Speaker
Description
For a fixed $T$ and $k\geq 2$, we analyze a $k$-dimensional vector stochastic differential equation over the time interval $[0,T]$:
$$dX_t=\mu(X_t, \theta)\,dt+\nu(X_t)\,dW_t,$$
where $\mu(X_t, \theta)$ is a $k$-dimensional vector and $\nu(X_t)$ is a $k \times k$-dimensional matrix, both consisting of sufficiently smooth functions. $\left(W_t, \, t \geq 0\right)$ is a $k$-dimensional standard Brownian motion whose components $W_t^1, W_t^2, \dots, W_t^k$ are independent scalar Brownian motions. Vector of drift parameters $\theta$ is unknown and its dependence is in general nonlinear. We prove that approximate maximum likelihood estimator of drift parameters $\hat{\theta}n$ obtained from discrete observations $(X{i\Delta_n}, 0 \leq i \leq n)$, when $\Delta_n=T/n$ tends to zero, is locally asymptotic mixed normal with covariance matrix that depends on maximum likelihood estimator $\hat{\theta}_T$ obtained from continuous observations $(X_t, 0\leq t\leq T)$, and on path $(X_t, 0 \leq t\leq T)$. To prove the desired result, we emphasize the importance of the so-called uniform ellipticity condition of diffusion matrix $S(x)=\nu(x)\nu^T(x)$.
Similar considerations can be made for any sufficiently large $T > 0$. The main assumptions are then that $X$ is an ergodic diffusion with stationary distribution and the diffusion matrix is diagonalizable so that only eigenvalues depend on $X$, i.e., there exists an orthogonal constant matrix $U$ such that $S(x)=U\Lambda(x) U^{-1}$.
