engleski

Local asymptotic mixed normality of AMLE of drift parameters in diffusion model

We assume that the diffusion $X$ satisfies a stochastic differential equation of the form: $dX_t=\mu(X_t, \theta)dt+\sigma_0\nu(X_t)dW_t$, with unknown drift parameter $\theta$ and known diffusion coefficient parameter $\sigma_0$. We prove that approximate maximum likelihood estimator of drift parameter $\bar{; ; ; ; ; \theta}; ; ; ; ; _n$ obtained from discrete observations $(X_{; ; ; ; ; i\Delta_n}; ; ; ; ; , 0\leq i\leq n)$ along fixed time interval $[0, T]$, and when $\Delta_n =\frac{; ; ; ; ; T}; ; ; ; ; {; ; ; ; ; n}; ; ; ; ; $ tends to zero, is locally asymptotic mixed normal, with covariance matrix which depends on MLE $\hat{; ; ; ; ; \theta}; ; ; ; ; $ obtained from continuous observations $(X_t, 0\leq t\leq T)$ along fixed time interval $[0, T]$, and on path $(X_t, 0\leq t\leq T)$.

asymptotic mixed normality ; diffusion processes ; discrete observation ; parameter estimation

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