Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=\alpha b(t)X_t dt + \sigma(t) dB_t$, $t\in[0,T)$, with a parameter $\alpha\in R$, where $T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=\alpha b(t)a(Y_t) dt + \sigma(t) dB_t$, $t\in[0,T)$, 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 the MLE of $\alpha$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics.