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

For a fixed \(T\) and \(k ≥ 2\), a \(k\)-dimensional vector stochastic differential equation \(dX_t=μ(X_t, θ)\,dt+ν(X_t)\,dW_t,\) is studied over a time interval \([0,T]\). Vector of drift parameters \(θ\) is unknown. The dependence in \(θ\) is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter \(θ̄_n≡ θ̄n,T\) obtained from discrete observations \((XiΔ_n, 0 ≤ i ≤ n)\) and maximum likelihood estimator \(θ̂≡ θ̂_T\) obtained from continuous observations \((X_t, 0≤ t≤ T)\), when \(Δ_n=T/n\) tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on \(θ̂\) and on path \((X_t, 0 ≤ t≤ T)\). The uniform ellipticity of diffusion matrix \(S(x)=ν(x)ν(x)^T\) emerges as the main assumption on the diffusion coefficient function.