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.