Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

First we consider a process (Xt(α))t∈[0,T) given by a SDE dXt(α)=αb(t)Xt(α)dt+σ(t)dBt, t∈[0,T)t∈[0,T), with a parameter α∈Rα∈R, where T∈(0,∞]T∈(0,∞] and (Bt)t∈[0,T)(Bt)t∈[0,T) is a standard Wiener process. We study asymptotic behavior of the MLE α^t(X(α)) of αα based on the observation (Xs(α))s∈[0,t] as t↑Tt↑T. We formulate sufficient conditions under which IX(α)(t)(α^t(X(α))−α) converges to the distribution of c∫01WsdWs/∫01(Ws)2ds, where IX(α)(t)IX(α)(t) denotes the Fisher information for αα contained in the sample (Xs(α))s∈[0,t], (Ws)s∈[0,1](Ws)s∈[0,1] is a standard Wiener process, and c=1/2 or c=−1/2. We also weaken the sufficient conditions due to Luschgy (1992, Section 4.2) under which IX(α)(t)(α^t(X(α))−α) converges to the Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of αα is asymptotically normal with some appropriate random normalizing factor.Next we study a SDE dYt(α)=αb(t)a(Yt(α))dt+σ(t)dBt, t∈[0,T)t∈[0,T), with a perturbed drift satisfying a(x)=x+O(1+|x|γ)a(x)=x+O(1+|x|γ) with some γ∈[0,1)γ∈[0,1). We give again sufficient conditions under which IY(α)(t)(α^t(Y(α))−α) converges to the distribution of c∫01WsdWs/∫01(Ws)2ds.We emphasize that our results are valid in both cases T∈(0,∞)T∈(0,∞) and T=∞T=∞, and we develop a unified approach to handle these cases.