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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 t (α) ) t[0,T) given by a SDE

dX t (α) =αb(t)X t (α) dt+σ(t)dB t ,t[0,T),

with parameter α, where T(0,] and (Bt) t[0,T) is a standard Wiener process. We study the asymptotic behavior of the MLE α ^ t (X (α) ) of α based on the observation (X s (α) ) s[0,T] as tT. 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 dW s / 0 1 (W s ) 2 ds, 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 H. Luschgy [Probab. Theory Relat. Fields 92, No. 2, 151–176 (1992; Zbl 0768.62067), Section 4.2)] under which I X (α) (t)(α ^ t (X (α) ) -α) converges to a Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of α is asymptotically normal with some appropriate random normalizing factor. Next we study a SDE

dY t (α) =αb(t)a(Y t (α) )dt+σ(t)dB 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 W s / 0 1 (W s ) 2 ds. We emphasize that our results are valid in both cases T(0,) and T=, and we develop a unified approach to handle these cases.

MSC:
62M05Markov processes: estimation
62F12Asymptotic properties of parametric estimators
60J60Diffusion processes
60H10Stochastic ordinary differential equations
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