An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model

Abstract

For a fixed $T$ and $k \geq 2$, a $k$-dimensional vector stochastic differential equation $dX_t=\mu(X_t, \theta)dt+\nu(X_t)dW_t,$ is studied over a time interval $[0,T]$. Vector of drift parameters $\theta$ is unknown. The dependence in $\theta$ is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter $\overline{\theta}_n\equiv \overline{\theta}_{n,T}$ obtained from discrete observations $(X_{i\Delta_n}, 0 \leq i \leq n)$ and maximum likelihood estimator $\hat{\theta}\equiv \hat{\theta}_T$ obtained from continuous observations $(X_t, 0\leq t\leq T)$, when $\Delta_n=T/n$ tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on $\hat{\theta}$ and on path $(X_t, 0 \leq t\leq T)$. The uniform ellipticity of diffusion matrix $S(x)=\nu(x)\nu(x)^T$ emerges as the main assumption on the diffusion coefficient function.