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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.

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)
Full Text: DOI
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