Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

First we consider a process ( X t ( α ) ) t ∈ [ 0 , T ) given by a SDE d X t ( α ) = α b ( t ) X t ( α ) d t + σ ( t ) d B t , t ∈ [ 0 , T ) , with a parameter α ∈ R , where T ∈ ( 0 , ∞ ] and ( B t ) t ∈ [ 0 , T ) is a standard Wiener process. We study asymptotic behavior of the MLE α ^ t ( X ( α ) ) of α based on the observation ( X s ( α ) ) s ∈ [ 0 , t ] as t ↑ T . We formulate sufficient conditions under which I X ( α ) ( t ) ( α ^ t ( X ( α ) ) − α ) converges to the distribution of c ∫ 0 1 W s d W s / ∫ 0 1 ( W s ) 2 d s , where I X ( α ) ( t ) denotes the Fisher information for α contained in the sample ( X s ( α ) ) s ∈ [ 0 , t ] , ( W s ) 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 I X ( α ) ( t ) ( α ^ t ( X ( α ) ) − α ) converges to the Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of α is asymptotically normal with some appropriate random normalizing factor. e study a SDE d Y t ( α ) = α b ( t ) a ( Y t ( α ) ) d t + σ ( t ) d B t , t ∈ [ 0 , T ) , with a perturbed drift satisfying a ( x ) = x + O ( 1 + | x | γ ) with some γ ∈ [ 0 , 1 ) . We give again sufficient conditions under which I Y ( α ) ( t ) ( α ^ t ( Y ( α ) ) − α ) converges to the distribution of c ∫ 0 1 W s d W s / ∫ 0 1 ( W s ) 2 d s . hasize that our results are valid in both cases T ∈ ( 0 , ∞ ) and T = ∞ , and we develop a unified approach to handle these cases.