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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 $$(X^{(\alpha)}_t)_{t \in [0,T)}$$ 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)
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