7: 153--188

[30] Uchida M. Estimation for discretely observed small diffusions based on approximate martingale estimating functions. Scand J Statist, 2004, 31: 553--566

[31] Uchida M. Approximate martingale estimating functions for stochastic differential equations with small noises. Stochastic Process Appl, 2008, 118: 1706--1721

[32] Uchida M, Yoshida N. Information criteria for small diffusions via the theory of Malliavin-Watanabe. Stat Inference Stoch Process, 2004, 7: 35--67

[33] Uchida M, Yoshida N. Asymptotic expansion for small diffusions applied to option pricing. Stat Inference Stoch Process, 2004, 7: 189--223

[34] Valdivieso L, Schoutens W, Tuerlinckx F. Maximum likelihood estimation in processes of Ornstein-Uhlenbeck type. Stat Infer Stoch Process, 2008, DOI:10.1007/s11203-008-9021-8

[35] Yoshida N. Asymptotic expansion of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe. Probab Theory Relat Fields, 1992, 92: 275--311

[36] Yoshida N. Asymptotic expansion for statistics related to small diffusions. J Japan Statist Soc, 1992, 22: 139--159

[37] Yoshida N. Conditional expansions and their applications. Stochastic Process Appl, 2003, 107: 53--81