Local asymptotic mixed normality of approximate maximum likelihood estimator of drift parameters in diffusion model

We assume that the diffusion X satisfies a stochastic differential equation of the form: dXt=μ(Xt,θ)dt+σ0ν(Xt)dWt, with unknown drift parameter θ and known diffusion coefficient parameter σ0. We prove that approximate maximum likelihood estimator of drift parameter θ n obtained from discrete observations (XiΔn,0≤ i≤ n) along fixed time interval [0,T], and when Δn =T/n tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE obtained from continuous observations (Xt,0≤ t≤ T) along fixed time interval [0,T], and on path (Xt,0≤ t≤ T)