×

zbMATH — the first resource for mathematics

Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes. (English) Zbl 1185.62147
Summary: First we consider a process \((X^{(\alpha)}_t)_{t \in [0,T)}\) given by a SDE
\[ dX^{(\alpha)}_t = \alpha b(t)X^{(\alpha)}_t\,dt+ \sigma(t)\,dB_t, \quad t \in [0,T), \]
with parameter \(\alpha \in \mathbb R\), where \(T \in (0,\infty ]\) and \((Bt)_{t \in [0,T)}\) is a standard Wiener process. We study the asymptotic behavior of the MLE \(\widehat {\alpha}^{(X^{(\alpha)})}_t\) of \(\alpha \) based on the observation \((X^{(\alpha)}_s)_{s \in [0,T]}\) as \(t \uparrow T\). We formulate sufficient conditions under which \(\sqrt {I_{X^{(\alpha)}}(t)} (\widehat {\alpha}^{(X^{(\alpha)})}_t - \alpha)\) converges to the distribution of \(c \int^1_0 W_s\,dW_s/ \int^1_0 (W_s)^2\,ds\), where \(I_{X^{(\alpha)}}(t)\) denotes the Fisher information for \(\alpha \) contained in the sample \((X^{(\alpha)}_s)_{s \in [0,t]}, (W_s)_{s \in [0,1]}\) is a standard Wiener process, and \(c =1/ \sqrt 2\) or \(c = -1/ \sqrt 2 \). We also weaken the sufficient conditions due to H. Luschgy [Probab. Theory Relat. Fields 92, No. 2, 151–176 (1992; Zbl 0768.62067), Section 4.2)] under which \(\sqrt {I_{X^{(\alpha)}}(t)} (\widehat {\alpha}^{(X^{(\alpha)})}_t - \alpha)\) converges to a Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of \(\alpha \) is asymptotically normal with some appropriate random normalizing factor. Next we study a SDE
\[ dY_t^{(\alpha)}= \alpha b(t)a(Y_t^{(\alpha)})\,dt+ \sigma(t)\,dB_t, \quad t \in [0,T), \]
with a perturbed drift satisfying \(a(x)=x+O(1+|x|^{\gamma})\) with some \(\gamma \in [0,1)\). We give again sufficient conditions under which \(\sqrt {I_{Y^{(\alpha)}}(t)} (\widehat {\alpha}^{(Y^{(\alpha)})}_t - \alpha)\) converges to the distribution of \(c \int^1_0 W_s \,W_s/ \int^1_0 (W_s)^2\,ds\). We emphasize that our results are valid in both cases \(T \in (0,\infty )\) and \(T=\infty \), and we develop a unified approach to handle these cases.

MSC:
62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bainov, D.; Simeonov, P., Integral inequalities and applications, (1992), Kluwer Academic Publishers Dordrecht · Zbl 0759.26012
[2] Barczy, M., Pap, G., 2009. Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes. arXiv, URL: \(\langle\)http://arxiv.org/abs/0810.2688〉.
[3] Basawa, I.V.; Rao Prakasa, B.L.S., Statistical inference for stochastic processes, (1980), Academic Press London · Zbl 0448.62070
[4] Basawa, I.V., Scott, D.J., 1983. Asymptotic optimal inference for non-ergodic models. In: Lecture Notes in Statistics, vol. 17, Springer, Berlin. · Zbl 0519.62039
[5] Bishwal, J.P.N., Parameter estimation in stochastic differential equations, (2007), Springer Berlin · Zbl 0936.60024
[6] Bobkoski, M.J., 1983. Hypothesis testing in nonstationary time series. Ph.D. Dissertation, University of Wisconsin.
[7] Dietz, H.M.; Kutoyants, YU.A., Parameter estimation for some non-recurrent solutions of SDE, Statistics and decisions, 21, 1, 29-46, (2003) · Zbl 1046.62081
[8] Feigin, P.D., Maximum likelihood estimation for continuous-time stochastic processes, Advances in applied probability, 8, 4, 712-736, (1976) · Zbl 0355.62086
[9] Gushchin, A.A., On asymptotic optimality of estimators of parameters under the LAQ condition, Theory of probability and its applications, 40, 2, 261-272, (1995) · Zbl 0898.62031
[10] Jacod, J.; Shiryaev, A.N., Limit theorems for stochastic processes, (2003), Springer Berlin · Zbl 0830.60025
[11] Karatzas, I.; Shreve, S.E., Brownian motion and stochastic calculus, (1991), Springer Berlin, Heidelberg · Zbl 0734.60060
[12] Kátai, I.; Mogyoródi, J., Some remarks concerning the stable sequences of random variables, Publicationes mathematicae debrecen, 14, 227-238, (1967) · Zbl 0183.20304
[13] Kutoyants, Yu.A., Parameter estimation for stochastic processes, (1984), Heldermann Verlag Berlin · Zbl 0542.62073
[14] Kutoyants, Yu.A., Identification of dynamical systems with small noise, (1994), Kluwer Academic Publisher Dordrecht · Zbl 0831.62058
[15] Kutoyants, Yu.A., Statistical inference for ergodic diffusion processes, (2004), Springer Berlin, Heidelberg · Zbl 1038.62073
[16] Lépingle, D., 1978. Sur les comportement asymptotique des martingales locales. In: Seminaire de Probabilites XII, Lecture Notes in Mathematics, vol. 649. Springer, Berlin, pp. 148-161.
[17] Liptser, R.S.; Shiryaev, A.N., Statistics of random processes I. general theory, (2001), Springer Berlin, Heidelberg
[18] Liptser, R.S.; Shiryaev, A.N., Statistics of random processes II. applications, (2001), Springer Berlin, Heidelberg · Zbl 0591.60039
[19] Luschgy, H., Local asymptotic mixed normality for semimartingale experiments, Probability theory and related fields, 92, 2, 151-176, (1992) · Zbl 0768.62067
[20] Luschgy, H., Asymptotic inference for semimartingale models with singular parameter points, Journal of statistical planning and inference, 39, 2, 155-186, (1994) · Zbl 0807.62062
[21] Mishra, M.N.; Prakasa Rao, B.L.S., Asymptotic study of maximum likelihood estimation for nonhomogeneous diffusion processes, Statistics and decisions, 3, 3-4, 193-203, (1985) · Zbl 0579.62069
[22] Revuz, D., Yor, M., 2001. Continuous Martingales and Brownian Motion, third ed., corrected 2nd printing. Springer, Berlin. · Zbl 0731.60002
[23] Shiryaev, A.N., Probability, (1989), Springer Berlin · Zbl 0682.60001
[24] Tanaka, K., Time series analysis, nonstationary and noninvertible distribution theory, wiley series in probability and statistics, (1996)
[25] van der Vaart, A.W., Asymptotic statistics, (1998), Cambridge University Press Cambridge · Zbl 0910.62001
[26] van Zanten, H., A multivariate central limit theorem for continuous local martingales, Statistics and probability letters, 50, 3, 229-235, (2000) · Zbl 1004.60020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.