Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

First we consider a process (X-r((alpha)))(t is an element of[0,T)) given by a SDE dX(t)((alpha)) = alpha b(t)X-t((alpha)) dt + sigma(t) dB(t), t is an element of [0, T), with a parameter alpha is an element of R, where T is an element of (0, infinity] and (B-t)(t is an element of[0, T)) is a standard Wiener process. We study asymptotic behavior of the MLE alpha((X(alpha)))(t) of a based on the observation (X-S((alpha)))(S is an element of[0,t]) as t up arrow T. We formulate sufficient conditions under which root I-X(alpha) (t) (alpha(X(alpha))-alpha) converges to the distribution of c integral(1)(0) W-s dW(s)/ integral(1)(0) (W-s)(2) ds, where I-X(alpha) denotes the Fisher information for alpha contained in the sample (X-s((alpha)))(s is an element of[0,t]), (W-s)(s is an element of[0,1]) is a standard Wiener process, and c= 1/root 2 or c=-1/root 2. We also weaken the sufficient conditions due to Luschgy (1992, Section 4.2) under which root I-X(alpha) (t) (alpha(X(alpha))-alpha) converges to the Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of a 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), t is an element of [0,T), with a perturbed drift satisfying a(x) = x+O(1+|x|(gamma)) with some gamma is an element of [0, 1). We give again sufficient conditions under which root I-Y(alpha) (t) (alpha(Y(alpha))-alpha) converges to the distribution of c integral(1)(0) W-s dW(s)/ integral(1)(0) (W-s)(2) ds. We emphasize that our results are valid in both cases T is an element of (0,infinity) and T = infinity, and we develop a unified approach to handle these cases. (C)2009 Elsevier B.V. All rights reserved.