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Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes. (English) Zbl 1185.62147

Summary: First we consider a process ${\left({X}_{t}^{\left(\alpha \right)}\right)}_{t\in \left[0,T\right)}$ given by a SDE

$d{X}_{t}^{\left(\alpha \right)}=\alpha b\left(t\right){X}_{t}^{\left(\alpha \right)}\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{t},\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right),$

with parameter $\alpha \in ℝ$, where $T\in \left(0,\infty \right]$ and ${\left(Bt\right)}_{t\in \left[0,T\right)}$ is a standard Wiener process. We study the asymptotic behavior of the MLE ${\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}$ of $\alpha$ based on the observation ${\left({X}_{s}^{\left(\alpha \right)}\right)}_{s\in \left[0,T\right]}$ as $t↑T$. We formulate sufficient conditions under which $\sqrt{{I}_{{X}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}-\alpha \right)$ converges to the distribution of $c{\int }_{0}^{1}{W}_{s}\phantom{\rule{0.166667em}{0ex}}d{W}_{s}/{\int }_{0}^{1}{\left({W}_{s}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}ds$, where ${I}_{{X}^{\left(\alpha \right)}}\left(t\right)$ denotes the Fisher information for $\alpha$ contained in the sample ${\left({X}_{s}^{\left(\alpha \right)}\right)}_{s\in \left[0,t\right]},{\left({W}_{s}\right)}_{s\in \left[0,1\right]}$ 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 $\sqrt{{I}_{{X}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({X}^{\left(\alpha \right)}\right)}-\alpha \right)$ 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

$d{Y}_{t}^{\left(\alpha \right)}=\alpha b\left(t\right)a\left({Y}_{t}^{\left(\alpha \right)}\right)\phantom{\rule{0.166667em}{0ex}}dt+\sigma \left(t\right)\phantom{\rule{0.166667em}{0ex}}d{B}_{t},\phantom{\rule{1.em}{0ex}}t\in \left[0,T\right),$

with a perturbed drift satisfying $a\left(x\right)=x+O\left(1+|x{|}^{\gamma }\right)$ with some $\gamma \in \left[0,1\right)$. We give again sufficient conditions under which $\sqrt{{I}_{{Y}^{\left(\alpha \right)}}\left(t\right)}\left({\stackrel{^}{\alpha }}_{t}^{\left({Y}^{\left(\alpha \right)}\right)}-\alpha \right)$ converges to the distribution of $c{\int }_{0}^{1}{W}_{s}\phantom{\rule{0.166667em}{0ex}}{W}_{s}/{\int }_{0}^{1}{\left({W}_{s}\right)}^{2}\phantom{\rule{0.166667em}{0ex}}ds$. We emphasize that our results are valid in both cases $T\in \left(0,\infty \right)$ and $T=\infty$, and we develop a unified approach to handle these cases.

##### MSC:
 62M05 Markov processes: estimation 62F12 Asymptotic properties of parametric estimators 60J60 Diffusion processes 60H10 Stochastic ordinary differential equations